# The atomic Leibniz rule

Ben Elias  Department of Mathematics, Fenton Hall, University of Oregon,
Eugene, OR, 97403-1222, USA<EMAIL_ADDRESS>,  Hankyung Ko  Department of
Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, Sweden
<EMAIL_ADDRESS>,  Nicolas Libedinsky  Departamento de Matemáticas,
Facultad de Ciencias, Universidad de Chile<EMAIL_ADDRESS>and
Leonardo Patimo  Dipartimento di Matematica, Università di Pisa, Italy
<EMAIL_ADDRESS>
###### Abstract.

The Demazure operator associated to a simple reflection satisfies the twisted
Leibniz rule. In this paper we introduce a generalization of the twisted
Leibniz rule for the Demazure operator associated to any atomic double coset.
We prove that this atomic Leibniz rule is equivalent to a polynomial forcing
property for singular Soergel bimodules.

##  1\. Introduction

We introduce a generalization of the twisted Leibniz rule which applies to
Demazure operators associated with certain double cosets. We call it the
atomic Leibniz rule, and it should play an important role in an eventual
description of the singular Hecke category by generators and relations. This
result is the algebraic heart which we extract from singular Soergel bimodules
and transplant to their diagrammatic calculus.

### 1.1.

In this paper, we work with general Coxeter systems and quite general actions
of these groups on polynomial rings. For ease of exposition, in this section,
we assume that $R=\mathbb{Q}[x_{1},\ldots,x_{n}]$, with the standard action of
the symmetric group $W=\mathbf{S}_{n}$. Let $s_{i}$ be the simple reflection
swapping $i$ and $i+1$, and let $S=\\{s_{1},\ldots,s_{n-1}\\}$ be the set of
simple reflections. There is a _Demazure operator_ $\partial_{s_{i}}\colon
R\to R$ defined by the formula

(1) $\partial_{s_{i}}(f)=\frac{f-s_{i}f}{x_{i}-x_{i+1}}.$

This operator famously satisfies a _twisted Leibniz rule_

(2) $\partial_{s_{i}}(fg)=s_{i}(f)\partial_{s_{i}}(g)+\partial_{s_{i}}(f)g.$

Because the operators $\partial_{s_{i}}$ satisfy the braid relations, one can
unambiguously define an operator $\partial_{w}$ associated with any
$w\in\mathbf{S}_{n}$ by composing the operators $\partial_{s_{i}}$ along a
reduced expression. When computing $\partial_{w}(fg)$, one could apply the
twisted Leibniz rule repeatedly to obtain a complicated generalization of (2)
for $\partial_{w}$. We discuss this in §2.2.

What we provide in this paper is the natural generalization of the twisted
Leibniz rule to the setting of double cosets. In double coset combinatorics,
the analogues of simple reflections are called atomic cosets. We prove a
version of (2) for Demazure operators attached to atomic cosets.

### 1.2.

We recall some definitions so as to precisely state our first theorem. Let
$(W,S)$ be any Coxeter system. For any subset $I\subset S$, let $W_{I}$ be the
parabolic subgroup of $W$ generated by $I$. We assume that $I$ is _finitary_ ,
i.e. that $W_{I}$ is a finite group. Let $R^{I}\subset R$ be the subring of
polynomials invariant under $W_{I}$. Let $w_{I}$ denote the longest element of
$W_{I}$. For two subsets $I,J\subset S$, a double coset $p\in W_{I}\backslash
W/W_{J}$ will be called an _$(I,J)$ -coset_. Any $(I,J)$-coset $p$ has a
unique minimal element $\underline{p}\in W$ and a unique maximal element
$\overline{p}\in W$ with respect to the Bruhat order.

In [EKLP23a, §3.4], we introduced a Demazure operator $\partial_{p}:R^{J}\to
R^{I}$ for any $(I,J)$-coset $p$. In fact, $\partial_{p}$ is equal to
$\partial_{w}$ for $w=\overline{p}w_{J}\in W$. Normally one views
$\partial_{w}$ as a map $R\to R$, but when the source of this map is
restricted from $R$ to $R^{J}$, then the image is contained in $R^{I}$. When
$I=J=\emptyset$, so that $p=\\{w\\}$ for some $w\in W$, then
$\partial_{p}=\partial_{w}$.

In [EKLP24b] we introduced the notion of an _atomic_ $(I,J)$-coset. Briefly
stated, ${\mathtt{a}}$ is atomic if

  * •

there exists $s,t\in S$ (possibly equal) with $s\notin I$ and $t\notin J$ such
that $I\cup\\{s\\}=J\cup\\{t\\}=:M$.

  * •

$W_{M}$ is a finite group, $\overline{{\mathtt{a}}}=w_{M}$ and
$w_{M}s=tw_{M}$.

In particular, the subsets $I$ and $J$ are conjugate under both
$\overline{{\mathtt{a}}}$ and $\underline{{\mathtt{a}}}$. If $f\in R^{J}$ then
$\underline{{\mathtt{a}}}(f)\in R^{I}$. We also have
$\underline{{\mathtt{a}}}=\overline{{\mathtt{a}}}w_{J}$, so that
$\partial_{\mathtt{a}}$ is the restriction of
$\partial_{\underline{{\mathtt{a}}}}$.

When $I=J=\emptyset$, the atomic $(I,J)$-cosets have the form $\\{s\\}$ for
$s\in S$. Atomic cosets in $\mathbf{S}_{n}$ can be described using cabled
crossings. For example, the permutation with one-line notation $(345671289)$
crosses the first two numbers past the next five while fixing the remaining
numbers. It is a prototypical example of $\underline{{\mathtt{a}}}$ for an
atomic coset ${\mathtt{a}}$; in this example
$W_{I}\cong\mathbf{S}_{2}\times\mathbf{S}_{5}\times\cdots$ and
$W_{J}\cong\mathbf{S}_{5}\times\mathbf{S}_{2}\times\cdots$ and
$W_{M}\cong\mathbf{S}_{7}\times\cdots$.

### 1.3.

Our story began with the defining representation of $\mathbf{S}_{n}$ over
$\mathbb{Q}$. Then we considered the polynomial ring $R$, its invariant
subrings, and the structure derived therefrom. Alternatively, one can start
with a _realization_ of a Coxeter system $(W,S)$, which is effectively a
“reflection representation” of $W$ over a commutative base ring $\Bbbk$,
equipped with a choice of roots and coroots. From this representation we
construct its polynomial ring $R$, Demazure operators, etcetera. See §3.1 for
details.

For each realization111Soergel’s construction depends only on a reflection
representation of $W$, and not a choice of roots and coroots. Elias and
Williamson gave a presentation of Soergel’s category in [EW16] which depends
on a choice of realization. The Demazure operators also depend on the choice
of roots and coroots., Soergel defined a monoidal category of $R$-bimodules
called Soergel bimodules, and proved that for certain
realizations222Specifically, the base ring $\Bbbk$ should be an infinite field
of characteristic not equal to $2$, and the representation should be (faithful
and) reflection-faithful. (that we call _SW-realizations_) one obtains a
categorification of the Hecke algebra. Soergel bimodules and related
categorifications of the Hecke algebra are objects of critical importance in
geometric representation theory. The following is the first main result of
this paper (Theorem 5.10).

######  Theorem A (The atomic Leibniz rule).

Let $(W,S)$ be any Coxeter system equipped with an SW-realization, or the
symmetric group $\mathbf{S}_{n}$ with $R=\mathbb{Z}[x_{1},\ldots,x_{n}].$ Let
${\mathtt{a}}$ be an atomic $(I,J)$-coset and $\leq$ the Bruhat order for
double cosets333The Bruhat order on $(I,J)$-cosets can be defined by
$q\leq{\mathtt{a}}$ if and only if
$\underline{q}\leq\underline{{\mathtt{a}}}$. See [EKLP23b, Thm. 2.16] for
equivalent definitions. Note that $q$ need not be atomic, so $\partial_{q}$
need not equal (the restriction of) $\partial_{\underline{q}}$.. For every
$q<{\mathtt{a}}$ there is a unique $R^{I\cup J}$-linear operator on
polynomials denoted $T^{\mathtt{a}}_{q}$, satisfying the equation

(3) $\partial_{\mathtt{a}}(f\cdot
g)=\underline{{\mathtt{a}}}(f)\partial_{\mathtt{a}}(g)+\sum_{q<{\mathtt{a}}}\partial_{q}(T^{\mathtt{a}}_{q}(f)\cdot
g)$

for all $f,g\in R^{J}.$ Polynomials in the image of $T^{\mathtt{a}}_{q}$ are
appropriately invariant, see Definition 3.8 for details. When ${\mathtt{a}}$
is fixed, we will often write $T_{q}:=T_{q}^{\mathtt{a}}$.

The motivation for this result will take some setup, so please bear with us.

######  Example 1.1.

We let $e$ denote the identity element. Let $s$ be a simple reflection and let
${\mathtt{a}}$ be the $(\emptyset,\emptyset)$-coset $\\{s\\}$. There is only
one coset less than ${\mathtt{a}}$, namely $q=\\{e\\}$. We set
$T^{\mathtt{a}}_{q}(f)=\partial_{s}(f)$. Note that
$\partial_{\mathtt{a}}=\partial_{s}$ and $\partial_{q}=\operatorname{id}$.
Then (3) becomes

(4) $\partial_{s}(fg)=s(f)\partial_{s}(g)+\partial_{s}(f)g,$

which recovers the twisted Leibniz rule.

For more examples see §2. There we also give an example of a non-atomic coset
$p$ for which no equality of the form (3) can hold (with $p$ replacing
${\mathtt{a}}$).

######  Remark 1.2.

Though we can prove the existence and uniqueness of such a formula, we do not
have an explicit description of the operators $T_{q}$. We consider this an
interesting open problem. A more accessible problem is to compute $T_{q}$ on a
(carefully chosen) set of generators of $R^{J}$, which we accomplish in type
$A$ in Theorem 6.3. This is useful computationally because it gives enough
information to apply the atomic Leibniz rule for any pair of elements in
$R^{J}$ (see the discussion of multiplicativity in §1.6).

######  Remark 1.3.

In §3.3 we prove that the atomic Leibniz rule for one realization implies the
atomic Leibniz rule for related realizations (e.g. obtained via base change,
enlargement, or quotient). The atomic Leibniz rule also depends only on the
restriction of the realization to the finite parabolic subgroup $W_{M}$
associated to ${\mathtt{a}}$. Thus Theorem A implies the atomic Leibniz rule
in broad generality for realizations of Coxeter systems in both finite and
affine type $A$, and in finite characteristic as well, see Example 3.19.

### 1.4.

Our second main result is a connection between the atomic Leibniz rule and the
theory of singular Soergel bimodules [Wil11]. Like Soergel bimodules, singular
Soergel bimodules are ubiquitous in geometric representation theory, appearing
e.g. in the geometric Satake equivalence and other situations where partial
flag varieties play a role. Specifically, we connect the atomic Leibniz rule
to a property called polynomial forcing, whose motivation we postpone a little
longer.

Let us first be more precise. Singular Soergel bimodules are graded bimodules
over graded rings, but we ignore all gradings in this paper. To an atomic
coset ${\mathtt{a}}$ as above, one can associate the $(R^{I},R^{J})$-bimodule
$B_{\mathtt{a}}:=R^{I}\otimes_{R^{M}}R^{J}$. This is an indecomposable
bimodule; there are also indecomposable $(R^{I},R^{J})$-bimodules $B_{q}$
associated to any $(I,J)$-coset $q$, which we will not try to describe here.
Then $B_{\mathtt{a}}$ has a submodule called the _submodule of lower terms_ ,
spanned by the images of all bimodule endomorphisms of $B_{\mathtt{a}}$ which
factor through $B_{q}$ for $q<{\mathtt{a}}$.

_Atomic polynomial forcing_ is the statement that $1\otimes f$ and
$\underline{{\mathtt{a}}}(f)\otimes 1$ are equal modulo lower terms in
$B_{\mathtt{a}}$, for any $f\in R^{J}$. Williamson has proven for SW-
realizations [Wil11, Theorem 6.4] that singular Soergel bimodules have
“standard filtrations,” which implies atomic polynomial forcing for SW-
realizations. _Polynomial forcing_ (Definition 5.7) is a generalization of
atomic polynomial forcing for a bimodule $B_{p}$ associated with an arbitrary
double coset $p$. Using [EKLP23b, EKLP24b] we prove Theorem 5.13, which says
that polynomial forcing for any double coset is a consequence of atomic
polynomial forcing.

The restrictions imposed on SW-realizations are significant as they rule out
some examples of great importance to modular representation theory, e.g.
affine Weyl groups in finite characteristic. An _almost SW-realization_
(Definition 4.21) is a realization over a domain $\Bbbk$ such that one obtains
an SW-realization after base change to the fraction field of $\Bbbk$. An
example is $\mathbb{Z}[x_{1},\ldots,x_{n}]$ with its action of
$\mathbf{S}_{n}$. The second main result of this paper is the following
equivalence (5.5):

######  Theorem B.

Let $(W,S)$ be a Coxeter system equipped with an almost SW-realization. Then
the atomic Leibniz rule is equivalent to atomic polynomial forcing.

We deduce the SW-realization case of Theorem A from Theorem B and Williamson’s
theory of standard filtrations.

### 1.5.

Let us now explain the motivation for our results.

The diagrammatic presentation for the category of Soergel bimodules, developed
by Elias, Khovanov, Libedinsky, and Williamson [Lib10],[EK10], [Eli16], [EW16]
has proven to be an important tool for both abstract and computational
reasons. This diagrammatically constructed category is typically called the
_Hecke category_ , and is equivalent to the category of Soergel bimodules for
SW-realizations. For non-SW-realizations the category of Soergel bimodules
need not behave well, but the diagrammatic category does behave well, and
continues to provide a categorification of the Hecke algebra. The ability to
compute within the Hecke category has led to advances such as [EW14],
[EW21],[Wil17],[RW18],[LW18],[Ach+19].

The authors are in the midst of a concerted effort to define the _singular
Hecke category_ , a diagrammatic presentation of singular Soergel bimodules.
The framework for such a presentation was developed in [ESW17], but this
framework lacks some of the relations needed. In [EKLP24a] we described what
should be the basis for morphisms in the diagrammatic category, called the
_double leaves basis_ , and proved that it descends to a basis for morphisms
between singular Soergel bimodules for SW-realizations.

Once the remaining relations are understood, proving that the diagrammatic
category is correctly presented reduces to proving that double leaves span. An
important part of that proof will be _diagrammatic polynomial forcing_ , which
is like polynomial forcing except that the submodule of lower terms is
replaced by the span of double leaves associated to smaller elements in the
Bruhat order. We abbreviate diagrammatic polynomial forcing to
_$\operatorname{DL}$ -forcing_. It is $\operatorname{DL}$-forcing which is our
true goal in this paper.

For an atomic coset ${\mathtt{a}}$ one can identify
$\operatorname{End}(B_{{\mathtt{a}}})$ with $B_{{\mathtt{a}}}$ as bimodules.
In Section 4.2.3, we explicitly compute the submodule
$\operatorname{DL}_{<{\mathtt{a}}}\subset B_{\mathtt{a}}$ corresponding to the
$\Bbbk$-linear subspace of $\mathrm{End}(B_{\mathtt{a}})$ spanned by double
leaves factoring through $q<{\mathtt{a}}$. The following theorem (Corollary
4.33 and Theorem 5.5) is proven by direct computation and is the reason we
discovered the atomic Leibniz rule in the first place.

######  Theorem C.

Let $(W,S)$ be a Coxeter system equipped with an almost SW-realization. Then
the atomic Leibniz rule is equivalent to atomic $\operatorname{DL}$-forcing.

In other words, for a given $f\in R^{J}$, we prove that $1\otimes
f-\underline{{\mathtt{a}}}(f)\otimes 1\in\operatorname{DL}_{<{\mathtt{a}}}$ if
and only if the atomic Leibniz rule holds for $f$ and any $g\in R^{J}$.

For an SW-realization, the results of [EKLP24a] can be used to prove that
$\operatorname{DL}_{<{\mathtt{a}}}$ agrees precisely with the submodule of
lower terms (see Corollary 4.19). In §4.4 we use some novel localization
tricks to extend this statement to almost-SW-realizations. Thus
$\operatorname{DL}$-forcing agrees with polynomial forcing for almost-SW-
realizations, thus Theorem C implies Theorem B.

### 1.6.

One is still motivated to prove $\operatorname{DL}$-forcing (or equivalently,
the atomic Leibniz rule) for almost-SW-realizations and more general
realizations. We now discuss how this process is simplified by Theorem B.

Let us say the atomic Leibniz rule _holds for $f\in R^{J}$_ if (3) holds for
that specific $f$ and all $g\in R^{J}$. It is obvious that if the rule holds
for $f_{1}$ and $f_{2}$ then it holds for $f_{1}+f_{2}$. It is not obvious
that if it holds for $f_{1}$ and $f_{2}$ then it holds for $f_{1}\cdot f_{2}$,
thus the formula (3) is not obviously _multiplicative_ in $f$. However, atomic
polynomial forcing is obviously multiplicative in $f$. Our equivalence in
Theorem B is proven on an element-by-element basis, from which we conclude
that the atomic Leibniz rule is actually multiplicative.

As a consequence, the atomic Leibniz rule can be proven without relying on
Williamson’s results, by checking it on each generator $f$ of $R^{J}$. We
perform this computation in type $A$ in §6, for carefully chosen generators
$f$, via a direct and elementary proof. The operators $T_{q}$ are simplified
dramatically when applied to these generators.

On the one hand, this proves Theorem A for the symmetric group over
$\mathbb{Z}[x_{1},\ldots,x_{n}]$, and consequently for other realizations (see
Remark 1.3). Via Theorem C, this implies $\operatorname{DL}$-forcing in the
same generality. On the other hand, it also gives a computationally effective
way to apply the atomic Leibniz rule (or polynomial forcing), even if one does
not know the operators $T_{q}$ in general: decompose $f$ as a linear
combination of products of generators, and apply the atomic Leibniz rule for
generators one term at a time.

Acknowledgments. BE was partially supported by NSF grants DMS-2201387 and
DMS-2039316. HK was partially supported by the Swedish Research Council. NL
was partially supported by FONDECYT-ANID grant 1230247.

##  2\. Examples and remarks

In this chapter we give some examples of atomic Leibniz rules, in the
relatively small Coxeter group $W=\mathbf{S}_{4}$. The reader should not be
disheartened if these examples are still quite difficult and technical to
verify by hand! Indeed, one purpose of this chapter is to showcase the
complexity involved in even the simplest examples of the atomic Leibniz rule.

The reader will not miss out on anything important if they skip directly to
Section 3.

###  2.1. Examples

Our examples take place in $W=\mathbf{S}_{4}$. We let $s=(12)$, $t=(23)$, and
$u=(34)$ denote the simple reflections. We give the examples without
justification first, and then discuss the verification afterwards. We require
that $T_{q}(f)\in R^{K}$ for some $K\subset S$ depending on $q$ (precisely:
$K=\underline{q}^{-1}I\underline{q}\cap J$).

######  Example 2.1.

There are three $(su,su)$-cosets in $\mathbf{S}_{4}$: the maximal one $p$
which is atomic, the submaximal coset $q$ with $\underline{q}=t$, and the
minimal coset $r$ containing the identity. We claim that for $f,g\in R^{su}$
we have

(5)
$\partial_{tsut}(fg)=tsut(f)\cdot\partial_{tsut}(g)+\partial_{sut}(T_{q}(f)\cdot
g)+T_{r}(f)\cdot g,$

or in other words

(6)
$\partial_{p}(fg)=\underline{p}(f)\cdot\partial_{p}(g)+\partial_{q}(T_{q}(f)\cdot
g)+\partial_{r}(T_{r}(f)\cdot g),$

where

(7) $T_{q}(f)=su\partial_{t}(f),\qquad
T_{r}(f)=\partial_{tsut}(f)-\partial_{sut}(T_{q}(f)).$

Note that (obviously) $T_{q}(f)\in R=R^{\emptyset}$ while (less obviously)
$T_{r}(f)\in R^{su}$. These are the invariance requirements.

######  Remark 2.2.

Iterating the ordinary twisted Leibniz rule, it is not hard to deduce the
existence of a formula of the form

$\partial_{tsut}(fg)=tsut(f)\cdot\partial_{tsut}(g)+\sum_{x<tsut}\partial_{x}(T_{x}(f)g)$

for some operators $T_{x}\colon R\to R$. What is not obvious is that, when
$f,g\in R^{su}$, this formula will simplify so that only the terms where
$x=sut$ and $x=1$ survive.

######  Example 2.3.

There are two $(tu,st)$-cosets in $\mathbf{S}_{4}$: the maximal coset $p$
which is atomic, and the minimal coset $q$ containing the identity element. We
claim that for $f,g\in R^{st}$ we have

(8) $\partial_{stu}(fg)=stu(f)\partial_{stu}(g)+\partial_{tu}(T_{q}(f)\cdot
g),$

or in other words

(9)
$\partial_{p}(fg)=\underline{p}(f)\partial_{p}(g)+\partial_{q}(T_{q}(f)\cdot
g),$

where

(10) $T_{q}(f)=st\partial_{u}(f).$

Note also that $T_{q}(f)\in R^{t}$.

It helps to look at an example of a non-atomic coset, to see that Leibniz
rules are not guaranteed.

######  Example 2.4.

We return to the notation of Example 2.1. Note that $r$ is the only coset less
than $q$, and $\partial_{r}$ is the identity map. Let us argue that there is
no naive analogue of (3) for $\partial_{q}$. If there were, it would have to
have the form

(11) $\partial_{sut}(f\cdot g)=sut(f)\partial_{sut}(g)+T(f)\cdot g$

for some operator $T$. By evaluating at $g=1$ we have
$T(f)=\partial_{sut}(f)$.

Note that $f,g\in R^{su}$. Iterating the twisted Leibniz rule we have

(12) $\displaystyle\quad\partial_{sut}(f\cdot g)=$ $\displaystyle
sut(f)\partial_{sut}(g)+\partial_{s}(ut(f))\partial_{ut}(g)+\partial_{u}(st(f))\partial_{st}(g)+\partial_{su}(t(f))\partial_{t}(g)+\partial_{sut}(f)g.$

Only two of these five terms are in (11). If $g$ is linear and
$\partial_{t}(g)=1$ then the sum of the three missing terms is nonzero for
some $f$. Thus (11) is false.

However, now suppose that $f\in t(R^{su})$ instead of $R^{su}$. Then many
terms in (12) vanish, yielding

(13) $\partial_{sut}(f\cdot g)=sut(f)\partial_{sut}(g)+\partial_{sut}(f)g.$

This is compatible with (11), with $T=\partial_{sut}$.

In view of this example, there is hope of finding some generalization of the
atomic Leibniz rule to some non-atomic $(I,J)$ cosets $q$, letting
$f\in\underline{q}^{-1}(R^{I})$. Be warned that the coset $q$ considered in
this example has various special properties, see Remark 3.12.

The examples above can easily be verified by computer (all the equations are
$R^{stu}$-linear so one need only check the result for $f,g$ in a basis). They
can also all be verified using (1) and (2), but this is a very tricky
exercise. Doing this exercise may be very instructive for the reader, and
emphasizes the difficult and subtlety in these formulae, so we encourage it,
and provide some helping hands.

We begin with a few helpful general properties of Demazure operators, which
hold in general when $m_{st}=3$ and $m_{su}=2$.

######  Lemma 2.5.

We have

(14) $\partial_{s}(s(f))=-\partial_{s}(f),\quad
s\partial_{s}(f)=\partial_{s}(f),\qquad\partial_{s}(\partial_{s}(f))=0,$ (15)
$st\partial_{s}(f)=\partial_{t}(stf),\quad
s\partial_{t}(sf)=t\partial_{s}(tf),\quad s\partial_{u}(f)=\partial_{u}(sf),$
(16) $\alpha_{s}\partial_{s}(f)=f-sf,$ (17)
$\partial_{st}(sf)+\partial_{ts}(f)=t\partial_{st}(f).$

Note that $\alpha_{s_{i}}=x_{i}-x_{i+1}$ above.

###### Proof.

The reader can verify these relations directly from (1). ∎

Most of these formulae are well-known. Meanwhile, we have not seen (17)
before; we only use it in Example 2.9 below. With these relations in hand, one
need not refer to the original definition (1) again, and need only use the
twisted Leibniz rule.

Here are some example computations using (14) and (15).

(18)
$\partial_{s}(t\partial_{s}(f))=-\partial_{s}(st\partial_{s}(f))=-\partial_{s}\partial_{t}(stf)=\partial_{s}\partial_{t}(tstf).$
(19)
$\partial_{s}(t\partial_{s}(f))=\partial_{s}(ts\partial_{s}(f))=ts\partial_{t}\partial_{s}(f).$

Let us consider example 2.3. It helps to observe that

(20) $\partial_{st}(u(f))\in R^{st},\qquad\partial_{stu}(f)\in R^{stu},$

under the assumption that $f\in R^{st}$. One way to see the equality (8) is to
expand both sides using the twisted Leibniz rule. Since $g\in R^{st}$, any
terms containing $\partial_{st}(g)$ or $\partial_{s}(g)$ or $\partial_{t}(g)$
or $\partial_{su}(g)$ are zero. Now compare the coefficients of $g$,
$\partial_{u}(g)$ and $\partial_{tu}(g)$ respectively. On the left the
coefficient of $g$ is $\partial_{stu}(f)$, while on the right the coefficient
of $g$ is

$\partial_{tu}st\partial_{u}(f)=\partial_{t}s\partial_{u}tu\partial_{u}(f)=\partial_{t}stu\partial_{tu}(f)=st\partial_{s}u\partial_{tu}(f)=stu\partial_{stu}(f)=\partial_{stu}(f).$

We have applied (15) repeatedly and used that $\partial_{stu}(f)\in R^{stu}$.
Thus the coefficients of $g$ are the same on both sides of (8). We leave the
other coefficients to the reader.

Example 2.1 is the hardest. One should first prove the following statements
under the critical assumption that $f,g\in R^{su}$.

(21a) $\partial_{tsut}(f)\in R^{stu},\qquad\partial_{ts}(ut(f))\in R^{st},$
(21b) $\partial_{su}(tsu(\partial_{t}(f)))=tsut(\partial_{sut}(f)),$ (21c)
$t(\partial_{sut}(f))=tsu(\partial_{sut}(f)),$ (21d)
$tsu\left[\partial_{sut}(f)-t\partial_{sut}(f)\right]=tsu(\alpha_{t}\partial_{tust}(f))=(\alpha_{s}+\alpha_{t}+\alpha_{u})\partial_{tsut}(f),$
(21e)
$\partial_{tsu}(tf)=\partial_{tsu}(-\alpha_{t}\partial_{t}(f))=-(\alpha_{s}+\alpha_{t}+\alpha_{u})\partial_{tsut}(f).$

We leave these verifications, and the deduction of (5) therefrom, to the
ambitious reader.

###  2.2. Leibniz rules for permutations

A natural question is raised: for which double cosets $p$ does one expect an
equality of the form (3) to hold? Given the discussion in Example 2.4, one
might ask instead: for which double cosets $q$ does (3) hold under the
alternate assumption that $f\in\underline{q}^{-1}(R^{I})$? Note that
$f\in\underline{q}^{-1}(R^{I})$ is equivalent to $f\in R^{J}$ for atomic
cosets, and also for more general cosets called _core cosets_. We believe
these are interesting questions, even though these more general Leibniz rules
currently lack a clear connection to the theory of singular Soergel bimodules.

A special case would be when $I=J=\emptyset$, so that double cosets are in
bijection with elements $w\in W$; all such cosets are core. We start with the
following well-known lemma.

######  Lemma 2.6.

Let $w\in W$ and $f,g\in R$. Let $w=s_{1}\ldots s_{n}$ be a reduced expression
and for $\mathbf{e}=\\{0,1\\}^{n}$ we define the element
$w^{\mathbf{e}}=s_{1}^{e_{1}}\ldots s_{n}^{e_{n}}$. For
$\mathbf{e}=\\{0,1\\}^{n}$ let

$\theta_{i}^{\mathbf{e}}=\begin{cases}s_{i}&\text{if }e_{i}=1\\\
\partial_{i}&\text{if }e_{i}=0\end{cases}\text{ and
}\Theta^{\mathbf{e}}(f)=\theta_{1}^{\mathbf{e}}\circ\theta_{2}^{\mathbf{e}}\circ\ldots\circ\theta_{n}^{\mathbf{e}}(f).$

Then we have

$\partial_{w}(fg)=\sum_{x\leq w}T^{\prime}_{x}(f)\partial_{x}(g)$

where

$T^{\prime}_{x}(f)=\sum_{\mathbf{e}\mid
w^{\mathbf{e}}=x}\Theta^{\mathbf{e}}(f).$

As a special case, $T_{w}^{\prime}(f)=w(f)$.

###### Proof.

This is just an iteration of the twisted Leibniz rule. ∎

To summarize, we obtain a generalized Leibniz rule of the form

(22)
$\partial_{w}(fg)=w(f)\partial_{w}(g)+\sum_{x<w}T^{\prime}_{x}(f)\partial_{x}(g),$

for operators $T^{\prime}_{x}\colon R\to R$ (depending on $w$). The _formula_
for $T^{\prime}_{x}$ one derives in this way is seemingly dependent on the
choice of reduced expression for $w$, though the _operator_ only
depends444Abstractly, the nilHecke algebra is the subalgebra of
$\operatorname{End}(R)$ generated by $R$ (i.e. multiplication by polynomials)
and by Demazure operators. It is well known that the operators
$\\{\partial_{w}\\}$ form a basis of the nilHecke algebra as a free left
$R$-module. Letting $m_{f}$ denote multiplication by $f$, (22) can be viewed
as an equality  $\partial_{w}\circ m_{f}=m_{w(f)}\circ\partial_{w}+\sum
m_{T^{\prime}_{x}(f)}\circ\partial_{x}$ in the nilHecke algebra, which
rewrites $\partial_{w}\circ m_{f}$ in this basis. Consequently, the
coefficients $T^{\prime}_{x}(f)$ in this linear combination depend only on $w$
and $f$. on $w$. In practice, confirming the independence of reduced
expression can be quite subtle. We are unaware of a formula for
$T^{\prime}_{x}$ which is obviously independent of the choice of reduced
expression.

Meanwhile, one can also deduce an equality of the form

(23)
$\partial_{w}(fg)=w(f)\partial_{w}(g)+\sum_{x<w}\partial_{x}(T_{x}(f)\cdot
g),$

for some operators $T_{x}\colon R\to R$. We are unaware of any previous study
of the operators $T_{x}$ and the formula (23).

######  Remark 2.7.

Later in the paper, we also discuss an atomic Leibniz rule similar to (22)
rather than (23).

Here are some examples.

######  Example 2.8.

Let $s=s_{1}$ and $t=s_{2}$. When $w=ts$, applying (2) twice gives

(24)
$\partial_{ts}(fg)=ts(f)\partial_{ts}(g)+\partial_{t}(sf)\partial_{s}(g)+t\partial_{s}(f)\partial_{t}(g)+\partial_{ts}(f)g.$

An equivalent formula is

(25)
$\partial_{ts}(fg)=ts(f)\partial_{ts}(g)+\partial_{t}(\partial_{s}(f)\cdot
g)+\partial_{s}(s\partial_{t}(sf)\cdot g)+\partial_{st}(sf)\cdot g.$

By applying (2) to the second and third terms on the RHS of equation (25), and
with a little help from (14), one obtains (24).

######  Example 2.9.

With notation as above, when $w=sts$, applying (2) thrice gives

$\displaystyle\partial_{sts}(fg)$ $\displaystyle=$ $\displaystyle
sts(f)\partial_{sts}(g)+\partial_{s}(tsf)\partial_{ts}(g)+\partial_{t}(stf)\partial_{st}(g)$
$\displaystyle+\partial_{s}(t\partial_{s}(f))\partial_{t}(g)+\partial_{t}(s\partial_{t}(f))\partial_{s}(g)+\partial_{sts}(f)g.$

More honestly, applying (2) thrice gives the above except that the coefficient
of $\partial_{s}(g)$ is $\partial_{st}(sf)+s\partial_{ts}(f)$. By applying $s$
to (17), one obtains

(27)
$\partial_{st}(sf)+s\partial_{ts}(f)=st\partial_{st}(f)=\partial_{t}(st\partial_{t}(f))=\partial_{t}(s\partial_{t}(f)).$

This is how one deduces (2.9).

An equivalent formula is

$\displaystyle\partial_{sts}(fg)$ $\displaystyle=$ $\displaystyle
sts(f)\partial_{sts}(g)+\partial_{ts}(\partial_{t}(f)\cdot
g)+\partial_{st}(\partial_{s}(f)\cdot g)$
$\displaystyle-\partial_{t}(\partial_{st}(f)\cdot
g)-\partial_{s}(\partial_{ts}(f)\cdot g)+\partial_{sts}(f)\cdot g.$

To verify that (2.9) and (2.9) agree, apply (25) to the term
$\partial_{ts}(\partial_{t}(f)\cdot g)$, and apply (25) with $s$ and $t$
swapped to $\partial_{st}(\partial_{s}(f)\cdot g)$. After some additional
massaging using (14) and (15), one will recover (2.9).

##  3\. Atomic Leibniz rules

###  3.1. Realizations

We fix a Coxeter system $(W,S)$. We let $e$ denote the identity element of
$W$. Recall the definition of a realization from [EW16, §3.1].

######  Definition 3.1.

A _realization_ of $(W,S)$ over $\Bbbk$ is the data
$(\Bbbk,V,\Delta,\Delta^{\vee})$ of a commutative ring $\Bbbk$, a free finite-
rank $\Bbbk$-module $V$, a set $\Delta=\\{\alpha_{s}\\}_{s\in S}$ of _simple
roots_ inside $V$, and a set $\Delta^{\vee}=\\{\alpha_{s}^{\vee}\\}_{s\in S}$
of _simple coroots_ inside $\operatorname{Hom}_{\Bbbk}(V,\Bbbk)$, satisfying
the following properties. One has $\alpha_{s}^{\vee}(\alpha_{s})=2$ for all
$s\in S$. The formula

$s(v):=v-\alpha_{s}^{\vee}(v)\cdot\alpha_{s}$

defines an action of $W$ on $V$. Also, the technical condition [EW16, (3.3)]
holds, which is redundant for most base rings $\Bbbk$.

For short, we often refer to the data of a realization simply by reference to
the $W$-representation $V$.

######  Example 3.2.

The _permutation realization_ of $\mathbf{S}_{n}$ over $\mathbb{Z}$ has
$V=\mathbb{Z}^{n}$ with basis $\\{x_{i}\\}_{i=1}^{n}$ and with dual bases
$\\{x_{i}^{*}\\}$. For $1\leq i\leq n-1$ one sets $\alpha_{i}=x_{i}-x_{i+1}$
and $\alpha_{i}^{\vee}=x_{i}^{*}-x_{i+1}^{*}$.

######  Example 3.3.

For a Weyl group $W$, the _root realization_ of $(W,S)$ over $\mathbb{Z}$ is
the free $\mathbb{Z}$-module with basis $\Delta$. One defines $\Delta^{\vee}$
so that the pairings $\alpha_{s}^{\vee}(\alpha_{t})$ agree with the usual
Cartan matrix of $W$.

######  Example 3.4.

Let $(\Bbbk,V,\Delta,\Delta^{\vee})$ be a realization of $(W,S)$, and
$I\subset S$. Then $(\Bbbk,V,\Delta_{I},\Delta_{I}^{\vee})$ is also a
realization of $(W_{I},I)$, the _restriction of the realization to a parabolic
subgroup_. Here $\Delta_{I}=\\{\alpha_{s}\\}_{s\in I}$, and similarly for
$\Delta_{I}^{\vee}$.

Given a realization, let $R$ be the polynomial ring whose linear terms are
$V$. We can associate Demazure operators $\partial_{s}\colon R\to R$ for $s\in
S$, which agree with $\alpha_{s}^{\vee}$ on $V\subset R$, and are extended by
the twisted Leibniz rule. For details, see [EW16, §3.1]. For each finitary
subset $I\subset S$, we also consider the subring $R^{I}$ of
$W_{I}$-invariants in $R$. The ring $R^{I}$ is graded, and all $R^{I}$-modules
will be graded, but we will not keep track of grading shifts in this paper as
they will play no significant role. The background on this material in
[EKLP23a] should be sufficient.

######  Definition 3.5.

A _(balanced) Frobenius realization_ is a realization satisfying the following
properties, see [EKLP23a, §3.1] for definitions.

  * •

It is balanced.

  * •

It satisfies generalized Demazure surjectivity.

  * •

It is faithful when restricted to each finite parabolic subgroup $W_{I}$.

We assume tacitly throughout this paper that we work with a Frobenius
realization. The main implication of these assumptions is that, when $I\subset
S$ is finitary, the Demazure operator $\partial_{w_{I}}\colon R\to R^{I}$ is
well-defined and equips the ring extension $R^{I}\subset R$ with the structure
of a Frobenius extension.

The left and right redundancy sets of an $(I,J)$-coset $p$ are defined and
denoted as

(29)
$\operatorname{LR}(p)=I\cap\underline{p}J\underline{p}^{-1},\qquad\operatorname{RR}(p)=\underline{p}^{-1}I\underline{p}\cap
J.$

An $(I,J)$-coset $p$ is a _core coset_ if $I=\operatorname{LR}(p)$ and
$J=\operatorname{RR}(p)$.

For any $(I,J)$-coset $q$, in [EKLP23a] we define a Demazure operator

$\partial_{q}:R^{J}\to R^{I}.$

By definition, $\partial_{q}$ is the restriction of the ordinary Demazure
operator

$\partial_{\overline{q}w_{J}^{-1}}\colon R\to R$

to the subring $R^{J}$. After restriction, the image is contained in $R^{I}$,
see [EKLP23a, Lemma 3.9]. Note that $\overline{q}w_{J}^{-1}=\underline{q}$ if
and only if $q$ is a core coset.

######  Remark 3.6.

Some results from [EKLP23a] require further that the realization is faithful,
rather than just faithful upon restriction to each finite parabolic subgroup.
In particular, the set $\\{\partial_{p}\\}$ as $p$ ranges over $(I,J)$-cosets
need not be linearly independent when the realization is not faithful.

A _multistep $(I,J)$-expression_ is a sequence of finitary subsets

$I_{\bullet}=[[I=I_{0}\subset K_{1}\supset I_{1}\subset\ldots K_{m}\supset
I_{m}=J]].$

The definition of a _reduced_ multistep expression, and of the $(I,J)$-coset
that it _expresses_ , can be found in [EK21, Definition 1.4]. When
$I_{\bullet}$ is a (reduced) expression which expresses $p$, we write
$p\leftrightharpoons I_{\bullet}$.

As in [EK21], for $x,y\in W$ we write $x.y$ for a reduced composition, where
$\ell(xy)=\ell(x)+\ell(y)$. We also use this notation for the reduced
composition of reduced expressions, or the reduced composition of double
cosets, see [EK21] for more details. Demazure operators compose well over
reduced compositions: one has $\partial_{p.q}=\partial_{p}\circ\partial_{q}$,
as proven in [EKLP23a, Corollary 3.19].

###  3.2. Precise statement of atomic Leibniz rules

######  Remark 3.7.

Note that $w_{J}$ is an involution, so $w_{J}=w_{J}^{-1}$. We write
$\overline{q}w_{J}^{-1}$ above to emphasize that
$\overline{q}=(\overline{q}w_{J}^{-1}).w_{J}$.

######  Definition 3.8.

Suppose $M$ is finitary, $s\in M$, and $t=w_{M}sw_{M}$. Let
$I=\widehat{s}:=M\setminus s$, and $J=\widehat{t}:=M\setminus t$. Let
${\mathtt{a}}$ be the (atomic) $(I,J)$-coset containing $w_{M}$. We say a
_(rightward) atomic Leibniz rule_ holds for ${\mathtt{a}}$ if there exist
$R^{M}$-linear operators $T^{\mathtt{a}}_{q}$ from $R^{J}$ to
$R^{\operatorname{RR}(q)}$ for each $(I,J)$-coset $q<{\mathtt{a}}$, such that
for any $f,g\in R^{J}$ we have

(30) $\partial_{\mathtt{a}}(f\cdot
g)=\underline{{\mathtt{a}}}(f)\partial_{\mathtt{a}}(g)+\sum_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}(T^{\mathtt{a}}_{q}(f)\cdot
g).$

We encourage the reader to confirm that $T^{\mathtt{a}}_{q}(f)\in
R^{\operatorname{RR}(q)}$ in the examples of §2. We continue to write $T_{q}$
instead of $T^{\mathtt{a}}_{q}$ when ${\mathtt{a}}$ is understood.

######  Remark 3.9.

We say “an atomic Leibniz rule” rather than “the atomic Leibniz rule” because
we are defining a prototype for a kind of formula. If one specifies operators
$T_{q}$ such that the formula holds, then one has produced “the” atomic
Leibniz rule for that coset ${\mathtt{a}}$ (indeed, we prove in 5.5 that such
operators are unique for certain realizations).

The difference between (30) and (3) is subtle: we have written
$\partial_{\overline{q}w_{J}^{-1}}$ instead of $\partial_{q}$. The difference
between $\partial_{\overline{q}w_{J}^{-1}}$ and $\partial_{q}$ is only a
matter of the domain and codomain of the functions: the former is a function
$R\to R$, while the latter is its restriction to a function $R^{J}\to R^{I}$.
Meanwhile, $T_{q}(f)$ lives in $R^{\operatorname{RR}(q)}$. The inclusion
$R^{J}\subset R^{\operatorname{RR}(q)}$ is proper unless $q$ is core. It is
therefore inappropriate to apply $\partial_{q}$ to $T_{q}(f)\cdot g$. Having
altered notation so that the domain of the operator is appropriate, we still
need to worry about the codomain, which we address in the following lemma.

######  Lemma 3.10.

With notation as in Definition 3.8, we have
$\partial_{\overline{q}w_{J}^{-1}}(T_{q}(f)\cdot g)\in R^{I}$.

###### Proof.

Recall from [EK21, Proposition 4.28] that any $(I,J)$-coset $q$ has a reduced
expression of the form

(31)
$q\leftrightharpoons[[I\supset\operatorname{LR}(q)]].q^{\operatorname{core}}.[[\operatorname{RR}(q)\subset
J]].$

Let $z$ be the $(I,\operatorname{RR}(q))$-coset with reduced expression

(32)
$z\leftrightharpoons[[I\supset\operatorname{LR}(q)]].q^{\operatorname{core}}.$

Since (31) is reduced, by [EK21, Proposition 4.3], we have
$\overline{q}=\overline{z}.(w_{\operatorname{RR}(q)}^{-1}w_{J}),$ so that

(33) $\overline{q}w_{J}^{-1}=\overline{z}w_{\operatorname{RR}(q)}^{-1}.$

Consequently, the same operator $\partial_{\overline{q}w_{J}^{-1}}\colon R\to
R$ restricts to both $\partial_{q}\colon R^{J}\to R^{I}$ and
$\partial_{z}\colon R^{\operatorname{RR}(q)}\to R^{I}$. In particular, this
operator sends $R^{\operatorname{RR}(q)}$ to $R^{I}$. ∎

Further elaboration will be helpful in subsequent chapters. By [EKLP23a, Lemma
3.17], the reduced expression (31) implies that the map $\partial_{q}$ is a
composition of three Demazure operators. Recall that
$\partial_{[[I\supset\operatorname{LR}(q)]]}$ is the Frobenius trace map
$R^{\operatorname{LR}(q)}\to R^{I}$ often denoted as
$\partial^{\operatorname{LR}(q)}_{I}$. Recall also that
$\partial_{[[\operatorname{RR}(q)\subset J]]}$ is the inclusion map
$R^{J}\subset R^{\operatorname{RR}(q)}$. We denote this inclusion map
$\iota^{\operatorname{RR}(q)}_{J}$ below. So we have

(34)
$\partial_{q}=\partial^{\operatorname{LR}(q)}_{I}\circ\partial_{q^{\operatorname{core}}}\circ\iota^{\operatorname{RR}(q)}_{J}.$

By (32) we have

(35)
$\partial_{z}=\partial^{\operatorname{LR}(q)}_{I}\circ\partial_{q^{\operatorname{core}}},$

which agrees with the restriction of $\partial_{\overline{q}w_{J}^{-1}}$ to
$R^{\operatorname{RR}(q)}$. Thus one has the following reformulation of (30):

(36) $\partial_{\mathtt{a}}(f\cdot
g)=\underline{{\mathtt{a}}}(f)\partial_{\mathtt{a}}(g)+\sum_{q<{\mathtt{a}}}\partial^{\operatorname{LR}(q)}_{I}\partial_{q^{\operatorname{core}}}(T_{q}(f)\cdot\iota^{\operatorname{RR}(q)}_{J}(g)).$

Now the polynomial $T_{q}(f)$ appears more appropriately in the “middle” of
this factorization of $\partial_{q}$. This discussion of the “placement” of
the polynomial $T_{q}(f)$ will play a role in our diagrammatic proof of
polynomial forcing.

We are now prepared to discuss another version of the atomic Leibniz rule,
using the factorization (34). It should not be obvious that these two atomic
Leibniz rules are related, though the equivalence with polynomial forcing will
shed light on this issue.

######  Definition 3.11.

Use the notation from Definition 3.8 and from (34). We say that a _leftward
atomic Leibniz rule_ holds for ${\mathtt{a}}$ if there exist $R^{M}$-linear
operators $T^{\prime}_{q}$ from $R^{J}$ to $R^{\operatorname{LR}(q)}$ for each
$(I,J)$-coset $q<{\mathtt{a}}$, such that for any $f,g\in R^{J}$ we have

(37) $\partial_{\mathtt{a}}(f\cdot
g)=\underline{{\mathtt{a}}}(f)\partial_{\mathtt{a}}(g)+\sum_{q<{\mathtt{a}}}\partial^{\operatorname{LR}(q)}_{I}(T^{\prime}_{q}(f)\cdot\partial_{q^{\operatorname{core}}}(\iota_{J}^{\operatorname{RR}(q)}(g))).$

######  Remark 3.12.

The fact that $T_{q}(f)$ lives in $R^{\operatorname{RR}(q)}$ and not in
$R^{J}$ is easy to overlook, but overlooking it is dangerous. We have
attempted to prove atomic Leibniz-style rules for more general families of
cosets (core cosets, cosets whose core is atomic, etcetera). Each time what
prevents one from bootstrapping from the atomic case to more general cases is
the fact that $T_{q}(f)$ does not live in $R^{J}$. The generalization in
Example 2.4 has the special feature that the lower cosets are all core, so
that their right redundancy equals $J$. (It also has the special feature that
$q^{\operatorname{core}}$ is atomic.)

###  3.3. Changing the realization

We argue that the atomic Leibniz rule for some realizations implies the atomic
Leibniz rule for others. Given a realization, one can obtain another
realization by applying base change $(-)\otimes_{\Bbbk}\Bbbk^{\prime}$ to $V$,
and choosing new roots and coroots in the natural way. We call this a
_specialization_. Here are two other common ways to alter the realization.

######  Definition 3.13.

Let $(V,\Delta,\Delta^{\vee})$ be a realization of $(W,S)$ over $\Bbbk$. Let
$N$ be a free $\Bbbk$-module acted on trivially by $W$. Then $(V\oplus
N,\Delta\oplus 0,\Delta^{\vee}\oplus 0)$ is a realization, called a _$W$
-invariant enlargement_ of the original. More precisely, the new roots are the
image of the old roots under the inclusion map, and the new coroots kill the
summand $N$.

######  Definition 3.14.

Let $(V,\Delta,\Delta^{\vee})$ be a realization of $(W,S)$ over $\Bbbk$.
Suppose one has a decomposition $V=X\oplus Y$ of free $\Bbbk$-modules, such
that $W$ acts trivially on $X$, though $W$ need not preserve $Y$. Note that
the coroots necessarily annihilate $X$. Then
$(Y,\overline{\Delta},\Delta^{\vee}_{Y})$ is a realization, called a _$W$
-invariant quotient_ of the original. Here, we identify $Y$ as the quotient
$V/X$, and $\overline{\Delta}$ represents the image of $\Delta$ under the
quotient map. The functionals $\Delta^{\vee}$ kill $X$, so they descend to
functionals $\Delta^{\vee}_{Y}$ on $Y$. We also make the technical
assumption555This assumption is required for the $W$-invariant quotient to
satisfy Demazure surjectivity. that $\alpha_{s}$ induces a surjective map
$Y^{*}\to\Bbbk$.

######  Example 3.15.

Let $(W,S)$ have type $\widetilde{A}_{n-1}$, with simple reflections $s_{i}$
for $1\leq i\leq n$. Let $V$ be the free $\mathbb{Z}$-module spanned by
$\\{x_{i}\\}_{i=1}^{n}$ and $\delta$. Let
$\\{x_{i}^{*}\\}\cup\\{\delta^{*}\\}$ denote the dual basis in
$\operatorname{Hom}_{\Bbbk}(V,\Bbbk)$. With indices considered modulo $n$, let
$\alpha_{i}=x_{i}-x_{i+1}+\delta$, and
$\alpha_{i}^{\vee}=x_{i}^{*}-x_{i+1}^{*}$. This is a realization of $(W,S)$
called the _affine permutation realization_. Note that
$\sum_{i=0}^{n-1}\alpha_{i}=n\delta$, which is $W$-invariant. Let $X$ be the
span of $\delta$, and $Y$ be the span of $\\{x_{i}\\}_{i=1}^{n}$. Note that
$W$ does not preserve $Y$, since the roots are not contained in $Y$. There is
a valid $W$-invariant quotient $(Y,\overline{\Delta},\Delta^{\vee}_{Y})$ which
agrees, upon restriction to the parabolic subgroup $\mathbf{S}_{n}$ generated
by $\\{s_{i}\\}_{i=1}^{n-1}$, with the permutation representation.

######  Example 3.16.

Continuing the previous example, let $y_{i}=x_{i}-i\delta$. Then we can also
view $V$ as having basis $\\{y_{i}\\}_{i=1}^{n}\cup\\{\delta\\}$, and
$\alpha_{i}=y_{i}-y_{i+1}$ for $i\neq n$. Upon restriction to the parabolic
subgroup $\mathbf{S}_{n}$, we see that $V$ is isomorphic to the $W$-invariant
enlargement of the permutation representation of $\mathbf{S}_{n}$ (with basis
$\\{y_{i}\\}$) by the $W$-invariant span of $\delta$.

Indeed, $W$ has $n$ distinct maximal parabolic subgroups isomorphic to
$\mathbf{S}_{n}$ as groups. A similar construction will show that the
restriction of $V$ to any maximal parabolic subgroup (a copy of
$\mathbf{S}_{n}$) will be isomorphic to an invariant enlargement of its
permutation representation.

######  Lemma 3.17.

If a rightward (resp. leftward) atomic Leibniz rule holds for a Frobenius
realization, then it also holds for specializations, $W$-invariant
enlargements, and $W$-invariant quotients.

###### Proof.

Let $R$ be the ring associated to the original realization, and
$R_{\operatorname{new}}$ be the realization associated to the specialization,
enlargement, or quotient. All three cases are united by the fact that
$R_{\operatorname{new}}$ is a tensor product of the form $R\otimes_{A}B$,
where $A\subset R$ is a subring on which $W$ acts trivially, and $B$ is a ring
on which $W$ acts trivially. For specializations we have
$R_{\operatorname{new}}=R\otimes_{\Bbbk}\Bbbk^{\prime}$; for enlargements we
have $R_{\operatorname{new}}=R\otimes_{\Bbbk}R_{N}$, where $R_{N}$ is the
polynomial ring of $N$; for quotients we have
$R_{\operatorname{new}}=R\otimes_{R_{X}}\Bbbk$, where $R_{X}$ is the
polynomial ring of $X$, and $\Bbbk$ is its quotient by the ideal of positive
degree elements. For $w\in W$, its action on $R_{\operatorname{new}}$ is given
by $w\otimes\operatorname{id}$. The roots in $R_{\operatorname{new}}$ are
given by $\alpha_{s}\otimes 1$, and the Demazure operators
$\partial^{\operatorname{new}}_{s}$ on $R_{\operatorname{new}}$ have the form
$\partial_{s}\otimes\operatorname{id}$.

The important point in all three cases is that for each $I\subset S$ finitary
we have $R_{\operatorname{new}}^{I}=R^{I}\otimes_{A}B$. We now prove this
somewhat subtle point. There is an obvious inclusion $R^{I}\otimes_{A}B\subset
R_{\operatorname{new}}^{I}$, so we need only show the other inclusion.

It is straightforward to verify that the new realization satisfies generalized
Demazure surjectivity. A consequence is that the typical properties of
Demazure operators are satisfied. For example, the kernel and the image of
$\partial^{\operatorname{new}}_{s}$ are both equal to
$R_{\operatorname{new}}^{s}$, and $\partial^{\operatorname{new}}_{s}$ is
$R^{s}_{\operatorname{new}}$-linear. It follows that
$\partial^{\operatorname{new}}_{I}$ is also
$R_{\operatorname{new}}^{I}$-linear.

Suppose that $g\in R_{\operatorname{new}}^{I}$, and write $g=\sum f_{i}\otimes
b_{i}$. Choose some $P\in R$ with $\partial_{I}(P)=1$, which exists by
generalized Demazure surjectivity. Then
$\partial^{\operatorname{new}}_{I}(P\otimes 1)=1\otimes 1$ in
$R_{\operatorname{new}}$. Thus

(38) $g=g\partial^{\operatorname{new}}_{I}(P\otimes
1)=\partial^{\operatorname{new}}_{I}(g\cdot(P\otimes
1))=\sum\partial_{I}(f_{i}\cdot P)\otimes b_{i}.$

Thus $g\in R^{I}\otimes_{A}B$.

The rest of the proof is straightforward. Fix an atomic coset ${\mathtt{a}}$.
For each $q<{\mathtt{a}}$, given operators $T_{q}$ for the original
realization satisfying (30), we define
$T_{q,\operatorname{new}}:=T_{q}\otimes\operatorname{id}$. By linearity, we
need only check (30) for $R_{\operatorname{new}}$ on elements in
$R_{\operatorname{new}}^{J}$ of the form $f\otimes b_{1}$ and $g\otimes b_{2}$
for $f,g\in R^{J}$. It is easy to verify (30) for $R_{\operatorname{new}}$ on
such elements, since all operators (like $\underline{{\mathtt{a}}}$ or
$\partial_{\overline{q}w_{J}^{-1}}$) are applied only to the first tensor
factor, where we can use the atomic Leibniz rule from $R$. We conclude by
noting that $T_{q,\operatorname{new}}$ has the appropriate codomain as well. ∎

We do not claim that any statements about the unicity of the operators $T_{q}$
will extend from a realization to its specializations, enlargements, or
quotients.

######  Lemma 3.18.

Let $(\Bbbk,V,\Delta,\Delta^{\vee})$ be a realization of $(W,S)$. If one can
prove an atomic Leibniz rule for the restriction of $V$ to $W_{M}$, for all
(maximal) finitary subsets $M\subset S$, then an atomic Leibniz rule holds for
$W$.

###### Proof.

Every atomic coset in $W$ lives within $W_{M}$ for some finitary $M$ (which
lives within a maximal finitary subset), and the same atomic Leibniz rule
which works for $W_{M}$ will work for $W$. ∎

######  Example 3.19.

Suppose one can prove an atomic Leibniz rule for the permutation realization
of $\mathbf{S}_{n}$ over $\mathbb{Z}$. Then by enlargement, one obtains an
atomic Leibniz rule for the affine permutation realization restricted to any
finite parabolic subgroup, see Example 3.16. By the previous lemma, an atomic
Leibniz rule holds for the affine permutation realization of the affine Weyl
group of type $\widetilde{A}_{n-1}$.

##  4\. Lower terms

In this section we give an explicit description of the ideal of lower terms
for an atomic coset using the technology of singular light leaves.

###  4.1. Definition of lower terms

######  Definition 4.1.

Let

$I_{\bullet}=[[I=I_{0}\subset K_{1}\supset I_{1}\subset\ldots K_{m}\supset
I_{m}=J]]$

be a multistep $(I,J)$ expression. To this expression we associate a
_(singular) Bott-Samelson bimodule_

$\operatorname{BS}(I_{\bullet}):=R^{I_{0}}\otimes_{R^{K_{1}}}R^{I_{1}}\otimes_{R^{K_{2}}}\cdots\otimes_{R^{K_{m}}}R^{I_{m}}.$

This is an $(R^{I},R^{J})$-bimodule. The collection of all Bott-Samelson
bimodules is closed under tensor product, and forms (the set of objects in) a
full sub-2-category of the 2-category of bimodules. This sub-2-category is
denoted $\mathbf{SBSBim}$.

For two $(R^{I},R^{J})$-bimodules $B$ and $B^{\prime}$,
$\operatorname{Hom}(B,B^{\prime})$ denotes the space of bimodule maps.
Moreover, $\operatorname{Hom}(B,B^{\prime})$ is itself an
$(R^{I},R^{J})$-bimodule in the usual way.

Inside any linear category, given a collection of objects, their identity maps
generate a two-sided ideal. This ideal consists of all morphisms which factor
through one of those objects, and linear combinations thereof. In the context
of $(R^{I},R^{J})$-bimodules, the actions of $R^{I}$ and $R^{J}$ commute with
any morphism, and thus preserve the factorization of morphisms. Hence the
morphisms within any such ideal form a sub-bimodule of the original Hom space.

######  Definition 4.2.

Let $p$ be an $(I,J)$-coset. Consider the set of reduced expressions
$M_{\bullet}$ for any $(I,J)$-coset $q$ with $q<p$. Let
$\operatorname{Hom}_{<p}$ denote the ideal in the category of
$(R^{I},R^{J})$-bimodules generated by the identity maps of
$\operatorname{BS}(M_{\bullet})$ for such expressions. Then
$\operatorname{Hom}_{<p}$ is a two-sided ideal, the _ideal of lower terms_
relative to $p$. The ideal $\operatorname{Hom}_{\leq p}$ is defined similarly.

So $\operatorname{Hom}_{<p}(B,B^{\prime})$ is a subset of
$\operatorname{Hom}(B,B^{\prime})$, and is a sub-bimodule for $(R^{I},R^{J})$.
We write $\operatorname{End}_{<p}(B)$ instead of
$\operatorname{Hom}_{<p}(B,B)\subset\operatorname{End}(B)$.

We now focus on the case of atomic cosets. We use the letter ${\mathtt{a}}$ to
denote an atomic coset and let $[[I\subset M\supset J]]$ denote the unique
reduced expression of ${\mathtt{a}}$. We let

(39) $B_{\mathtt{a}}:=\operatorname{BS}([[I\subset M\supset
J]])=R^{I}\otimes_{R^{M}}R^{J}.$

Because $B_{\mathtt{a}}$ is generated by $1\otimes 1$ as a bimodule, any
endomorphism is determined by where it sends this element. Thus

(40) $\operatorname{End}(B_{\mathtt{a}})\cong R^{I}\otimes_{R^{M}}R^{J}$

as $(R^{I},R^{J})$-bimodules, via the operations of left and right
multiplication. Hence $\operatorname{End}(B_{\mathtt{a}})\cong B_{\mathtt{a}}$
as $(R^{I},R^{J})$-bimodules666We have ignored gradings in this paper. Using
traditional grading conventions for Bott-Samelson bimodules,
$\operatorname{End}(B_{\mathtt{a}})$ and $B_{\mathtt{a}}$ are only isomorphic
up to shift. The identity map of $\operatorname{End}(B_{\mathtt{a}})$ is in
degree zero, while $1\otimes 1\in B_{\mathtt{a}}$ is not.. It is easy to
deduce that $B_{\mathtt{a}}$ is indecomposable (when $\Bbbk$ is a domain)
since there are no non-trivial idempotents in
$\operatorname{End}(B_{\mathtt{a}})\cong B_{\mathtt{a}}$.

###  4.2. Atomic double leaves

The goal of the section is to describe a large family of morphisms in
$\operatorname{End}(B_{{\mathtt{a}}})$ called double leaves, most of which are
in $\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})$ by construction. We
use the diagrammatic technology originally found in [ESW17] and developed
further in [EKLP24a].

We assume a Frobenius realization, see Definition 3.5. In particular, the ring
inclusions $R^{I}\subset R^{J}$ are Frobenius extensions. Under these
assumptions, a diagammatic 2-category $\mathbf{Frob}$ is constructed in
[ESW17], and it comes equipped with a 2-functor to $\mathbf{SBSBim}$. This
2-functor is essentially surjective, but is not expected to be an equivalence;
the category $\mathbf{Frob}$ is missing a number of relations.

Double leaves are to be constructed either as morphisms in $\mathbf{Frob}$, or
as their images in $\mathbf{SBSBim}$, depending on the context.

The objects in $\mathbf{Frob}$ are indexed not by multistep expressions but by
singlestep expressions. An _$(I,J)$ singlestep expression_ is a sequence

$I_{\bullet}=[I=I_{0},I_{1},\ldots,I_{d}=J]$

where each $I_{i}$ is a finitary subset of $S$, and each $I_{i}$ and $I_{i+1}$
differ by the addition or removal of a single simple reflection. We use single
brackets for singlestep expressions, and double brackets for multistep
expressions.

Throughout this section, we fix $I\subset M=Is\subset S$ finitary, and let
$t=w_{M}sw_{M}$ and $J=M\setminus t$, so that

${\mathtt{a}}\leftrightharpoons[I,M,J]$

is an atomic coset. We also fix the $(I,M)$-coset

$n=W_{I}eW_{M}.$

####  4.2.1. Elementary light leaves for atomic Grassmannian pairs

By definition of atomic, $\overline{{\mathtt{a}}}=w_{M}$. Then for an
$(I,J)$-coset $q$, then condition $q\leq{\mathtt{a}}$ is equivalent (see
[EKLP23b, Thm 2.16]) to $\overline{q}\leq w_{M}$, which in turn is equivalent
to $q\subseteq n=W_{M}.$

For an $(I,J)$-coset $q$ contained in $W_{M}$, the pair $q\subset n$ is
Grassmannian in the sense of [EKLP24a, Definition 2.7]. Associated to such a
pair, [EKLP24a, Section 7.3] constructs a distinguished map called an
elementary light leaf. The map (and codomain of the map) depends on a choice
we make now: we fix a reduced expression $X_{q}$ of the form

(41) $X_{q}=[[I\supset\operatorname{LR}(q)]]\circ
X_{q}^{\operatorname{core}}\circ[[\operatorname{RR}(q)\subset J]]$

where $X_{q}^{\operatorname{core}}$ is a reduced expression of
$q^{\operatorname{core}}$.

######  Definition 4.3.

Let $q$ be an $(I,J)$-coset contained in $n$. The _elementary light leaf
associated to $[n,q]$ (and $X_{q}$)_ is the $(R^{I},R^{J})$-bimodule morphism

$\mathbb{ELL}([n,q]):\operatorname{BS}([I,M,J])\to\operatorname{BS}(X_{q})$

sending the generator $1\otimes 1\in
R^{I}\otimes_{R^{M}}R^{J}=\operatorname{BS}([I,M,J])$ to the element
$1^{\otimes}:=1\otimes\cdots\otimes 1\in\operatorname{BS}(X_{q})$.
Equivalently, $\mathbb{ELL}([n,q])$ is defined by the diagrammatic
construction in [EKLP24a, Section 7.3] (see also [EKLP24a, Lemma 8.10]).

We refer to [EKLP24a] and [ESW17] for a diagrammatic exhibition of morphisms
between Bott-Samelson bimodules. The morphism $\mathbb{ELL}([n,q])$ is
determined by the condition that its diagram consists only of counterclockwise
cups and right-facing crossings, as in the following diagram.

${\labellist\small\hair
1pt\pinlabel{\operatorname{LR}(q)}[]at6050\pinlabel{\operatorname{RR}(q)}[]at24550\pinlabel{I}[]at3514\pinlabel{M}[]at15014\pinlabel{J}[]at28014\endlabellist\centering\vbox{\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1.1}]{./EPS/ELLat}}$}}}}}}}}}}\@add@centering}$

In our examples, we color the simple reflection
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}s}$ in
strawberry, and
${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}t}$ in
teal. Sometimes $s=t$, which will force us to change our convention.

In type $A$ the expression $X_{q}^{\operatorname{core}}$ takes a simple form,
and thus $\mathbb{ELL}([n,q])$ could be described more explicitly.

######  Example 4.4.

Let $W_{M}=\mathbf{S}_{a+b}$ be a symmetric group, for some $a\neq b$, and let
$I=\widehat{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}s}}$
be such that
$W_{I}=\mathbf{S}_{b}\times\mathbf{S}_{a}\subset\mathbf{S}_{a+b}$. Then
$J=\widehat{{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}t}}$
is such that
$W_{J}=\mathbf{S}_{a}\times\mathbf{S}_{b}\subset\mathbf{S}_{a+b}$. As will be
explained in Section 6 (Equation (88)), each $(I,J)$-coset $q$ in $W_{M}$ has
a unique reduced expression $X_{q}$ of the form (41).

  1. (1)

If $q=W_{I}eW_{J}$ then we have

$q\leftrightharpoons
X_{q}=[[\widehat{s}\supset\widehat{s}\widehat{t}]]\circ[\widehat{s}\widehat{t}]\circ[[\widehat{s}\widehat{t}\subset\widehat{t}]]=[\widehat{s}-{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}t}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}s}].$

Here we have

$\mathbb{ELL}([n,q])=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={.6}]{./EPS/rightcross}}$}}}}}}}}}}.$

  2. (2)

If $q=W_{I}w_{M}W_{J}$ then we have

$q={\mathtt{a}}\leftrightharpoons X_{q}=[I,M,J].$

Here we have
$\mathbb{ELL}([n,q])=\operatorname{id}_{\operatorname{BS}([I,M,J])}$.

  3. (3)

Otherwise, we have

$q\leftrightharpoons
X_{q}=[[\widehat{s}\supset\widehat{s}{\widehat{k}\widehat{\ell}}]]\circ[\widehat{s}{\widehat{k}\widehat{\ell}}\subset\widehat{k}\widehat{\ell}\supset\widehat{t}\widehat{k}\widehat{\ell}]\circ[[\widehat{t}\widehat{k}\widehat{\ell}\subset\widehat{t}]]$

for distinct $s,k,\ell,t\in M$. Here we have

$\mathbb{ELL}([n,q])=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={.6}]{./EPS/qELL1}}$}}}}}}}}}}.$

Here is a non-type A example.

######  Example 4.5.

Let $(W,S)$ be of type $E_{6}$ where $S$ is indexed as in the Dynkin diagram
$6$$5$$4$$3$$2$$1$. Let $M=S$ and $s=3$. Then $w_{M}3w_{M}=5$ and thus

${\mathtt{a}}\leftrightharpoons[\widehat{3}+3-5]=[\widehat{3},M,\widehat{5}]$

is an atom. For the $(\widehat{3},\widehat{5})$-coset $q<{\mathtt{a}}$ with
reduced expression

$q\leftrightharpoons
X_{q}=[[\widehat{3}\supset\\{4,6\\}]]\circ[+3-4+5+2+1-6-2-3+4-5]\circ[[\\{1,4\\}\subset\widehat{5}]],$

the elementary light leaf $\mathbb{ELL}([n,q])$ is

${\labellist\small\hair
2pt\pinlabel{1}[]at683\pinlabel{2}[]at1883\pinlabel{5}[]at3083\pinlabel{3}[]at4683\pinlabel{4}[]at5883\pinlabel{5}[]at7083\pinlabel{2}[]at8283\pinlabel{1}[]at9483\pinlabel{6}[]at10683\pinlabel{2}[]at11883\pinlabel{3}[]at13083\pinlabel{4}[]at14283\pinlabel{5}[]at15483\pinlabel{3}[]at17083\pinlabel{2}[]at18283\pinlabel{6}[]at19483\endlabellist\centering\vbox{\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/qELL2}}$}}}}}}}}}}\@add@centering}$

####  4.2.2. Atomic double leaves

There is a contravariant (but monoidally-covariant) “duality” functor
$\mathcal{D}$ from $\mathbf{Frob}$ to itself defined as follows:

  * •

it preserves objects and $1$-morphisms,

  * •

on $2$-morphisms, it flips each diagram upside-down and reverses all the
orientations.

This functor is an involution.

######  Definition 4.6.

Given $q\leq{\mathtt{a}}$ and $b\in R^{\operatorname{RR}(q)}$, the associated
_(right-sprinkled) double leaf_ $\mathbb{DLL}_{r}(q,b)$ is the composition

(42)
$B_{\mathtt{a}}\xrightarrow{\mathbb{ELL}([n,q])}\operatorname{BS}(X_{q})\xrightarrow{b}\operatorname{BS}(X_{q})\xrightarrow{\mathcal{D}(\mathbb{ELL}([n,q]))}B_{\mathtt{a}}.$

The middle map in (42) uses that $X_{q}$ has the form (41): the map is
multiplication by the element

$1^{\otimes}\otimes b\otimes
1\in\operatorname{BS}([[I\supset\operatorname{LR}(q)]]\circ
X_{q}^{\operatorname{core}})\otimes_{R^{\operatorname{RR}(q)}}R^{\operatorname{RR}(q)}\otimes_{R^{J}}R^{J}.$

In diagrams, we have

$\mathbb{DLL}_{r}(q,b)={\labellist\small\hair
1pt\pinlabel{b}[]at24564\endlabellist\centering\vbox{\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={.9}]{./EPS/DLLat}}$}}}}}}}}}}\@add@centering}$

######  Definition 4.7.

Given $q\leq{\mathtt{a}}$ and $b\in R^{\operatorname{LR}(q)}$, the associated
_left-sprinkled double leaf_ $\mathbb{DLL}_{l}(q,b)$ is the composition

(43)
$B_{\mathtt{a}}\xrightarrow{\mathbb{ELL}([n,q])}\operatorname{BS}(X_{q})\xrightarrow{b}\operatorname{BS}(X_{q})\xrightarrow{\mathcal{D}(\mathbb{ELL}([n,q]))}B_{\mathtt{a}},$

whose diagram is

${\labellist\small\hair
1pt\pinlabel{b}[]at6064\endlabellist\centering\vbox{\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={.9}]{./EPS/DLLat}}$}}}}}}}}}}\<EMAIL_ADDRESS>
The maps provided here (when $b$ ranges over a basis for
$R^{\operatorname{RR}(q)}$ or $R^{\operatorname{LR}(q)}$) are the same as the
“double leaves basis” from [EKLP24a]. This is verified in the following
remark, intended for a reader familiar with [EKLP24a].

######  Remark 4.8.

Let us verify that

(44) $\mathbb{DLL}_{l}(q,b)=\mathbb{DLL}(q,([p,n,q],1),([p,n,q],1),b),$

where $p$ is the identity $(I,I)$-coset, by following the stages in [EKLP24a,
Chapter 7]. First, we construct $\mathbb{LL}(q,([p,n,q],1))$. The single step
light leaf for the first step $[p,n]$ is the identity map since $[p,n]$ is
reduced. Since the left redundancy for $p$ and $n$ are the same, In the second
step, we have the coset pair $[n,q]$ which is already a Grassmannian pair,
thus the single step light leaf is the elementary light leaf
$\mathbb{ELL}([n,q])$.

Then we choose $X_{q}=Y_{q}$ and take the dual map for the upside-down light
leaf from $\operatorname{BS}(X_{q})$ to $B_{\mathtt{a}}$. Altogether we obtain
the double leaf of the form (43). No rex moves were used at any stage of the
process.

Moreover, for each $q\leq{\mathtt{a}}$, there is one subordinate path with
terminus $q$, namely $[p,n,q]$. It follow that the morphisms
$\mathbb{DLL}(q,([p,n,q],1),([p,n,q],1),b)$ form a double leaves basis in the
sense of [EKLP24a].

######  Remark 4.9.

We know that $\operatorname{End}(B_{{\mathtt{a}}})$ is an
$(R^{I},R^{J})$-bimodule, so it is natural to ask how the actions of $R^{I}$
and $R^{J}$ interact with the bases presented in Propositions 4.18 and 4.20.
For $g\in R^{J}$ we claim that

(45) $\mathbb{DLL}_{r}(q,b)\cdot g=\mathbb{DLL}_{r}(q,b\cdot g).$

Consider the diagram in Definition 4.6, and right-multiply by $g\in R^{J}$.
Since $g$ is also in $R^{\operatorname{RR}(q)}$, it can be slid from the right
side to the region where $b$ lives.

However, the left action of $f\in R^{I}$ is more mysterious. For fixed $q$, it
does not preserve the span of $\\{\mathbb{DLL}_{r}(q,b)\mid b\in
R^{\operatorname{RR}(q)}\\}$. Indeed, the comparison between the left action
and the right action is controlled by polynomial forcing for
$\operatorname{BS}(X_{q})$, which involves lower terms.

Similarly, for $f\in R^{I}$, the left action on left-sprinkled double leaves
is straightforward,

$f\cdot\mathbb{DLL}_{l}(q,b)=\mathbb{DLL}_{l}(q,f\cdot b),$

whereas the right action of $R^{J}$ is mysterious.

####  4.2.3. Evaluation of double leaves

The following crucial computation links double leaves with the description
$\operatorname{End}(B_{\mathtt{a}})\cong R^{I}\otimes_{R^{M}}R^{J}$. Let
$\Delta^{J}_{M,(1)}$ and $\Delta^{J}_{M,(2)}$ be dual bases of $R^{J}$ over
$R^{M}$, where we use Sweedler notation.

######  Lemma 4.10.

The double leaf $\mathbb{DLL}_{r}(q,b)$ coincides with multiplication by the
element

(46)
$\partial_{\overline{q}w_{J}^{-1}}\left(b\cdot\Delta^{J}_{M,(1)}\right)\otimes\Delta^{J}_{M,(2)}.$

This can also be written as

(47)
$\partial_{I}^{\operatorname{LR}(q)}\partial_{q^{\operatorname{core}}}(b\cdot\iota_{J}^{\operatorname{RR}(q)}\Delta^{J}_{M,(1)})\otimes\Delta^{J}_{M,(2)}.$

###### Proof.

The proof is immediate from [EKLP24a, Algorithm 8.12]. It is proven exactly as
[EKLP24a, Lemma 6.10]. ∎

A similar computation involving left-sprinkled double leaves gives the
following.

######  Lemma 4.11.

The double leaf $\mathbb{DLL}_{l}(q,b)$ coincides with multiplication by the
element

(48)
$\partial_{I}^{\operatorname{LR}(q)}(b\cdot\partial_{q^{\operatorname{core}}}(\iota_{J}^{\operatorname{RR}(q)}\Delta^{J}_{M,(1)}))\otimes\Delta^{J}_{M,(2)}.$

######  Example 4.12.

If $q\leq{\mathtt{a}}$ is the minimal $(I,J)$-coset, namely $q=W_{I}eW_{J}$,
then we have two cases.

  1. (1)

When
${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}t}:=w_{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}s}w_{M}\neq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}s}$,
we have

$q\leftrightharpoons[I,K,J]=[I-{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}t}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}s}]$

for $K=I\cap J$. In this case, both left and right redundancies are $K$, and
for $b\in R^{K}$ the double leaf $\mathbb{DLL}_{r}(q,b)=\mathbb{DLL}_{l}(q,b)$
is the left diagram in the equality

(49) ${\labellist\small\hair
2pt\pinlabel{b}[]at3132\endlabellist\centering\vbox{\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={.9}]{./EPS/R2hard1_colorswap}}$}}}}}}}}}}\@add@centering}=\partial^{K}_{I}(b\cdot\Delta^{J}_{M,(1)})\vbox{\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={.9}]{./EPS/ups}}$}}}}}}}}}}\vbox{\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={.9}]{./EPS/downt}}$}}}}}}}}}}\Delta^{J}_{M,(2)}.$

Thus we have

$\mathbb{DLL}_{r}(q,b)=\mathbb{DLL}_{l}(q,b)=\partial^{K}_{I}(b\cdot\Delta^{J}_{M,(1)})\otimes\Delta^{J}_{M,(2)}$

(see [EKLP24a, Equations (89) and (91)]).

  2. (2)

When
$w_{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}s}w_{M}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}s}$,
we have $q\leftrightharpoons[I]$. In this case, we have
$\operatorname{LR}(q)=I=\operatorname{RR}(q)$, and for $b\in R^{I}$ the double
leaf has a capcup diagram

(50) ${\labellist\small\hair
2pt\pinlabel{b}[]at1532\endlabellist\centering\vbox{\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={.9}]{./EPS/cupcap}}$}}}}}}}}}}\<EMAIL_ADDRESS>
Thus we have
$\mathbb{DLL}_{r}(q,b)=\mathbb{DLL}_{l}(q,b)=b\cdot\Delta_{M}^{I}=\Delta_{M}^{I}\cdot
b$.

######  Definition 4.13.

For an atomic coset ${\mathtt{a}}\leftrightharpoons[I,M,J]$, let
$\operatorname{DL}_{<{\mathtt{a}}}$ denote the $\Bbbk$-linear subspace of
$\operatorname{End}(B_{{\mathtt{a}}})$ spanned by right-sprinkled double
leaves factoring through $q<{\mathtt{a}}$. More explicitly we have

$\operatorname{DL}_{<{\mathtt{a}}}:=\operatorname{Span}_{q<{\mathtt{a}}}\\{\mathbb{DLL}_{r}(q,b)\mid
b\in
R^{\operatorname{RR}(q)}\\}=\operatorname{Span}_{q<{\mathtt{a}}}\\{\partial_{\overline{q}w_{J}^{-1}}(R^{\operatorname{RR}(q)}\cdot\Delta_{M,(1)}^{J})\otimes\Delta_{M,(2)}^{J}\\}.$

######  Lemma 4.14.

We have
$\operatorname{DL}_{<{\mathtt{a}}}\subset\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})$.

###### Proof.

By construction, every double leaf associated to $q<{\mathtt{a}}$ factors
through a reduced expression for $q$, and thus lives in
$\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})$. ∎

We also note a consequence of Remark 4.9.

######  Corollary 4.15.

For each $g\in R^{J}$ and $b\in R^{\operatorname{RR}(q)}$ we have

(51)
$\partial_{\overline{q}w_{J}^{-1}}(b\cdot\Delta_{M,(1)}^{J})\otimes\Delta_{M,(2)}^{J}\cdot
g=\partial_{\overline{q}w_{J}^{-1}}(b\cdot
g\cdot\Delta_{M,(1)}^{J})\otimes\Delta_{M,(2)}^{J}.$

In particular, $\operatorname{DL}_{<{\mathtt{a}}}$ is a right $R^{J}$-module.

###### Proof.

This follows from (45) and Lemma 4.10. ∎

###  4.3. Double leaves and lower terms: part I

The main result of [EKLP24a] is that double leaves form a basis for morphisms
between Bott-Samelson bimodules. However, [EKLP24a] relies on Williamson’s
theory of standard filtrations, which relies on several assumptions originally
made by Soergel.

######  Definition 4.16.

We call a realization a _Soergel-Williamson realization_ or an _SW-
realization_ for short, if it is a Frobenius realization (see Definition 3.5),
and it also satisfies the following assumptions.

  * •

The realization is reflection faithful, i.e. it is faithful, and the
reflections in $W$ are exactly those elements that fix a codimension-one
subspace.

  * •

The ring $\Bbbk$ is an infinite field of characteristic not equal to $2$.

######  Remark 4.17.

Abe [Abe24] has recently developed a theory of singular Soergel bimodules that
works for Frobenius realizations, without the extra restrictions of an SW-
realization. One expects that the results of [EKLP24a] can be
straightforwardly generalized to Abe’s setting. For simplicity and because of
the current state of the literature, we will work with Williamson’s category
of bimodules.

######  Proposition 4.18.

Assume an SW-realization. Let $\mathbb{B}_{q}$ be a $\Bbbk$-basis of
$R^{\operatorname{RR}(q)}$, for each $q\leq{\mathtt{a}}$. Then

(52) $\\{\mathbb{DLL}_{r}(q,b)\\}_{b\in\mathbb{B}_{q}}$

gives a basis of $\operatorname{End}_{\leq
q}(B_{{\mathtt{a}}})/\operatorname{End}_{<q}(B_{{\mathtt{a}}})$ over $\Bbbk$.
In particular,

(53) $\\{\mathbb{DLL}_{r}(q,b)\ |\ q\leq{\mathtt{a}},b\in\mathbb{B}_{q}\\}$

is a $\Bbbk$-basis of $\operatorname{End}(B_{{\mathtt{a}}})$, and the subset
indexed by $q<{\mathtt{a}}$ is a basis for
$\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})$.

###### Proof.

This is a special case of [EKLP24a, Lemma 8.35 and Theorem 7.49], as confirmed
in Remark 4.8. ∎

######  Corollary 4.19.

Assume an SW-realization. We have

(54)
$\operatorname{DL}_{<{\mathtt{a}}}=\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})=\bigoplus_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}(R^{\operatorname{RR}(q)}\cdot\Delta_{M,(1)}^{J})\otimes\Delta_{M,(2)}^{J}.$

In particular, $\operatorname{DL}_{<{\mathtt{a}}}$ is an
$(R^{I},R^{J})$-bimodule.

###### Proof.

As $\operatorname{DL}_{<{\mathtt{a}}}$ is the span of double leaves factoring
through $q<{\mathtt{a}}$, the first equality follows from Proposition 4.18.
The second equality follows from the linear independence of double leaves.
Since $\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})$ is an
$(R^{I},R^{J})$-bimodule, so is $\operatorname{DL}_{<{\mathtt{a}}}$. ∎

Similarly, double leaves provides a left-sprinkled basis.

######  Proposition 4.20.

Assume an SW-realization. Let $\mathbb{B}_{q}$ be a $\Bbbk$-basis of
$R^{\operatorname{LR}(q)}$, for each $q\leq{\mathtt{a}}$. Then

(55) $\\{\mathbb{DLL}_{l}(q,b)\\}_{b\in\mathbb{B}_{q}}$

gives a basis of $\operatorname{End}_{\leq
q}(B_{{\mathtt{a}}})/\operatorname{End}_{<q}(B_{{\mathtt{a}}})$ over $\Bbbk$.
In particular,

(56) $\\{\mathbb{DLL}_{l}(q,b)\ |\ q\leq{\mathtt{a}},b\in\mathbb{B}_{q}\\}$

is a $\Bbbk$-basis of $\operatorname{End}(B_{{\mathtt{a}}})$, and the subset
indexed by $q<{\mathtt{a}}$ is a basis for
$\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})$.

###  4.4. Double leaves and lower terms: part II

######  Definition 4.21.

An _almost-SW realization_ is a Frobenius realization, together with the
following assumptions.

  * •

The ring $\Bbbk$ is a domain with fraction field ${\mathbb{F}}$.

  * •

After base change to ${\mathbb{F}}$, the result is an SW-realization.

  * •

Finitely-generated projective modules over $\Bbbk$ are free.

######  Example 4.22.

The defining representation of $\mathbf{S}_{n}$ over $\mathbb{Z}$ is an
almost-SW realization [Dem73, Lemma 5].

######  Example 4.23.

The root realization of a Weyl group is almost-SW when defined over
$\Bbbk=\mathbb{Z}[1/N]$ for small $N$ ($N=30$ will suffice for all Weyl groups
by [Dem73, Proposition 8]).

######  Lemma 4.24.

Let $I\subset S$ be finitary. Then $R^{I}$ is a free $\Bbbk$-module.

###### Proof.

As a polynomial ring over a free $\Bbbk$-module $V$, $R$ is a free
$\Bbbk$-module. By the assumption of generalized Demazure surjectivity, $R$ is
free as an $R^{I}$-module when $I\subset S$ is finitary. Thus $R^{I}$ is a
direct summand of $R$, and is therefore projective as a $\Bbbk$-module. Since
both $R$ and $R^{I}$ are finitely-generated as $\Bbbk$-modules in each graded
degree, we deduce that $R^{I}$ is also a free module over $\Bbbk$. ∎

Our goal in this section is to generalize the results of the previous section
to almost-SW realizations. In all the lemmas in this section, we assume an
almost-SW realization. First we note the compatibility of base change with
most of the constructions above.

We let $R$ be the polynomial ring of the realization over $\Bbbk$, and let
$R_{\mathbb{F}}:=R\otimes_{\Bbbk}{\mathbb{F}}$ be the polynomial ring of the
realization after base change. Let $R^{I}_{\mathbb{F}}\subset
R_{{\mathbb{F}}}$ be the invariant subring.

######  Lemma 4.25.

We have $R^{I}_{{\mathbb{F}}}\cong R^{I}\otimes_{\Bbbk}{\mathbb{F}}$.

###### Proof.

There is a natural map $R^{I}\otimes_{\Bbbk}{\mathbb{F}}\to R_{{\mathbb{F}}}$,
and since scalars are $W$-invariant, the image lies within
$R_{{\mathbb{F}}}^{I}$. The map is injective since ${\mathbb{F}}$ is flat over
$\Bbbk$. We now argue that the map $R^{I}\otimes_{\Bbbk}{\mathbb{F}}\to
R_{{\mathbb{F}}}^{I}$ is surjective. If $f\in R_{{\mathbb{F}}}^{I}$, then
there is some $c\in\Bbbk$ such that $cf\in R$ (e.g. letting $c$ be the product
of the denominators of each monomial in $f$). Clearly $cf\in R^{I}$, whence
$f$ is the image of $cf\otimes\frac{1}{c}$.

∎

Let
$B_{{\mathtt{a}},{\mathbb{F}}}=R^{I}_{\mathbb{F}}\otimes_{R^{M}_{\mathbb{F}}}R^{J}_{\mathbb{F}}$.
If $I_{\bullet}=[[I=I_{0}\subset K_{1}\supset I_{1}\subset\ldots K_{m}\supset
I_{m}=J]]$ is a multistep $(I,J)$-expression, let

$\operatorname{BS}_{\mathbb{F}}(I_{\bullet}):=R^{I_{0}}_{\mathbb{F}}\otimes_{R_{\mathbb{F}}^{K_{1}}}R_{\mathbb{F}}^{I_{1}}\otimes_{R_{\mathbb{F}}^{K_{2}}}\cdots\otimes_{R_{\mathbb{F}}^{K_{m}}}R_{\mathbb{F}}^{I_{m}}.$

######  Lemma 4.26.

The natural inclusion map
$\operatorname{BS}(I_{\bullet})\to\operatorname{BS}(I_{\bullet})\otimes_{\Bbbk}{\mathbb{F}}$
is injective. We have
$\operatorname{BS}_{\mathbb{F}}(I_{\bullet})\cong\operatorname{BS}(I_{\bullet})\otimes_{\Bbbk}{\mathbb{F}}$.
As a consequence we have an injective map

(57)
$\operatorname{Hom}(\operatorname{BS}(I_{\bullet}),\operatorname{BS}(I^{\prime}_{\bullet}))\to\operatorname{Hom}(\operatorname{BS}_{{\mathbb{F}}}(I_{\bullet}),\operatorname{BS}_{{\mathbb{F}}}(I^{\prime}_{\bullet})).$

###### Proof.

By our assumptions from §3.1, $R^{I}$ is free over $R^{K}$ whenever $I\subset
K$. We fix a basis $\\{b_{i}^{I,K}\\}$ for this extension. Hence the Bott-
Samelson bimodule $\operatorname{BS}(I_{\bullet})$ is free as a right
$R^{I_{m}}$-module with basis

(58) $\left\\{b_{i_{1}}^{I_{0},K_{1}}\otimes\ldots\otimes
b_{i_{m}}^{I_{m-1},K_{m}}\otimes 1\right\\}.$

It is also free as a right $\Bbbk$-module by 4.24. So base change is injective
on bimodules. Notice that $\\{b_{i}^{I,K}\otimes 1\\}$ is a basis of
$R^{I}_{\mathbb{F}}$ over $R^{K}_{\mathbb{F}}$. Hence (58) gives also a basis
of $\operatorname{BS}_{{\mathbb{F}}}(I_{\bullet})$ over
$R^{I_{m}}_{\mathbb{F}}$. Since it sends a basis to a basis, we deduce that
the natural map
$\operatorname{BS}(I_{\bullet})\otimes_{\Bbbk}{\mathbb{F}}\to\operatorname{BS}_{\mathbb{F}}(I_{\bullet})$
is an isomorphism.

The localization functor gives the map in (57). For a morphism $\phi$ between
bimodules over $\Bbbk$, let $\phi\otimes 1$ denote its image, a morphism
between bimodules over ${\mathbb{F}}$. The restriction of $\phi\otimes 1$ to
the subset
$\operatorname{BS}(I_{\bullet})\subset\operatorname{BS}_{{\mathbb{F}}}(I_{\bullet})$
is the original morphism $\phi$. Hence if $\phi\otimes 1$ is the zero
morphism, so is $\phi$. ∎

######  Lemma 4.27.

For $q\leq{\mathtt{a}}$ and $b\in R^{\operatorname{RR}(q)}$, let us
temporarily write $\mathbb{DLL}_{r,\Bbbk}(q,b)$ for the double leaf as a
morphism between Bott-Samelson bimodules over $\Bbbk$, and
$\mathbb{DLL}_{r,{\mathbb{F}}}(q,b)$ for the double leaf as a morphism between
Bott-Samelson bimodules over ${\mathbb{F}}$, where for the latter we identify
$b$ with its image in $R_{{\mathbb{F}}}^{\operatorname{RR}(q)}$. Under the map
(57), we have

$\mathbb{DLL}_{r,\Bbbk}(q,b)\mapsto\mathbb{DLL}_{r,{\mathbb{F}}}(q,b).$

###### Proof.

The calculus from [ESW17] for interpreting diagrams is invariant under base
change. Alternatively, $B_{{\mathtt{a}}}$ is generated as a bimodule by
$1\otimes 1$, and elementary light leaves $\mathbb{ELL}([n,q])$ are determined
uniquely in their Hom space by the fact that they send $1\otimes 1$ to
$1\otimes\ldots\otimes 1$. This property is preserved by base change. ∎

######  Lemma 4.28.

Let $\mathbb{B}_{q}$ be a basis of $R^{\operatorname{RR}(q)}$ over $\Bbbk$.
Then the set $\\{\mathbb{DLL}_{r,\Bbbk}(q,b)\ |\
q\leq{\mathtt{a}},b\in\mathbb{B}_{q}\\}$ is linearly independent. Let
$\operatorname{DL}_{<{\mathtt{a}},\Bbbk}$ and
$\operatorname{DL}_{<{\mathtt{a}},{\mathbb{F}}}$ be defined as before for
their respective realizations. The map (57) induces an isomorphism

(59)
$\operatorname{DL}_{<{\mathtt{a}},\Bbbk}\otimes_{\Bbbk}\,{\mathbb{F}}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{DL}_{<{\mathtt{a}},{\mathbb{F}}}.$

###### Proof.

By definition $\operatorname{DL}_{<{\mathtt{a}}}$ is the span (over $\Bbbk$ or
${\mathbb{F}}$) of the double leaf morphisms. The map (57) restricts to a map
$\operatorname{DL}_{<{\mathtt{a}},\Bbbk}\to\operatorname{Hom}(\operatorname{BS}_{{\mathbb{F}}}(I_{\bullet}),\operatorname{BS}_{{\mathbb{F}}}(I^{\prime}_{\bullet}))$.
By the previous lemma, the image of this map is contained in
$\operatorname{DL}_{<{\mathtt{a}},{\mathbb{F}}}$. Thus one has an induced map
$\operatorname{DL}_{<{\mathtt{a}},\Bbbk}\otimes_{\Bbbk}{\mathbb{F}}\to\operatorname{DL}_{<{\mathtt{a}},{\mathbb{F}}}$.

Note that $\mathbb{B}_{q}$ is sent by base change to a basis of
$R^{\operatorname{RR}(q)}_{{\mathbb{F}}}$ over ${\mathbb{F}}$. The elements
$\\{\mathbb{DLL}_{r,\Bbbk}(q,b)\otimes 1\\}$ (ranging over the appropriate
index set) form an ${\mathbb{F}}$-spanning set for the left-hand side of (59),
and are sent to $\\{\mathbb{DLL}_{r,{\mathbb{F}}}(q,b)\\}$, which form a
${\mathbb{F}}$-basis for $\operatorname{DL}_{<{\mathtt{a}},{\mathbb{F}}}$ by
Proposition 4.18. Thus the map (59) is an isomorphism, and the elements
$\\{\mathbb{DLL}_{r,\Bbbk}(q,b)\otimes 1\\}$ are linearly independent over
${\mathbb{F}}$. Consequently, $\\{\mathbb{DLL}_{r,\Bbbk}(q,b)\\}$ are linearly
independent over $\Bbbk$. ∎

######  Lemma 4.29.

We have

$\operatorname{DL}_{<{\mathtt{a}},\Bbbk}=\bigoplus_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}(R^{\operatorname{RR}(q)}\cdot\Delta_{M,(1)}^{J})\otimes\Delta_{M,(2)}^{J}.$

###### Proof.

The span in Definition 4.13 in indeed a direct sum of subspaces, by the linear
independence shown in the previous lemma. ∎

Henceforth we return to the $\Bbbk$-linear setting by default (i.e. in the
absence of a subscript). Now the question remains: is the inclusion
$\operatorname{DL}_{<{\mathtt{a}}}\subset\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})$
an equality over $\Bbbk$, knowing that the result holds over ${\mathbb{F}}$?
In Corollary 4.33 below, we prove that the answer is yes.

######  Lemma 4.30.

For any $\phi\in\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})$ there is
some $n\in\Bbbk$ such that $n\phi\in\operatorname{DL}_{<{\mathtt{a}}}$.

###### Proof.

In view of 4.26 and (57), we can regard both
$\operatorname{DL}_{<{\mathtt{a}}}$ and $\operatorname{End}(B_{{\mathtt{a}}})$
as $\Bbbk$-submodules of $\operatorname{End}(B_{{\mathtt{a}},{\mathbb{F}}})$,
which we identify with $B_{{\mathtt{a}},{\mathbb{F}}}$. We have
$\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})\subset\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}},{\mathbb{F}}})$
by definition. Meanwhile, 4.19 holds over ${\mathbb{F}}$ and thus

(60)
$\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}},{\mathbb{F}}})\cong\bigoplus_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}(R_{\mathbb{F}}^{\operatorname{RR}(q)}\cdot\Delta_{M,(1)}^{J})\otimes\Delta_{M,(2)}^{J}.$

So any $\phi\in\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}})$ is a
${\mathbb{F}}$-multiple of an element of $\operatorname{DL}_{<{\mathtt{a}}}$.
Multiplying by the denominator, there exists $n\in\Bbbk$ such that
$n\phi\in\operatorname{DL}_{<{\mathtt{a}}}$. ∎

We continue with a divisibility lemma on Demazure operators, which ensures
that Demazure operators do not “produce” additional divisibility by elements
of $\Bbbk$.

######  Lemma 4.31.

Let $b\in R$ and let $n\in\Bbbk$. If $n\mid\partial_{w_{M}w_{J}^{-1}}(bg)$ for
all $g\in R^{J}$, then $n\mid b$.

###### Proof.

Assume that $n\nmid b$. We need to find $g\in R^{J}$ such that
$n\nmid\partial_{w_{M}w_{J}^{-1}}(bg)$.

Recall from Lemma 2.6 that for $w\in W$ and $f,g\in R$ we have

$\partial_{w}(fg)=\sum_{x\leq w}T^{\prime}_{x}(f)\partial_{x}(g)$

for certain operators $T^{\prime}_{x}$ defined over $\Bbbk$, where
$T^{\prime}_{w}(f)=w(f)$. Let $W^{J}$ be the subset of elements in $W$ which
are minimal (for the Bruhat order) in their right $W_{J}$-coset. We have
$\partial_{x}(g)=0$ when $x\not\in W^{J}$ and $g\in R^{J}$, so we have

$\partial_{w}(fg)=\sum_{x\leq w,x\in W^{J}}T^{\prime}_{x}(f)\partial_{x}(g).$

We apply this formula when $w=w_{M}w_{J}^{-1}$. Let $W^{J}_{M}=W^{J}\cap
W_{M}$. Since $n\nmid b$, then also $n\nmid
w_{M}w_{J}^{-1}(b)=T_{w_{M}w_{J}^{-1}}^{\prime}(b)$. So there exists some
$y\in W_{M}^{J}$ (not necessarily unique) which is minimal with respect to the
property that $n\nmid T^{\prime}_{y}(b)$. Now we have

(61) $\partial_{w_{M}w_{J}^{-1}}(bg)=\sum_{x\in
W_{M}^{J}}T^{\prime}_{x}(b)\partial_{x}(g)\equiv\sum_{x\not<y,x\in
W_{M}^{J}}T^{\prime}_{x}(b)\partial_{x}(g)\pmod{n}.$

Recall that $\ell(xw_{M})=\ell(w_{M})-\ell(x)$ for all $x\in W_{M}$. Let
$z=y^{-1}w_{M}$ so that $y.z=w_{M}$. We have

$\ell(w_{J}y^{-1}w_{M})=\ell(w_{M})-\ell(w_{J}y^{-1})=\ell(w_{M})-\ell(y^{-1})-\ell(w_{J})=\ell(z)-\ell(w_{J}),$

thus the left descent set of $z$ contains $J$. By [EKLP23a, Lemma 3.2] we have
$\operatorname{Im}(\partial_{z})\subset R^{J}$. By our assumption of
generalized Demazure surjectivity, we can choose $P_{M}\in R$ such that
$\partial_{w_{M}}(P_{M})=1$. Set $g=\partial_{z}(P_{M})\in R^{J}$ and note
that $\partial_{y}(g)=1$.

Let $x\in W_{M}$. If $x.z$ is not reduced, then
$\partial_{x}(g)=\partial_{x}\partial_{z}(P_{M})=0$ by [EKLP23a, (27))]. If
$x.z$ is reduced, then

$\ell(w_{M})-\ell(yx^{-1})=\ell(x.y^{-1}w_{M})=\ell(x)+\ell(w_{M})-\ell(y).$

It follows that $\ell(yx^{-1})+\ell(x)=\ell(y)$, so $yx^{-1}.x=y$ and, in
particular, $x\leq y$. This means that $\partial_{x}(g)\neq 0$ only if $x\leq
y$.

Finally, we plug $g=\partial_{z}(P_{M})$ into (61) and we observe that
$\partial_{w_{M}w_{J}^{-1}}(bg)\equiv T^{\prime}_{y}(b)\not\equiv 0\pmod{n}$.
∎

Now we can prove that $\operatorname{DL}_{<{\mathtt{a}}}$, as a submodule of
$\operatorname{End}(B_{\mathtt{a}})$, is closed under division by elements in
$\Bbbk$ (when that makes sense).

######  Proposition 4.32.

Let $\phi\in\operatorname{DL}_{<{\mathtt{a}}}$ and assume there exists
$n\in\Bbbk$ such that $\frac{1}{n}\phi\in\operatorname{End}(B_{\mathtt{a}})$.
Then $\frac{1}{n}\phi\in\operatorname{DL}_{<{\mathtt{a}}}$.

###### Proof.

If $n$ is a unit in $\Bbbk$ the result is trivial, so assume otherwise.

Let $\phi\in\operatorname{DL}_{<{\mathtt{a}}}$ and assume that
$\frac{1}{n}\phi\in\operatorname{End}(B_{\mathtt{a}})\cong
R^{I}\otimes_{R^{M}}R^{J}$. We can write

$\phi=\sum_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}(b_{q}\cdot\Delta_{M,(1)}^{J})\otimes\Delta_{M,(2)}^{J}$

for some unique $b_{q}\in R^{\operatorname{RR}(q)}$. For clarity we choose to
unravel Sweedler’s notation. Choose dual bases $\\{c_{i}\\}$ and $\\{d_{i}\\}$
for $R^{J}$ over $R^{M}$ relative to the Frobenius trace map
$\partial^{J}_{M}$. We have

$\phi=\sum_{i}\sum_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}(b_{q}\cdot
c_{i})\otimes d_{i}.$

Since $d_{i}$ is a basis of $R^{J}$ over $R^{M}$, any element of
$R^{I}\otimes_{R^{M}}R^{J}$ is uniquely expressible as $\sum_{i}f_{i}\otimes
d_{i}$ for $f_{i}\in R^{I}$. In particular, if $n$ divides
$\sum_{i}f_{i}\otimes d_{i}$ we have

$\frac{1}{n}\sum_{i}f_{i}\otimes d_{i}=\sum_{i}f^{\prime}_{i}\otimes d_{i}$

for some unique $f^{\prime}_{i}$. Then $\sum f_{i}\otimes
d_{i}=\sum_{i}nf^{\prime}_{i}\otimes d_{i}$, and by the unicity mentioned
before, this implies that $nf_{i}^{\prime}=f_{i}$ for all $i.$

Consequently, $\phi$ is divisible by $n$ if and only if for all $i$ we have

$n\mid\sum_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}(b_{q}\cdot
c_{i}).$

We want to show that all the $b_{q}$ are actually divisible by $n$, so that
$\frac{1}{n}\phi\in\operatorname{DL}_{<{\mathtt{a}}}$.

Assume for contradiction that there exists a minimal $r$ such that $n\nmid
b_{r}$. Let $z$ be such that $z.\overline{r}w_{J}^{-1}=w_{M}w_{J}^{-1}$. Then
by similar arguments to the previous lemma we have

$0\equiv\partial_{z}\left(\sum_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}(b_{q}\cdot
c_{i})\right)\equiv\partial_{w_{M}w_{J}^{-1}}(b_{r}\cdot c_{i})\pmod{n}$

for each $i$. The subset of $R^{J}$ consisting of those $c$ for which
$\partial_{w_{M}w_{J}^{-1}}(b_{r}\cdot c)\equiv 0\pmod{n}$ is evidently an
$R^{M}$-submodule. Since this submodule contains a basis $\\{c_{i}\\}$ for
$R^{J}$ over $R^{M}$, it must contain all of $R^{J}$. Thus

$n\mid\partial_{w_{M}w_{J}^{-1}}(b_{r}\cdot g)$

for all $g\in R^{J}$. By 4.31 we deduce $n\mid b_{r}$, leading to a
contradiction. ∎

######  Corollary 4.33.

For an almost SW-realization we have
$\operatorname{DL}_{<{\mathtt{a}}}=\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}})$.
In particular, $\operatorname{DL}_{<{\mathtt{a}}}$ is an
$(R^{I},R^{J})$-bimodule.

###### Proof.

The containment
$\operatorname{DL}_{<{\mathtt{a}}}\subset\operatorname{End}_{{\mathtt{a}}}(B_{{\mathtt{a}}})$
was already shown in 4.14. Now pick an arbitrary element
$\psi\in\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}})$. By 4.30, there is
some $n\in\Bbbk$ such that $n\psi\in\operatorname{DL}_{<{\mathtt{a}}}$. Then
$\psi=\frac{1}{n}(n\psi)\in\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}})$,
so by 4.32 we deduce that $\psi\in\operatorname{DL}_{<{\mathtt{a}}}$. ∎

##  5\. Polynomial forcing and atomic Leibniz

###  5.1. Polynomial forcing for atomic cosets

Now we explain the concept of polynomial forcing. We consider first the case
of an atomic coset ${\mathtt{a}}\leftrightharpoons[[I\subset M\supset J]]$.

Recall that $\underline{{\mathtt{a}}}J=I\underline{{\mathtt{a}}}$ and
$\operatorname{LR}({\mathtt{a}})=I$ since ${\mathtt{a}}$ is core. Thus there
is an isomorphism

(62) $R^{J}\to R^{I},\qquad f\mapsto\underline{{\mathtt{a}}}(f).$

######  Definition 5.1.

Let ${\mathtt{a}}$ be an atomic coset with reduced expression $[[I\subset
M\supset J]]$ and let $f\in R^{J}$. We say that _polynomial forcing_ holds for
$f$ and ${\mathtt{a}}$ if we have

(63) $1\otimes f-\underline{{\mathtt{a}}}(f)\otimes
1\in\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}}).$

We say that _polynomial forcing_ holds for ${\mathtt{a}}$ if (63) holds for
all $f\in R^{J}$.

######  Lemma 5.2.

Suppose that (63) holds for $f_{1}$ and for $f_{2}$, with both $f_{1},f_{2}\in
R^{J}$. Then it holds for $f_{1}+f_{2}$ and $f_{1}f_{2}$.

###### Proof.

Additivity is trivial, because $\operatorname{Hom}_{<{\mathtt{a}}}$ is closed
under addition. Now consider the following:

(64) $1\otimes f_{1}\cdot f_{2}-\underline{{\mathtt{a}}}(f_{1}\cdot
f_{2})\otimes 1=(1\otimes f_{1}-\underline{{\mathtt{a}}}(f_{1})\otimes 1)\cdot
f_{2}+\underline{{\mathtt{a}}}(f_{1})\cdot(1\otimes
f_{2}-\underline{{\mathtt{a}}}(f_{2})\otimes 1).$

Since $\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}})$ is closed under
right and left multiplication, both terms on the right-hand side above are in
$\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}})$, and the result is
proven. ∎

Before continuing, let us contrast polynomial forcing with an a priori
different notion.

######  Definition 5.3.

Consider $\operatorname{DL}_{<{\mathtt{a}}}$ from Definition 4.13. We say that
_$\operatorname{DL}$ -forcing_ holds for ${\mathtt{a}}$ and $f$ if $1\otimes
f-\underline{{\mathtt{a}}}(f)\otimes
1\in\operatorname{DL}_{<{\mathtt{a}}}\subset\operatorname{End}(B_{{\mathtt{a}}})$.
We say that _$\operatorname{DL}$ -forcing_ holds for ${\mathtt{a}}$ if it
holds for ${\mathtt{a}}$ and $f$, for all $f\in R^{J}.$

For an almost-SW realization, $\operatorname{DL}$-forcing is equivalent to
polynomial forcing, since
$\operatorname{DL}_{<{\mathtt{a}}}=\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})$
by Corollary 4.33. In general, it is not obvious that
$\operatorname{DL}$-forcing is multiplicative. The proof of multiplicativity
in Lemma 5.2 relied on the fact that
$\operatorname{End}_{<{\mathtt{a}}}(B_{{\mathtt{a}}})$ is an
$(R^{I},R^{J})$-bimodule, whereas $\operatorname{DL}_{<{\mathtt{a}}}$ is only
a priori a right $R^{J}$-module.

###  5.2. Equivalence

Now we prove the equivalence between atomic Leibniz rules and polynomial
forcing. To formulate an intermediate condition in the proof, which is also of
importance for the next section, we agree to say the following. Given an
atomic $(I,J)$-coset ${\mathtt{a}}$ and an element $f\in R^{J}$, _an atomic
Leibniz rule for ${\mathtt{a}}$ and $f$_ is said to hold if there exist
elements $T_{q}(f)$ such that equation (30) is satisfied for all $g\in R^{J}$.
Since this condition is stated for one polynomial $f$ at a time, there is no
requirement that $T_{q}$ is an $R^{M}$-linear operator.

######  Proposition 5.4.

Let ${\mathtt{a}}\leftrightharpoons[I,M,J]$ be an atomic $(I,J)$-coset, and
$f\in R^{J}$. We have a rightward atomic Leibniz rule for ${\mathtt{a}}$ and
$f$ if and only if $\operatorname{DL}$-forcing holds for ${\mathtt{a}}$ and
$f$. Moreover, for an almost-SW realization, if $\operatorname{DL}$-forcing
holds for ${\mathtt{a}}$ and $f$, then the atomic Leibniz rule is unique,
i.e., the elements $T_{q}(f)\in R^{\operatorname{RR}(q)}$ in (30) are uniquely
determined.

###### Proof.

Note that $R^{J}\subset R^{M}$ is a Frobenius extension, see [EMTW20, Section
24.3.2]. The trace map is

$\partial^{J}_{M}:=\partial_{w_{M}w_{J}^{-1}}=\partial_{\mathtt{a}}.$

Let $\Delta^{J}_{M}\in R^{J}\otimes_{R^{M}}R^{J}$ denote the coproduct element
(the image of $1\in R^{J}$ under the coproduct map), which we often denote
using Sweedler notation. Then [ESW17, (2.2) with $f=1$] implies that

(65) $1\otimes
1=\partial_{\mathtt{a}}(\Delta^{J}_{M,(1)})\otimes\Delta^{J}_{M,(2)}.$

Multiplying $\underline{{\mathtt{a}}}(f)$ on the left we get

$\underline{{\mathtt{a}}}(f)\otimes
1=\underline{{\mathtt{a}}}(f)\cdot\partial_{\mathtt{a}}(\Delta^{J}_{M,(1)})\otimes\Delta^{J}_{M,(2)}.$

Meanwhile, [ESW17, (2.2)] implies that

(66) $1\otimes
f=\partial_{\mathtt{a}}(f\cdot\Delta^{J}_{M,(1)})\otimes\Delta^{J}_{M,(2)}.$

Thus we have

(67) $1\otimes f-\underline{{\mathtt{a}}}(f)\otimes
1=\left[\partial_{\mathtt{a}}(f\cdot\Delta^{J}_{M,(1)})-\underline{{\mathtt{a}}}(f)\cdot\partial_{\mathtt{a}}(\Delta^{J}_{M,(1)})\right]\otimes\Delta^{J}_{M,(2)}.$

Letting $g=\Delta^{J}_{M,(1)}$, a rightward atomic Leibniz rule for
${\mathtt{a}}$ and $f$ gives

(68) $\partial_{{\mathtt{a}}}(f\cdot
g)-\underline{{\mathtt{a}}}(f)\cdot\partial_{{\mathtt{a}}}(g)=\sum_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}(T_{q}(f)\cdot
g).$

Thus we have

(69) $1\otimes f-\underline{{\mathtt{a}}}(f)\otimes
1=\sum_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}\left(T_{q}(f)\cdot\Delta^{J}_{M,(1)}\right)\otimes\Delta^{J}_{M,(2)}.$

which lies in $\operatorname{DL}_{<{\mathtt{a}}}$ by definition.

We prove now the other direction. We have

$1\otimes f-\underline{{\mathtt{a}}}(f)\otimes
1\in\operatorname{DL}_{<{\mathtt{a}}}.$

By (67), we obtain

$\left[\partial_{\mathtt{a}}(f\cdot\Delta^{J}_{M,(1)})-\underline{{\mathtt{a}}}(f)\cdot\partial_{\mathtt{a}}(\Delta^{J}_{M,(1)})\right]\otimes\Delta^{J}_{M,(2)}\in\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}}).$

By definition of $\operatorname{DL}_{<{\mathtt{a}}}$ we deduce that

$\displaystyle\partial_{\mathtt{a}}(f\cdot\Delta^{J}_{M,(1)})$
$\displaystyle\otimes\Delta^{J}_{M,(2)}=$ (70)
$\displaystyle\underline{{\mathtt{a}}}(f)\cdot\partial_{\mathtt{a}}(\Delta^{J}_{M,(1)})\otimes\Delta^{J}_{M,(2)}+\sum_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}(T_{q}(f)\cdot\Delta_{M,(1)}^{J})\otimes\Delta_{M,(2)}^{J}$

for some $T_{q}(f)\in R^{\operatorname{RR}(q)}$. For an almost-SW realization,
Lemma 4.29 implies that the $T_{q}(f)$ are unique.

Note that $\Delta^{J}_{M,(1)}$ and $\Delta^{J}_{M,(2)}$ run over dual bases of
$R^{J}$ over $R^{M}$. The elements
$\mathbb{B}=\\{1\otimes\Delta^{J}_{M,(2)}\\}$ form a basis for
$B_{\mathtt{a}}$, when viewed as a left $R^{I}$-module. Thus in order for the
equation (70) to hold, it must be an equality for each coefficient with
respect to the basis $\mathbb{B}$. Hence we conclude

(71)
$\partial_{\mathtt{a}}(f\cdot\Delta)=\underline{{\mathtt{a}}}(f)\cdot\partial_{\mathtt{a}}(\Delta)+\sum_{q<{\mathtt{a}}}\partial_{q}(T_{q}(f)\cdot\Delta)$

for all $\Delta$ ranging through a basis of $R^{J}$ over $R^{M}$.

Using the linearity of (71) over $R^{M}$, we deduce that it continues to hold
when $\Delta$ is replaced by any element $g\in R^{J}$. Thus the atomic Leibniz
rule for $f$ is proven. ∎

######  Theorem 5.5.

Assume an almost-SW realization (see Definition 4.21). Let
${\mathtt{a}}\leftrightharpoons[I,M,J]$ be an atomic $(I,J)$-coset. Then the
following are equivalent.

  1. (1)

A rightward atomic Leibniz rule holds for ${\mathtt{a}}$.

  2. (2)

A leftward atomic Leibniz rule holds for ${\mathtt{a}}$.

  3. (3)

For a set of generators $\\{c_{i}\\}$ of the $R^{M}$-algebra $R^{J}$, a
rightward atomic Leibniz rule holds for ${\mathtt{a}}$ and each $c_{i}$.

  4. (4)

For a set of generators $\\{c_{i}\\}$ of the $R^{M}$-algebra $R^{J}$, a
leftward atomic Leibniz rule holds for ${\mathtt{a}}$ and each $c_{i}$.

  5. (5)

Polynomial forcing holds for ${\mathtt{a}}$.

Moreover, in this case, there are unique operators $T_{q},T^{\prime}_{q}$ that
satisfy atomic Leibniz rules.

###### Proof.

First, we observe that polynomial forcing holds for $f\in R^{M}$. Clearly
$1\otimes f=f\otimes 1$. Moreover, ${\mathtt{a}}\subset W_{M}$ so
$\underline{{\mathtt{a}}}(f)=f$.

That (1) implies (3) is clear.

Suppose that (3) holds. By Proposition 5.4, $\operatorname{DL}$ forcing holds
for all $c_{i}$. By Corollary 4.33, $\operatorname{DL}$-forcing for $c_{i}$ is
equivalent to polynomial forcing for $c_{i}$. By Lemma 5.2, the subset of
$R^{J}$ consisting of those $f$ for which polynomial forcing holds is a
subring. As explained above, this subring includes $R^{M}$, so if it includes
$\\{c_{i}\\}$ then it must be all of $R^{J}$. In this way, (3) implies (5).

Suppose that (5) holds. Once again, Proposition 5.4 and Corollary 4.33 imply
that, for each $f\in R^{J}$, a rightward atomic Leibniz rule holds for $f$,
with the elements $T_{q}(f)\in R^{\operatorname{RR}(q)}$ being unique. To
prove that a rightward atomic Leibniz rule holds, it remains to prove that the
operators $T_{q}\colon R^{J}\to R^{\operatorname{RR}(q)}$ are $R^{M}$-linear.
We do this below, finishing the proof that (5) implies (1).

Let $g\in R^{M}$. Multiplying both sides of equation (70) on the left by $g$,
and pulling $g$ into various $R^{M}$-linear operators (namely
$\partial_{{\mathtt{a}}}$ and $\underline{{\mathtt{a}}}$ and
$\partial_{\overline{q}w_{J}^{-1}}$), we obtain

$\displaystyle\partial_{\mathtt{a}}(gf\cdot\Delta^{J}_{M,(1)})$
$\displaystyle\otimes\Delta^{J}_{M,(2)}=$
$\displaystyle\underline{{\mathtt{a}}}(gf)\cdot\partial_{\mathtt{a}}(\Delta^{J}_{M,(1)})\otimes\Delta^{J}_{M,(2)}+\sum_{q<{\mathtt{a}}}\partial_{\overline{q}w_{J}^{-1}}(gT_{q}(f)\cdot\Delta_{M,(1)}^{J})\otimes\Delta_{M,(2)}^{J}$

This is exactly (70) with $gf$ replacing $f$, except that $gT_{q}(f)$ appears
instead of $T_{q}(gf)$. By uniqueness, we deduce that $T_{q}(gf)=gT_{q}(f)$.

We have thus shown the equivalence of (1),(3) and (5). A similar argument will
imply the equivalence of (2) and (4) and (5), and the uniqueness of
$T^{\prime}_{q}$. This similar argument replaces $\mathbb{DLL}_{r}(q,b)$ with
$\mathbb{DLL}_{l}(q,b)$, using Proposition 4.20 and Lemma 4.11. The left
analogue of the remaining arguments (e.g. Proposition 5.4 and Corollary 4.33)
is left to the reader. ∎

######  Remark 5.6.

The intermediate conditions (3) and (4) do not play a significant role in the
proof. We have included them to make it easier to prove the atomic Leibniz
rule by establishing it on a set of generators.

###  5.3. Polynomial forcing for general cosets

Now let $p$ be an arbitrary $(I,J)$-coset, with a reduced expression
$I_{\bullet}$. We wish to avoid the technicalities of changing the reduced
expression $I_{\bullet}$ in this paper. Instead we focus on the special case
when $I_{\bullet}$ is an _atomic-factored reduced expression_ , i.e. it has
the following form:

(72) $I_{\bullet}=[[I\supset\operatorname{LR}(p)]]\circ
I^{\prime}_{\bullet}\circ[[\operatorname{RR}(p)\subset J]]$

where $I^{\prime}_{\bullet}$ is an _atomic reduced expression_ (see below) for
$p^{\operatorname{core}}$.

An atomic reduced expression for a core coset $p^{\operatorname{core}}$ is a
reduced expression of the form

(73) $I^{\prime}_{\bullet}=[[\operatorname{LR}(p)=N_{0}\subset M_{1}\supset
N_{1}\subset\cdots\subset M_{m}\supset N_{m}=\operatorname{RR}(p)]],$

where each $[[N_{i}\subset M_{i+1}\supset N_{i+1}]]$ is a reduced expression
for an atomic coset ${\mathtt{a}}_{i+1}$. In particular,
$p={\mathtt{a}}_{1}.{\mathtt{a}}_{2}.\cdots.{\mathtt{a}}_{m}$. Any core coset
has an atomic reduced expression, see [EKLP24b, Cor. 2.17] and thus any coset
has an atomic-factored reduced expression by [EK21, Proposition 4.28].

We have

(74)
$\operatorname{BS}(I_{\bullet})=R^{\operatorname{LR}(p)}\otimes_{R^{M_{1}}}R^{N_{1}}\otimes_{R^{M_{2}}}\cdots\otimes_{R^{M_{m}}}R^{\operatorname{RR}(p)}$

viewed as an $(R^{I},R^{J})$-bimodule. Meanwhile,
$\operatorname{BS}(I^{\prime}_{\bullet})$ is the same abelian group, but is
viewed as an $(R^{\operatorname{LR}(p)},R^{\operatorname{RR}(p)})$-bimodule.
There is an action of each $R^{N_{i}}$ on $\operatorname{BS}(I_{\bullet})$ by
multiplication in the $i$-th tensor factor of (74). Indeed, this induces an
injective map

(75)
$\operatorname{BS}(I_{\bullet})\to\operatorname{End}(\operatorname{BS}(I_{\bullet}))$

which is not surjective in general.

An arbitrary reduced expression for $p$ might never factor through the subset
$\operatorname{LR}(p)$ or $\operatorname{RR}(p)$. The first advantage of an
atomic-factored expression is that there is an obvious action of
$R^{\operatorname{LR}(p)}$ on $\operatorname{BS}(I_{\bullet})$ by left-
multiplication, and an obvious action of $R^{\operatorname{RR}(p)}$ by right-
multiplication. The goal is to prove that these two actions agree up to a
twist by $\underline{p}$, modulo lower terms.

We denote by $\operatorname{Id}_{I_{\bullet}}$ the identity morphism of
$\operatorname{BS}(I_{\bullet})$.

######  Definition 5.7.

Let $I_{\bullet}$ be an atomic-factored reduced expression as in (72). We say
that _polynomial forcing_ holds for $I_{\bullet}$ if for all $f\in
R^{\operatorname{RR}(p)}$, within
$\operatorname{End}(\operatorname{BS}(I_{\bullet}))$ as described in (74) and
(75), we have

(76)
$\underline{p}(f)\cdot\operatorname{Id}_{I_{\bullet}}\equiv\operatorname{Id}_{I_{\bullet}}\cdot
f\text{ modulo }\operatorname{End}_{<p}(\operatorname{BS}(I_{\bullet})).$

We say that _polynomial forcing_ holds for a double coset $p$ if it holds for
all atomic-factored reduced expressions $I_{\bullet}$ satisfying
$I_{\bullet}\leftrightharpoons p$.

This definition generalizes Definition 5.1 because atomic cosets have only one
reduced expression.

Let $p$ be an arbitrary $(I,J)$-coset. Since $\operatorname{LR}(p)\subset I$,
there is an inclusion of rings $R^{I}\subset R^{\operatorname{LR}(p)}$, and
$R^{\operatorname{LR}(p)}$ is naturally an $R^{I}$-module. Similarly,
$R^{\operatorname{RR}(p)}$ is an $R^{J}$-module. If $f\in
R^{\operatorname{RR}(p)}$ then $\underline{p}(f)\in R^{\operatorname{LR}(p)}$.
We can identify the rings $R^{\operatorname{LR}(p)}$ and
$R^{\operatorname{RR}(p)}$ via $\underline{p}$. In this way,
$R^{\operatorname{LR}(p)}$ becomes an $(R^{I},R^{J})$-bimodule.

######  Definition 5.8.

Let $p$ be an $(I,J)$-coset. The _standard bimodule associated to $p$_,
denoted $R_{p}$, is $R^{\operatorname{LR}(p)}$ as a left $R^{I}$-module. If
$f\in R^{J}$ and $m\in R_{p}$ then

(77) $m\cdot f:=\underline{p}(f)m.$

We identify $R_{p}$ with either $R^{\operatorname{LR}(p)}$ (with right action
twisted) or $R^{\operatorname{RR}(p)}$ (with left action twisted), as is more
convenient.

Let
$\operatorname{qu}:\operatorname{End}(\operatorname{BS}(I_{\bullet}))\to\operatorname{End}(\operatorname{BS}(I_{\bullet}))/\operatorname{End}_{<p}(\operatorname{BS}(I_{\bullet}))$
denote the quotient map.

######  Lemma 5.9.

Let $p$ be a core $(I,J)$-coset and let $I_{\bullet}$ be a reduced expression
for $p$. The bimodule map

(78)
$R_{p}\to\operatorname{End}(\operatorname{BS}(I_{\bullet}))/\operatorname{End}_{<p}(\operatorname{BS}(I_{\bullet})),\qquad
1\mapsto\operatorname{qu}(\operatorname{Id}_{I_{\bullet}})$

is well-defined if and only if polynomial forcing holds for $I_{\bullet}$.

###### Proof.

The right action of $f\in R^{J}$ on $1\in R^{J}=R_{p}$ yields $f\in R^{J}$,
and the right action on $\operatorname{qu}(\operatorname{Id}_{I_{\bullet}})$
yields $\operatorname{qu}(\operatorname{Id}_{I_{\bullet}}\cdot f)$. The left
action of $\underline{p}(f)\in R^{I}$ on $1\in R^{J}=R_{p}$ yields $f\in
R^{J}$, and the left action on
$\operatorname{qu}(\operatorname{Id}_{I_{\bullet}})$ yields
$\operatorname{qu}(\underline{p}(f)\cdot\operatorname{Id}_{I_{\bullet}})$.
These agree if and only if the bimodule map is well-defined, and if and only
if (76) holds. ∎

In conclusion, we have shown the equivalence of three ideas (for almost SW-
realizations) for an atomic coset ${\mathtt{a}}$: the well-definedness of the
morphism (78) when $p={\mathtt{a}}\leftrightharpoons I_{\bullet}$, atomic
polynomial forcing, and the atomic Leibniz rule.

For SW-realizations, (78) is an isomorphism by the theory of singular Soergel
bimodules. We can thus prove one of our main theorems.

######  Theorem 5.10.

For an SW-realization, atomic polynomial forcing and atomic Leibniz hold.
Moreover, the operators $T_{q},T_{q}^{\prime}$ in the Leibniz formulas are
unique.

###### Proof.

Assume ${\mathtt{a}}\leftrightharpoons[I,M,J]$ is an atomic coset. Then
$B_{\mathtt{a}}\cong\operatorname{BS}([I,M,J])$.

Recall from [Wil11, §4.5] the definition of the submodule
$\Gamma_{<{\mathtt{a}}}B_{\mathtt{a}}$ of elements supported on lower cosets.
By [EKLP23b, Lemma 3.31] we have a short exact sequence

(79)
$0\to\operatorname{End}(B_{\mathtt{a}},\Gamma_{<{\mathtt{a}}}B_{\mathtt{a}})\to\operatorname{End}(B_{\mathtt{a}})\to\operatorname{Hom}(B_{\mathtt{a}},B_{\mathtt{a}}/\Gamma_{<{\mathtt{a}}}B_{\mathtt{a}})\to
0$

and, by [EKLP23b, Theorem 3.30], the first term in (79) is isomorphic to
$\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}})$. Moreover, since
$B_{\mathtt{a}}$ is indecomposable, by [Wil11, Theorem 7.10] we have
$B_{\mathtt{a}}/\Gamma_{<{\mathtt{a}}}B_{{\mathtt{a}}}\cong R_{\mathtt{a}}$.
The Soergel–Williamson hom formula [Wil11, Theorem 7.9] implies that we have
an isomorphism $\operatorname{Hom}(B_{\mathtt{a}},R_{\mathtt{a}})\cong
R_{\mathtt{a}}$777As in the rest of this paper, we are ignoring degrees here.
given by $f\mapsto f(1\otimes 1)$. Putting all together, we obtain an
isomorphism

(80)
$\operatorname{End}(B_{\mathtt{a}})/\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}R_{\mathtt{a}}$

which sends $\operatorname{id}_{\mathtt{a}}$ to $1\in R_{\mathtt{a}}$

By Lemma 5.9, the existence of the isomorphism implies polynomial forcing for
${\mathtt{a}}$. By Theorem 5.5, this is in turn equivalent to the atomic
Leibniz rule for ${\mathtt{a}}$. Moreover, as proven in 5.5, the operators
$T_{q},T_{q}^{\prime}$ in the Leibniz formulas are unique. ∎

######  Remark 5.11.

For a SW-realization, there is an equivalent module-theoretic (rather than
morphism-theoretic) version of polynomial forcing. We first recall from
[EKLP24b, Definition 3.23] the filtration on Soergel bimodules

$N_{<p}(B)=\sum_{f\in\operatorname{Hom}(\operatorname{BS}(I_{\bullet}),B),I_{\bullet}\leftrightharpoons
q<p}\operatorname{Im}(f).$

In [EKLP24b, Proposition 3.25] we have showed that this coincides with the
support filtration $\Gamma_{<p}$ introduced in [Wil11].

Let ${\mathtt{a}}\leftrightharpoons[I,M,J]$ be an atomic coset. We say that
_(module-theoretic) polynomial forcing_ holds for ${\mathtt{a}}$ and $f$ if

(81) $\underline{{\mathtt{a}}}(f)\otimes 1-1\otimes f\in
N_{<{\mathtt{a}}}(B_{\mathtt{a}}).$

There is an isomorphism
$B_{\mathtt{a}}\cong\operatorname{End}(B_{\mathtt{a}})$, where $b\otimes
b^{\prime}\in B_{\mathtt{a}}$ is sent to multiplication by $b\otimes
b^{\prime}$. Moreover, by [EKLP24b, Theorem 3.30] we have

$\operatorname{Hom}(B_{\mathtt{a}},N_{<{\mathtt{a}}}B_{\mathtt{a}})\cong\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}}).$

Hence, (81) holds if and only if multiplication by
$\underline{{\mathtt{a}}}(f)\otimes 1-1\otimes f$ induces a morphism in
$\operatorname{End}_{<{\mathtt{a}}}(B_{\mathtt{a}})$, that is, if an only if
(morphism-theoretic) polynomial forcing holds for $f$.

###  5.4. Polynomial forcing: atomic and general

In the diagrammatic category, we intend to use the atomic Leibniz rule to
prove polynomial forcing, and not vice versa. In that context, polynomial
forcing is to be interpreted as the morphism-theoretic statement that (78) is
a well-defined morphism, when $I_{\bullet}$ is an atomic-factored reduced
expression. The goal of this section is to prove that atomic polynomial
forcing implies general polynomial forcing.

In [EKLP23b], a compatibility between the Bruhat order and concatenation of
reduced expression is proven, which implies the following result.

######  Proposition 5.12 ([EKLP23b, Prop. 3.7]).

Let $P_{\bullet}\leftrightharpoons p$ and $Q_{\bullet}\leftrightharpoons q$
and $R_{\bullet}\leftrightharpoons r$ be reduced expressions such that
$P_{\bullet}\circ Q_{\bullet}\circ R_{\bullet}\leftrightharpoons p.q.r$ is
reduced. Then

(82)
$\operatorname{id}_{P_{\bullet}}\otimes\operatorname{End}_{<q}(Q_{\bullet})\otimes\operatorname{id}_{R_{\bullet}}\subset\operatorname{End}_{<p.q.r}(P_{\bullet}\circ
Q_{\bullet}\circ R_{\bullet}).$

######  Theorem 5.13.

Assume an SW-realization. Then polynomial forcing holds for all double cosets.

###### Proof.

We first treat the case where $p=p^{\operatorname{core}}$ is a core
$(I,J)$-coset. Consider an atomic reduced expression $I_{\bullet}$ for $p$,
yielding atomic cosets ${\mathtt{a}}_{i}$ such that
$p={\mathtt{a}}_{1}.{\mathtt{a}}_{2}.\cdots.{\mathtt{a}}_{m}$. Since
${\mathtt{a}}_{i}$ are core cosets, by [EKLP24b, Lem. 2.10] we have
$\underline{p}=\underline{{\mathtt{a}}_{1}}\cdot\underline{{\mathtt{a}}_{2}}\cdots\underline{{\mathtt{a}}_{m}}$.
Now within $\operatorname{BS}(I_{\bullet})$ we have

(83)
$\operatorname{Id}_{{\mathtt{a}}_{1}}\otimes\cdots\otimes\operatorname{Id}_{{\mathtt{a}}_{m}}f\equiv\operatorname{Id}_{{\mathtt{a}}_{1}}\otimes\cdots\otimes\underline{{\mathtt{a}}_{m}}(f)\operatorname{Id}_{{\mathtt{a}}_{m}}\equiv\ldots\equiv\underline{{\mathtt{a}}_{1}}(\cdots(\underline{{\mathtt{a}}_{m}}(f)))\operatorname{Id}_{{\mathtt{a}}_{1}}\otimes\cdots\otimes\operatorname{Id}_{{\mathtt{a}}_{m}},$

where $\equiv$ indicates equality modulo lower terms. At each step we applied
polynomial forcing for an atomic coset as proved in 5.10, and used Proposition
5.12 to argue that lower terms for ${\mathtt{a}}_{i}$ embed into lower terms
for $p$. Thus polynomial forcing holds for $p$.

If $p$ is not a core coset, let $I_{\bullet}$ be a special reduced expression
for $p$ as in (72). Polynomial forcing for $p^{\operatorname{core}}$ implies
that

$\underline{p}(f)\cdot\operatorname{Id}_{I_{\bullet}}\equiv\operatorname{Id}_{I_{\bullet}}\cdot
f$

modulo

$\operatorname{id}_{[[I\supset\operatorname{LR}(p)]]}\otimes\operatorname{End}_{<p^{\operatorname{core}}}(I^{\prime}_{\bullet})\otimes\operatorname{id}_{[[\operatorname{RR}(p)\subset
J]]}.$

By Proposition 5.12, they are also equivalent modulo
$\operatorname{End}_{<p}(I_{\bullet})$, as desired. ∎

The reader familiar with Soergel bimodules might be familiar with the
following example, which showcases how atomic polynomial forcing implies the
general case. It also relates our new concept of polynomial forcing to the
concept previously in the literature.

######  Example 5.14.

Consider the $(\emptyset,\emptyset)$-coset $p=\\{s\\}$ for a simple reflection
$s$, and the reduced expression $I_{\bullet}=[\emptyset,s,\emptyset]$. Inside
$B_{s}:=\operatorname{BS}(I_{\bullet})=R\otimes_{R^{s}}R$ we have

(84) $1\otimes f-s(f)\otimes
1=\partial_{s}(f)\cdot\frac{1}{2}(\alpha_{s}\otimes 1+1\otimes\alpha_{s}).$

The term on the right hand side is in $\operatorname{End}_{<s}(I_{\bullet})$,
a consequence of the so-called _polynomial forcing relation_ in the Hecke
category, see e.g. [EW16, (5.2)].

Now consider the $(\emptyset,\emptyset)$ coset $p=\\{w\\}$ for some $w\in W$,
and a reduced expression
$I_{\bullet}=[\emptyset,s_{1},\emptyset,s_{2},\ldots,\emptyset,s_{d},\emptyset]$.
By applying the polynomial forcing relation for $B_{s_{d}}$, we see that
$1\otimes\cdots\otimes 1\otimes f\equiv 1\otimes\cdots\otimes s_{d}(f)\otimes
1$ modulo maps which factor through
$[\emptyset,s_{1},\emptyset,\ldots,\emptyset,s_{d-1},\emptyset]$. Continuing,
we can apply polynomial forcing for each $B_{s_{i}}$ to force $f$ across all
the tensors, at the cost of maps which factor through subexpressions of
$s_{1}\cdots s_{d}$. By the subexpression property of the Bruhat order, such
maps consist of lower terms.

##  6\. Atomic Leibniz rule in type $A$

In this section we explicitly prove condition (3) in Theorem 5.5 in type $A$.
We establish this result for $\Bbbk=\mathbb{Z}$ in this section, rather than
over a field. In addition to extending our results over $\mathbb{Z}$, we feel
the ability to be explicit in a key example is its own reward.

In order to achieve this, we first prove in 6.3 explicitly an atomic Leibniz
rule for a specific set of generators when $W_{M}$ is the entire symmetric
group. In Section 6.3 we extend our results to the case when $W_{M}$ is a
product of symmetric groups. This handles all atomic cosets in type $A$.

###  6.1. Notation in type $A$

We fix notation under the assumption that $W_{M}$ is an irreducible Coxeter
group of type $A$. For $s,t\in M$ we write $\widehat{s}:=M\setminus\\{s\\}$
and $\widehat{s}\widehat{t}:=M\setminus\\{s,t\\}$, etcetera.

Fix $a,b\geq 1$ and let $n=a+b$. Let $W_{M}=\mathbf{S}_{n}$. Let $t=s_{a}$ and
$s=s_{b}=w_{M}tw_{M}$, so that $W_{J}=\mathbf{S}_{a}\times\mathbf{S}_{b}$ and
$W_{I}=\mathbf{S}_{b}\times\mathbf{S}_{a}$. Let ${\mathtt{a}}$ be the
$(\mathbf{S}_{b}\times\mathbf{S}_{a},\mathbf{S}_{a}\times\mathbf{S}_{b})$-coset
containing $w_{M}$. The coset ${\mathtt{a}}$ is depicted as follows, with its
minimal element $\underline{{\mathtt{a}}}$ being the string diagram visible.

(85) ${\mathtt{a}}=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/cosetq0}}$}}}}}}}}}}.$

Drawn is the example $a=3$ and $b=5$.

For each $0\leq k\leq\min(a,b)$, there is an $(\widehat{s},\widehat{t})$-coset
$q_{k}$ depicted as follows.

(86) $q_{1}=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/cosetq1}}$}}}}}}}}}},\qquad
q_{2}=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/cosetq2}}$}}}}}}}}}},\qquad
q_{3}=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/cosetq3}}$}}}}}}}}}}.$

Then $p=q_{0}$, and $\\{q_{k}\\}_{0\leq k\leq\min(a,b)}$ is an enumeration of
all the $(\widehat{s},\widehat{t})$-cosets. The Bruhat order is a total order
in this case:

$q_{0}>q_{1}>q_{2}>\ldots>q_{\min(a,b)}.$

The left redundancy subgroup (see [EKLP24b, Section 1.2] to see how to
calculate redundancies and cores) of $q_{k}$ is
$\mathbf{S}_{k}\times\mathbf{S}_{b-k}\times\mathbf{S}_{a-k}\times\mathbf{S}_{k}$.
For brevity, let $\ell:=n-k$. Then (with one exception)
$\operatorname{LR}(q_{k})=\widehat{b}\widehat{k}\widehat{\ell}:=M\setminus\\{b,k,\ell\\}$
and $\operatorname{RR}(q_{k})=\widehat{a}\widehat{k}\widehat{\ell}$. The core
of $q_{k}$ is the double coset depicted as

(87) $q_{1}^{\operatorname{core}}=\vbox{\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/coreq1}}$}}}}}}}}}},\qquad
q_{2}^{\operatorname{core}}=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/coreq2}}$}}}}}}}}}},\qquad
q_{3}^{\operatorname{core}}=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/coreq3}}$}}}}}}}}}}.$

Note that the core of $q_{k}$ is itself atomic, except for $k=\min(a,b)$ when
the core is an identity coset.

The case $k=a=b=\ell$ is relatively special. We denote this special coset as
$q_{a=b}$. We have
$\operatorname{LR}(q_{a=b})=\operatorname{RR}(q_{a=b})=I=J$. Unlike other
$q_{i}$, $q_{a=b}$ is core.

######  Remark 6.1.

In type $A$ the following statement is always true: if ${\mathtt{a}}$ is
atomic and $q<{\mathtt{a}}$ then $q^{\operatorname{core}}$ is either atomic or
an identity coset. We do not know for which atomic cosets ${\mathtt{a}}$ this
property holds in other types.

A reduced expression for $q_{k}$, which factors through the core, is

(88)
$q_{k}\leftrightharpoons[[\widehat{b}\supset\widehat{b}{\widehat{k}\widehat{\ell}}\subset\widehat{k}\widehat{\ell}\supset\widehat{a}\widehat{k}\widehat{\ell}\subset\widehat{a}]].$

The exception is when $a=b=k$, in which case

(89) $q_{a=b}\leftrightharpoons[\widehat{b}],$

that is, the identity expression of $I=\widehat{b}$ is a reduced expression
for the length zero coset $q_{a=b}$.

Finally, let us write $y_{k}=\overline{q_{k}}w_{J}^{-1}$. Then
$\partial_{q_{k}}=\partial_{y_{k}}$. Here are examples of $y_{k}$.

(90) $y_{0}=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/y0}}$}}}}}}}}}},\qquad
y_{1}=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/y1}}$}}}}}}}}}},\qquad
y_{2}=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/y2}}$}}}}}}}}}},\qquad
y_{3}=\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 0.0pt\hbox{{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\includegraphics[scale={1}]{./EPS/y3}}$}}}}}}}}}}.$

Of course $y_{0}=\underline{{\mathtt{a}}}$. Meanwhile $y_{k}$ is obtained from
$y_{0}$ by removing a $k\times k$ square of crossings from the top. In the
special case of the coset $q_{a=b}$, we have $y_{a=b}=e$, the identity of $W$.

###  6.2. Complete symmetric polynomials

Fix $a,b\geq 1$ and continue to use the notation from the previous section.
The standard action of $\mathbf{S}_{n}$ on $\mathbb{Z}^{n}$ (with the standard
choice of roots and coroots) we call the _permutation realization_. Let
$R=\mathbb{Z}[x_{1},\ldots,x_{n}]$. All Demazure operators preserve $R$. By a
result of Demazure [Dem73, Lemme 5], Frobenius surjectivity holds in type $A$
over $\mathbb{Z}$, that is, for any $I\subset M$ we can find $P_{I}\in R$ such
that $\partial_{I}(P_{I})=1$.

Moreover, for any $J\subset I$ the ring $R^{I}$ is Frobenius over $R^{J}$ and
we can choose dual bases $\Delta^{J}_{I,(1)}$ and $\Delta^{J}_{I,(1)}$
accordingly.

It is well-known that the subring
$R^{J}=R^{\mathbf{S}_{a}\times\mathbf{S}_{b}}$ is generated over
$R^{\mathbf{S}_{n}}$ by the complete symmetric polynomials

$h_{i}(x_{1},\ldots,x_{a})=\sum_{1\leq k_{1}\leq k_{2}\leq\ldots\leq k_{i}\leq
a}x_{k_{1}}x_{k_{2}}\cdots x_{k_{i}}$

in the first $a$ variables. In this section, we directly prove the atomic
Leibniz rule for ${\mathtt{a}}$ when $f=h_{i}(x_{1},\ldots,x_{a})$.

One of the great features of complete symmetric polynomials is their behavior
under Demazure operators. For example, we have

(91)
$\partial_{3}(h_{i}(x_{1},x_{2},x_{3}))=h_{i-1}(x_{1},x_{2},x_{3},x_{4}).$

As a consequence, $\partial_{2}\partial_{3}(h_{i}(x_{1},x_{2},x_{3}))=0$, a
fact which is false if $h_{i}$ is replaced by some general polynomial inside
$R^{\mathbf{S}_{3}\times\mathbf{S}_{n-3}}$. Indeed, the only elements $w\leq
s_{1}s_{2}s_{3}$ for which $\partial_{w}(h_{i}(x_{1},x_{2},x_{3}))\neq 0$ are
$w=s_{3}$ and $w=e$. This will simplify the computation considerably.

Below we shall use letters like $X$, $Y$, and $Z$ to denote subsets of
$\\{1,\ldots,n\\}$. We write $h_{i}(X)$ for the $i$-th complete symmetric
polynomial in the variables $x_{j}$ for $j\in X$.

######  Lemma 6.2.

We have

(92) $\partial_{j}(h_{i}(X))=\begin{cases}h_{i-1}(X\cup\\{j+1\\})&\text{if
}j\in X\text{and }j+1\not\in X\\\ -h_{i-1}(X\cup\\{j\\})&\text{if }j+1\in
X\text{and }j\not\in X\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

Clearly, $h_{i}(X)$ is $s_{j}$-invariant if both $j$ and $j+1$ are inside or
outside $X$, so $\partial_{j}(h_{i}(X))=0$. Assume now that $j\in X$ and
$j+1\not\in X$. We have $h_{i}(X)=h_{i}(X\setminus\\{j\\})+h_{i-1}(X)x_{j}$.
Hence,

$\partial_{j}(h_{i}(X))=\partial_{j}(h_{i-1}(X)x_{j}).$

Applying the twisted Leibniz rule, by induction on $|X|$, we obtain

$\displaystyle\partial_{j}(h_{i}(X))$
$\displaystyle=h_{i-2}(X\cup\\{j+1\\})x_{j}+s_{j}(h_{i-1}(X))\partial_{j}(x_{j})$
$\displaystyle=h_{i-2}(X\cup\\{j+1\\})x_{j}+h_{i-1}(X\cup\\{j+1\\}\setminus\\{j\\})$
$\displaystyle=h_{i-1}(X\cup\\{j+1\\}).$

The case where $j+1\in X$ and $j\notin X$ follows because
$\partial_{j}(s_{j}(f))=-\partial_{j}(f)$, so we have

$\partial_{j}(h_{i}(X))=-\partial_{j}(h_{i}(X\cup\\{j\\}\setminus\\{j+1\\}))=-h_{i-1}(X\cup\\{j\\}).\qed$

######  Theorem 6.3.

Use the notation of §6.1. Fix $a,b\geq 1$, and let $n=a+b.$ Let
$X=\\{1,\ldots,a\\}$ and $Y=\\{n,n-1,\ldots,n+1-a\\}$. Recall
$y_{0},y_{1}\in\mathbf{S}_{n}$ from (90).

Then for any $i\geq 0$ and any $g\in
R^{\mathbf{S}_{a}\times\mathbf{S}_{b}}=R^{J}$ we have

(93) $\partial_{y_{0}}(h_{i}(X)\cdot
g)=h_{i}(Y)\cdot\partial_{y_{0}}(g)+\partial_{y_{1}}(h_{i-1}(X\cup n)\cdot
g).$

As $\partial_{p}=\partial_{y_{0}},$ $\underline{p}=y_{0},$ and
$y_{0}(h_{i}(X))=h_{i}(Y)$, this is compatible with (30), where
$T_{q_{1}}(h_{i}(X))=h_{i-1}(X\cup n)$, and $T_{q_{k}}(h_{i}(X))$ is zero for
all $k>1$. Most of the terms in (30) are zero for complete symmetric
polynomials, making the formula much easier than the general case.

###### Proof.

We will do a proof by example, for the example $a=3$ and $b=4$. The general
proof is effectively the same only the notation is more cumbersome.

In this proof, we write $\partial_{123}$ for
$\partial_{1}\circ\partial_{2}\circ\partial_{3}$ (and not the Frobenius trace
associated to the longest element $w_{123}$). We use parenthesization for
emphasis, so that $\partial_{(12)3}$ is the same thing as $\partial_{123}$,
but emphasizes that $\partial_{(12)3}=\partial_{12}\circ\partial_{3}$.

Remember that $g$ is invariant under anything except $s_{3}$, so
$\partial_{j}(g)=0$ if $j\neq 3$. This implies, for example, that
$\partial_{3}$ kills $\partial_{23}(g)$, and that $\partial_{2}$ kills
$\partial_{3243}(g)$, etcetera.

We claim that

(94)
$\partial_{123}(h_{i}(123)g)=h_{i-1}(1234)\partial_{23}(g)+h_{i}(234)\partial_{123}(g).$

One proof is to apply the ordinary twisted Leibniz rule repeatedly using (92).
After one step we obtain

$\partial_{123}(h_{i}(123)g)=\partial_{12}\left(h_{i-1}(1234)g+h_{i}(124)\partial_{3}(g)\right).$

The first term on the right-hand side is invariant under $s_{2}$, so it is
killed by $\partial_{2}$. Thus we have

$\partial_{123}(h_{i}(123)g)=\partial_{1}(\partial_{2}(h_{i}(124)\partial_{3}(g)))=\partial_{1}(h_{i-1}(1234)\partial_{3}(g)+h_{i}(134)\partial_{23}(g)).$

Again the first term on the right-hand side is invariant under $s_{1}$, so it
is killed by $\partial_{1}$. A final application of the twisted Leibniz rule
to $\partial_{1}(h_{i}(134)\partial_{23}(g))$ gives (94). Essentially, this
proof is by iterating the twisted Leibniz rule and arguing that the first term
vanishes in every application but the last, because the first term is
appropriately invariant. We call this the _easy invariance argument_.

Note that $\partial_{123}(g)$ is invariant under everything but $s_{4}$. More
generally, $\partial_{12\ldots a}(g)$ is killed by $\partial_{j}$ for all
$j\neq a+1$. This is true when $j=1$ since $\partial_{1}\partial_{1}=0$. This
is true when $j>a+1$ since $\partial_{j}\partial_{12\ldots
a}=\partial_{12\ldots a}\partial_{j}$ and $\partial_{j}(g)=0$. This is true
when $2\leq j\leq a$ because $\partial_{j}\partial_{12\ldots
a}=\partial_{12\ldots a}\partial_{j-1}$ and $\partial_{j-1}(g)=0$.

By the easy invariance argument again, but with indices shifted and $g$
replaced by $\partial_{123}(g)$, we have

(95)
$\partial_{234}(h_{i}(234)\partial_{123}(g))=h_{i-1}(2345)\partial_{34123}(g)+h_{i}(345)\partial_{234123}(g).$

This is how we treat the second term in the right side of (94).

Note that all computations above are unchanged by adding new variables to our
complete symmetric polynomials which are untouched by any of the simple
reflections used by the formula. For example, adding $7$ to every $h_{i}$ in
(94) we get

(96)
$\partial_{123}(h_{i}(1237)g)=h_{i-1}(12347)\partial_{23}(g)+h_{i}(2347)\partial_{123}(g).$

In this example, we call $7$ an _irrelevant index_.

Now we examine the first term in the right side of (94). Note that
$\partial_{23}(g)$ is invariant under all simple reflections except $s_{1}$
and $s_{4}$. For the next computation, the index $1$ is irrelevant. The easy
invariance argument again implies that

(97)
$\partial_{234}(h_{i-1}(1234)\partial_{23}(g))=h_{i-2}(12345)\partial_{3423}(g)+h_{i-1}(1345)\partial_{23423}(g).$

However, as $23423=32434,$ we have $\partial_{23423}(g)=0$, so one has the
simpler formula

(98)
$\partial_{234}(h_{i-1}(1234)\partial_{23}(g))=h_{i-2}(12345)\partial_{3423}(g).$

Overall, we see that

$\displaystyle\partial_{(234)(123)}(h_{i}(123)g)$ $\displaystyle=$
$\displaystyle h_{i}(345)\partial_{(234)(123)}(g)$ $\displaystyle+$
$\displaystyle h_{i-1}(2345)\partial_{(34)(123)}(g)$ $\displaystyle+$
$\displaystyle h_{i-2}(12345)\partial_{(34)(23)}(g).$

The pattern is relatively straightforward. Here’s the next one in the pattern:

$\displaystyle\partial_{(345)(234)(123)}(h_{i}(123)g)$ $\displaystyle=$
$\displaystyle h_{i}(456)\partial_{(345)(234)(123)}(g)$ $\displaystyle+$
$\displaystyle h_{i-1}(3456)\partial_{(45)(234)(123)}(g)$ $\displaystyle+$
$\displaystyle h_{i-2}(23456)\partial_{(45)(34)(123)}(g)$ $\displaystyle+$
$\displaystyle h_{i-3}(123456)\partial_{(45)(34)(23)}(g).$

The word whose Demazure is applied to $g$ is obtained from the concatenation
of triples $(345)(234)(123)$ by removing the first index from some of the
triples; more specifically, from a prefix of the set of triples. The indices
that get removed are instead added to the complete symmetric polynomial. The
reason triples appear is because $a=3$.

The inductive proof of this pattern is the same as above. One takes (6.2) and
applies $\partial_{345}$. The first term splits in two, giving the first two
terms of (6.2), similar to (94) or (95). Each other term contributes one term
in (6.2), similar to (98).

Note that we could have added the irrelevant index $7$ to every set in sight
within (6.2), without any issues. This will be important later.

Repeating until one applies $\partial_{y_{0}}$, we calculate
$\partial_{y_{0}}(h_{i}(X)\cdot g)$:

$\displaystyle\partial_{(456)(345)(234)(123)}(h_{i}(123)g)$ $\displaystyle=$
$\displaystyle h_{i}(567)\partial_{(456)(345)(234)(123)}(g)$ $\displaystyle+$
$\displaystyle h_{i-1}(4567)\partial_{(56)(345)(234)(123)}(g)$
$\displaystyle+$ $\displaystyle
h_{i-2}(34567)\partial_{(56)(45)(234)(123)}(g)$ $\displaystyle+$
$\displaystyle h_{i-3}(234567)\partial_{(56)(45)(34)(123)}(g)$
$\displaystyle+$ $\displaystyle
h_{i-4}(1234567)\partial_{(56)(45)(34)(23)}(g).$

The fact that there were four triples is because $b=4$. Note that the first
term in the RHS is $h_{i}(Y)\partial_{y_{0}}(g).$

Let us now compute $\partial_{y_{1}}(h_{i-1}(1237)\cdot g)$. Note that
$y_{1}=(56)(345)(234)(123)$. To compute
$\partial_{(345)(234)(123)}(h_{i-1}(1237)g)$, we take (6.2), add the
irrelevant index $7$ to all variable lists, and reduce $i$ by one. Now we need
only apply $\partial_{(56)}$ to the result. The key thing to note here is that
each $h_{\bullet}(\cdots 567)$ is invariant already under $s_{5}$ and $s_{6}$.
Thus both operators in $\partial_{(56)}$ simply apply to the $g$ term. From
this we can compute $\partial_{y_{1}}(h_{i-1}(X\cup n)\cdot g)$:

$\displaystyle\partial_{(56)(345)(234)(123)}(h_{i-1}(1237)g)$ $\displaystyle=$
$\displaystyle h_{i-1}(4567)\partial_{(56)(345)(234)(123)}(g)$
$\displaystyle+$ $\displaystyle
h_{i-2}(34567)\partial_{(56)(45)(234)(123)}(g)$ $\displaystyle+$
$\displaystyle h_{i-3}(234567)\partial_{(56)(45)(34)(123)}(g)$
$\displaystyle+$ $\displaystyle
h_{i-4}(1234567)\partial_{(56)(45)(34)(23)}(g).$

This exactly matches all terms from (6.2) except the first term. Thus the
theorem is proven! ∎

######  Theorem 6.4.

The atomic Leibniz rule and atomic polynomial forcing both hold for atomic
cosets ${\mathtt{a}}\leftrightharpoons[I,M,J]$ when $W_{M}=\mathbf{S}_{n}$
when $R=\mathbb{Z}[x_{1},\ldots,x_{n}]$.

###### Proof.

Theorem 6.3 proved property (3) from Theorem 5.5 in this case. Thus conditions
(1) and (5) also hold in this case. ∎

###  6.3. Reduction to the connected case

The previous section proves an atomic Leibniz rule under the assumption