On Some Quadratic Algebras

On Some Quadratic Algebras I $\boldsymbol{\frac{1}{2}}$:
Combinatorics of Dunkl and Gaudin Elements,
Schubert, Grothendieck, Fuss–Catalan,
Universal Tutte and Reduced Polynomials

Anatol N. KIRILLOV †‡§

A.N. Kirillov

† Research Institute of Mathematical Sciences (RIMS), Kyoto, Sakyo-ku
606-8502, Japan<EMAIL_ADDRESS>http://www.kurims.kyoto-u.ac.jp/~kirillov/

‡ The Kavli Institute for the Physics and Mathematics of the Universe (IPMU),
‡ 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan

§ Department of Mathematics, National Research University Higher School of
Economics,
§ 7 Vavilova Str., 117312, Moscow, Russia

Received March 23, 2015, in final form December 27, 2015; Published online
January 05, 2016

We study some combinatorial and algebraic properties of certain quadratic
algebras related with dynamical classical and classical Yang–Baxter equations.

braid and Yang–Baxter groups; classical and dynamical Yang–Baxter relations;
classical Yang–Baxter, Kohno–Drinfeld and $3$-term relations algebras; Dunkl,
Gaudin and Jucys–Murphy elements; small quantum cohomology and $K$-theory of
flag varieties; Pieri rules; Schubert, Grothendieck, Schröder, Ehrhart,
Chromatic, Tutte and Betti polynomials; reduced polynomials; Chan–Robbins–Yuen
polytope; $k$-dissections of a convex $(n+k+1)$-gon, Lagrange inversion
formula and Richardson permutations; multiparameter deformations of
Fuss–Catalan and Schröder polynomials; Motzkin, Riordan, Fine, poly-Bernoulli
and Stirling numbers; Euler numbers and Brauer algebras; VSASM and CSTCPP;
Birman–Ko–Lee monoid; Kronecker elliptic sigma functions

14N15; 53D45; 16W30

To the memory of Alain Lascoux 1944–2013, the great Mathematician, from whom I
have learned a lot about the Schubert and Grothendieck polynomials.

###### Contents

  1. 1 Introduction
  2. 2 Dunkl elements
    1. 2.1 Some representations of the algebra $6DT_{n}$
      1. 2.1.1 Dynamical Dunkl elements and equivariant quantum cohomology
      2. 2.1.2 Step functions and the Dunkl–Uglov representations of the degenerate affine Hecke algebras [138]
      3. 2.1.3 Extended Kohno–Drinfeld algebra and Yangian Dunkl–Gaudin elements
    2. 2.2 “Compatible” Dunkl elements, Manin matrices and algebras related with weighted complete graphs $rK_{n}$
    3. 2.3 Miscellany
      1. 2.3.1 Non-unitary dynamical classical Yang–Baxter algebra ${\rm DCYB}_{n}$
      2. 2.3.2 Dunkl and Knizhnik–Zamolodchikov elements
      3. 2.3.3 Dunkl and Gaudin operators
      4. 2.3.4 Representation of the algebra $3T_{n}$ on the free algebra $\mathbb{Z}\langle t_{1},\ldots,t_{n}\rangle$
      5. 2.3.5 Kernel of Bruhat representation
      6. 2.3.6 The Fulton universal ring [47], multiparameter quantum cohomology of flag varieties [45] and the full Kostant–Toda lattice [29, 80]
  3. 3 Algebra $3HT_{n}$
    1. 3.1 Modified three term relations algebra $3MT_{n}(\beta,\psi)$
      1. 3.1.1 Equivariant modified three term relations algebra
    2. 3.2 Multiplicative Dunkl elements
    3. 3.3 Truncated Gaudin operators
    4. 3.4 Shifted Dunkl elements $\mathfrak{d}_{i}$ and $\mathfrak{D}_{i}$
  4. 4 Algebra $3T_{n}^{(0)}(\Gamma)$ and Tutte polynomial of graphs
    1. 4.1 Graph and nil-graph subalgebras, and partial flag varieties
      1. 4.1.1 Nil-Coxeter and affine nil-Coxeter subalgebras in $3T_{n}^{(0)}$
      2. 4.1.2 Parabolic 3-term relations algebras and partial flag varieties
        1. 4.1.3 Universal Tutte polynomials
      3. 4.1.4 Quasi-classical and associative classical Yang–Baxter algebras of type $B_{n}$
    2. 4.2 Super analogue of 6-term relations and classical Yang–Baxter algebras
      1. 4.2.1 Six term relations algebra $6T_{n}$, its quadratic dual $(6T_{n})^{!}$, and algebra $6HT_{n}$
      2. 4.2.2 Algebras $6T_{n}^{(0)}$ and $6T_{n}^{\bigstar}$
      3. 4.2.3 Hilbert series of algebras ${\rm CYB}_{n}$ and $6T_{n}$
      4. 4.2.4 Super analogue of 6-term relations algebra
    3. 4.3 Four term relations algebras / Kohno–Drinfeld algebras
      1. 4.3.1 Kohno–Drinfeld algebra $4T_{n}$ and that ${\rm CYB}_{n}$
      2. 4.3.2 Nonsymmetric Kohno–Drinfeld algebra $4NT_{n}$, and McCool algebras ${\cal P}\Sigma_{n}$ and ${\cal P}\Sigma_{n}^{+}$
      3. 4.3.3 Algebras $4TT_{n}$ and $4ST_{n}$
    4. 4.4 Subalgebra generated by Jucys–Murphy elements in $4T_{n}^{0}$
    5. 4.5 Nonlocal Kohno–Drinfeld algebra $NL4T_{n}$
      1. 4.5.1 On relations among JM-elements in Hecke algebras
    6. 4.6 Extended nil-three term relations algebra and DAHA, cf. [24]
    7. 4.7 Braid, affine braid and virtual braid groups
      1. 4.7.1 Yang–Baxter groups
      2. 4.7.2 Some properties of braid and Yang–Baxter groups
      3. 4.7.3 Artin and Birman–Ko–Lee monoids
  5. 5 Combinatorics of associative Yang–Baxter algebras
    1. 5.1 Combinatorics of Coxeter element
      1. 5.1.1 Multiparameter deformation of Catalan, Narayana and Schröder numbers
    2. 5.2 Grothendieck and $q$-Schröder polynomials
      1. 5.2.1 Schröder paths and polynomials
      2. 5.2.2 Grothendieck polynomials and $k$-dissections
      3. 5.2.3 Grothendieck polynomials and $q$-Schröder polynomials
      4. 5.2.4 Specialization of Schubert polynomials
      5. 5.2.5 Specialization of Grothendieck polynomials
    3. 5.3 The “longest element” and Chan–Robbins–Yuen polytope575757Some results of this section, e.g., Theorems 5.63 and 5.65, has been proved independently and in greater generality in [102].
      1. 5.3.1 The Chan–Robbins–Yuen polytope ${\cal{CRY}}_{n}$
      2. 5.3.2 The Chan–Robbins–Mészáros polytope ${\cal{P}}_{n,m}$
    4. 5.4 Reduced polynomials of certain monomials
      1. 5.4.1 Reduced polynomials, Motzkin and Riordan numbers
      2. 5.4.2 Reduced polynomials, dissections and Lagrange inversion formula
  6. A Appendixes
    1. A.1 Grothendieck polynomials
    2. A.2 Cohomology of partial flag varieties
    3. A.3 Multiparamater 3-term relations algebras
      1. A.3.1 Equivariant multiparameter 3-term relations algebras
      2. A.3.2 Algebra $3QT_{n}(\beta,h)$, generalized unitary case
    4. A.4 Koszul dual of quadratic algebras and Betti numbers
    5. A.5 On relations in the algebra $Z_{n}^{0}$
      1. A.5.1 Hilbert series ${\rm Hilb}\big{(}3T_{n}^{0},t\big{)}$ and ${\rm Hilb}\big{(}\big{(}3T_{n}^{0}\big{)}^{!},t\big{)}$: Examples
    6. A.6 Summation and Duality transformation formulas [63]
    7. Acknowledgments

### Extended abstract

We introduce and study a certain class of quadratic algebras, which are
nonhomogeneous in general, together with the distinguish set of mutually
commuting elements inside of each, the so-called Dunkl elements. We describe
relations among the Dunkl elements in the case of a family of quadratic
algebras corresponding to a certain splitting of the universal classical
Yang–Baxter relations into two three term relations. This result is a further
extension and generalization of analogous results obtained in [45, 117] and
[76]. As an application we describe explicitly the set of relations among the
Gaudin elements in the group ring of the symmetric group, cf. [108]. We also
study relations among the Dunkl elements in the case of (nonhomogeneous)
quadratic algebras related with the universal dynamical classical Yang–Baxter
relations. Some relations of results obtained in papers [45, 72, 75] with
those obtained in [54] are pointed out. We also identify a subalgebra
generated by the generators corresponding to the simple roots in the extended
Fomin–Kirillov algebra with the $\rm DAHA$, see Section 4.3.

The set of generators of algebras in question, naturally corresponds to the
set of edges of the complete graph $K_{n}$ (to the set of edges and loops of
the complete graph with (simple) loops ${\widetilde{K}}_{n}$ in dynamical and
equivariant cases). More generally, starting from any subgraph $\Gamma$ of the
complete graph with simple loops ${\widetilde{K}}_{n}$ we define a (graded)
subalgebra $3T_{n}^{(0)}(\Gamma)$ of the (graded) algebra
$3T_{n}^{(0)}({\widetilde{K}}_{n})$ [70]. In the case of loop-less graphs
$\Gamma\subset K_{n}$ we state conjecture, Conjecture 35 in the main text,
which relates the Hilbert polynomial of the abelian quotient
$3T_{n}^{(0)}(\Gamma)^{ab}$ of the algebra $3T_{n}^{(0)}(\Gamma)$ and the
chromatic polynomial of the graph $\Gamma$ we are started with111We expect
that a similar conjecture is true for any finite (oriented) matroid $\cal{M}$.
Namely, one (A.K.) can define an analogue of the three term relations algebra
$3T^{(0)}({\cal{M}})$ for any (oriented) matroid $\cal{M}$. We expect that the
abelian quotient $3T^{(0)}({\cal{M}})^{ab}$ of the algebra
$3T^{(0)}({\cal{M}})$ is isomorphic to the Orlik–Terao algebra [114], denoted
by ${\rm OT}({\cal{M}})$ (known also as even version of the Orlik–Solomon
algebra, denoted by ${\rm OS}^{+}({\cal{M}})$ ) associated with matroid
$\cal{M}$ [28]. Moreover, the anticommutative quotient of the odd version of
the algebra $3T^{(0)}({\cal{M}})$, as we expect, is isomorphic to the
Orlik–Solomon algebra ${\rm OS}({\cal{M}})$ associated with matroid
${\cal{M}}$, see, e.g., [11, 49]. In particular,  $\displaystyle{\rm
Hilb}(3T^{(0)}\big{(}{\cal{M}})^{ab},t\big{)}=t^{r({\cal{M}})}{\rm
Tutte}\big{(}{\cal{M}};1+t^{-1},0\big{)}.$ We expect that the Tutte polynomial
of a matroid, ${\rm Tutte}({\cal{M}},x,y)$, is related with the Betti
polynomial of a matroid $\cal{M}$. Replacing relations $u_{ij}^{2}=0$,
$\forall\,i,j$, in the definition of the algebra $3T^{(0)}(\Gamma)$ by
relations $u_{ij}^{2}=q_{ij}$, $\forall\,i,j$, $(i,j)\in E(\Gamma)$, where
$\\{q_{ij}\\}_{(i,j)\in E(\Gamma)}$, $q_{ij}=q_{ji}$, is a collection of
central elements, give rise to a quantization of the Orlik–Terao algebra ${\rm
OT}(\Gamma)$. It seems an interesting task to clarify combinatorial/geometric
significance of noncommutative versions of Orlik–Terao algebras (as well as
Orlik–Solomon ones) defined as follows:
${\cal{OT}}(\Gamma):=3T^{(0)}(\Gamma)$, its “quantization”
$3T^{({\boldsymbol{q}})}(\Gamma)^{ab}$ and $K$-theoretic analogue
$3T^{({\boldsymbol{q}})}(\Gamma,\beta)^{ab}$, cf. Definition 3.1, in the
theory of hyperplane arrangements. Note that a small modification of arguments
in [89] as were used for the proof of our Conjecture 35, gives rise to a
theorem that the algebra $3T_{n}(\Gamma)^{ab}$ is isomorphic to the
Orlik–Terao algebra ${\rm OT}(\Gamma)$ studied in [126].222In the case of
simple graphs our Conjecture 35 has been proved in [89].. We check our
conjecture for the complete graphs $K_{n}$ and the complete bipartite graphs
$K_{n,m}$. Besides, in the case of complete multipartite graph
$K_{n_{1},\ldots,n_{r}}$, we identify the commutative subalgebra in the
algebra $3T_{N}^{(0)}(K_{n_{1},\ldots,n_{r}})$, $N=n_{1}+\cdots+n_{r}$,
generated by the elements

$\displaystyle\theta_{j,k_{j}}^{(N)}:=e_{k_{j}}\big{(}\theta_{N_{j-1}+1}^{(N)},\ldots,\theta_{N_{j}}^{(N)}\big{)},$
$\displaystyle 1\leq j\leq r,\quad 1\leq k_{j}\leq n_{j},\quad
N_{j}:=n_{1}+\cdots+n_{j},\quad N_{0}=0,$

with the cohomology ring $H^{*}({\cal{F}}l_{n_{1},\ldots,n_{r}},\mathbb{Z})$
of the partial flag variety ${\cal{F}}l_{n_{1},\dots,n_{r}}$. In other words,
the set of (additive) Dunkl elements
$\big{\\{}\theta_{N_{j-1}+1}^{(N)},\ldots,\theta_{N_{j}}^{(N)}\big{\\}}$ plays
a role of the Chern roots of the tautological vector bundles $\xi_{j}$,
$j=1,\ldots,r$, over the partial flag variety
${{\cal{F}}l}_{n_{1},\ldots,n_{r}}$, see Section 4.1.2 for details. In a
similar fashion, the set of multiplicative Dunkl elements
$\big{\\{}\Theta_{N_{j-1}+1}^{(N)},\ldots,\Theta_{N_{j}}^{(N)}\big{\\}}$ plays
a role of the $K$-theoretic version of Chern roots of the tautological vector
bundle $\xi_{j}$ over the partial flag variety
${\cal{F}}l_{n_{1},\ldots,n_{r}}$. As a byproduct for a given set of weights
${\boldsymbol{\ell}}=\\{\ell_{ij}\\}_{1\leq i<j\leq r}$ we compute the Tutte
polynomial $T(K_{n_{1},\ldots,n_{k}}^{({\boldsymbol{\ell}})},x,y)$ of the
${\boldsymbol{\ell}}$-weighted complete multipartite graph
$K_{n_{1},\ldots,n_{k}}^{({\boldsymbol{\ell}})}$, see Section 4, Definition
4.4 and Theorem 4.3. More generally, we introduce universal Tutte polynomial

$\displaystyle T_{n}(\\{q_{ij}\\},x,y)\in\mathbb{Z}[\\{q_{ij}\\}][x,y]$

in such a way that for any collection of non-negative integers
${\boldsymbol{m}}=\\{m_{ij}\\}_{1\leq i<j\leq n}$ and a subgraph
$\Gamma\subset K_{n}^{({\boldsymbol{m}})}$ of the weighted complete graph on
$n$ labeled vertices with each edge $(i,j)\in K_{n}^{({\boldsymbol{m}})}$
appears with multiplicity $m_{ij}$, the specialization

$\displaystyle q_{ij}\longrightarrow 0\quad\text{if edge}\ \
(i,j)\notin\Gamma,\qquad
q_{ij}\longrightarrow[m_{ij}]_{y}:=\frac{y^{m_{ij}}-1}{y-1}\quad\text{if
edge}\ \ (i,j)\in\Gamma$

of the universal Tutte polynomial is equal to the Tutte polynomial of graph
$\Gamma$ multiplied by $(x-1)^{\kappa(\Gamma)}$, see Section 4.1.2, Theorem
4.24, and comments and examples, for details.

We also introduce and study a family of $($super$)$ $6$-term relations
algebras, and suggest a definition of “multiparameter quantum deformation” of
the algebra of the curvature of $2$-forms of the Hermitian linear bundles over
the complete flag variety ${\cal{F}}l_{n}$. This algebra can be treated as a
natural generalization of the (multiparameter) quantum cohomology ring
$QH^{*}({\cal{F}}l_{n})$, see Section 4.2. In a similar fashion as in the case
of three term relations algebras, for any subgraph $\Gamma\subset K_{n}$, one
(A.K.) can also define an algebra $6T^{(0)}(\Gamma)$ and projection333We treat
this map as an algebraic version of the homomorphism which sends the curvature
of a Hermitian vector bundle over a smooth algebraic variety to its cohomology
class, as well as a splitting of classical Yang–Baxter relations (that is six
term relations) in a couple of three term relations.

$\displaystyle\text{Ch}\colon\ 6T^{(0)}(\Gamma)\longrightarrow
3T^{(0)}(\Gamma).$

Note that subalgebra
${\cal{A}}(\Gamma):={\mathbb{Q}}[\theta_{1},\ldots,\theta_{n}]\subset
6T^{(0)}(\Gamma)^{ab}$ generated by additive Dunkl elements

$\displaystyle\theta_{i}=\sum_{j\atop(ij)\in E(\Gamma)}u_{ij}$

is closely related with problems have been studied in [118, 129], …, and [137]
in the case $\Gamma=K_{n}$, see Section 4.2.2. We want to draw attention of
the reader to the following problems related with arithmetic Schubert444See
for example [137] and the literature quoted therein. and Grothendieck calculi:

  1. (i)

Describe (natural) quotient $6T^{\dagger}(\Gamma)$ of the algebra
$6T^{(0)}(\Gamma)$ such that the natural epimorphism ${\rm
pr}\colon{\mathbb{A}}(\Gamma)\longrightarrow{\cal{A}}(\Gamma)$ turns out to be
isomorphism, where we denote by ${\mathbb{A}}(\Gamma)$ a subalgebra of
$6T^{\dagger}(\Gamma)$ generated over ${\mathbb{Q}}$ by additive Dunkl
elements.

  2. (ii)

It is not difficult to see [72] that multiplicative Dunkl elements
$\\{\Theta_{i}\\}_{1\leq i\leq n}$ also mutually commute in the algebra
$6T^{(0)}$, cf. Section 3.2. Problem we are interested in is to describe
commutative subalgebras generated by multiplicative Dunkl elements in the
algebras $6T^{\dagger}(\Gamma)$ and $6T^{(0)}(\Gamma)^{ab}$. In the latter
case one will come to the $K$-theoretic version of algebras studied in [118],
….

Yet another objective of our paper555This part of our paper had its origin in
the study/computation of relations among the additive and multiplicative Dunkl
elements in the quadratic algebras we are interested in, as well as the
author’s attempts to construct a monomial basis in the algebra $3T_{n}^{(0)}$
and find its Hilbert series for $n\geq 6$. As far as I’m aware these problems
are still widely open. is to describe several combinatorial properties of some
special elements in the associative quasi-classical Yang–Baxter algebras [72],
including among others, the so-called Coxeter element and the longest element.
In the case of Coxeter element we relate the corresponding reduced polynomials
introduced in [133, Exercise 6.C5(c)], and independently in [72], cf. [70],
with the $\beta$-Grothendieck polynomials [42] for some special permutations
$\pi_{k}^{(n)}$. More generally, we identify the $\beta$-Grothendieck
polynomial $\mathfrak{G}_{\pi_{k}^{(n)}}^{(\beta)}(X_{n})$ with a certain
weighted sum running over the set of $k$-dissections of a convex
$(n+k+1)$-gon. In particular we show that the specialization
$\mathfrak{G}_{\pi_{k}^{(n)}}^{(\beta)}(1)$ of the $\beta$-Grothendieck
polynomial $\mathfrak{G}_{\pi_{k}^{(n)}}^{(\beta)}(X_{n})$ counts the number
of $k$-dissections of a convex $(n+k+1)$-gon according to the number of
diagonals involved. When the number of diagonals in a $k$-dissection is the
maximal possible (equals to $n(2k-1)-1$), we recover the well-known fact that
the number of $k$-triangulations of a convex $(n+k+1)$-gon is equal to the
value of a certain Catalan–Hankel determinant, see, e.g., [129]. In Section
5.4.2 we study multiparameter generalizations of reduced polynomials
associated with Coxeter elements.

We also show that for a certain $5$-parameters family of vexillary
permutations, the specialization $x_{i}=1$, $\forall\,i\geq 1$, of the
corresponding $\beta$-Schubert polynomials
${{\mathfrak{S}}}_{w}^{(\beta)}(X_{n})$ turns out to be coincide either with
the Fuss–Narayana polynomials and their generalizations, or with a
$(q,\beta)$-deformation of $\rm VSASM$ or that of $\rm CSTCPP$ numbers, see
Corollary 5.33B. As examples we show that

  1. (a)

the reduced polynomial corresponding to a monomial $x_{12}^{n}x_{23}^{m}$
counts the number of $(n,m)$-Delannoy paths according to the number of
$NE$-steps, see Lemma 5.81;

  2. (b)

if $\beta=0$, the reduced polynomial corresponding to monomial
$(x_{12}x_{23})^{n}x_{34}^{k}$, $n\geq k$, counts the number of $n$ up, $n$
down permutations in the symmetric group ${\mathbb{S}}_{2n+k+1}$, see
Proposition 5.82; see also Conjecture 5.83.

We also point out on a conjectural connection between the sets of maximal
compatible sequences for the permutation $\sigma_{n,2n,2,0}$ and that
$\sigma_{n,2n+1,2,0}$ from one side, and the set of ${\rm VSASM}(n)$ and that
of ${\rm CSTCPP}(n)$ correspondingly, from the other, see Comments 5.48 for
details. Finally, in Sections 5.1.1 and 5.4.1 we introduce and study a
multiparameter generalization of reduced polynomials considered in [133,
Exercise 6.C5(c)], as well as that of the Catalan, Narayana and (small)
Schröder numbers.

In the case of the longest element we relate the corresponding reduced
polynomial with the Ehrhart polynomial of the Chan–Robbins–Yuen polytope, see
Section 5.3. More generally, we relate the $(t,\beta)$-reduced polynomial
corresponding to monomial

$\displaystyle\prod_{j=1}^{n-1}x_{j,j+1}^{a_{j}}\prod_{j=2}^{n-2}\left(\prod_{k=j+2}^{n}x_{jk}\right),\qquad
a_{j}\in\mathbb{Z}_{\geq 0},\qquad\forall\,j,$

with positive $t$-deformations of the Kostant partition function and that of
the Ehrhart polynomial of some flow polytopes, see Section 5.3.

In Section 5.4 we investigate reduced polynomials associated with certain
monomials in the algebra $({\widehat{{\rm ACYB}}})^{ab}_{n}(\beta)$, known
also as Gelfand–Varchenko algebra [67, 72], and study its combinatorial
properties. Our main objective in Section 5.4.2 is to study reduced
polynomials for Coxeter element treated in a certain multiparameter
deformation of the (noncommutative) quadratic algebra ${\widehat{{\rm
ACYB}}}_{n}(\alpha,\beta)$. Namely, to each dissection of a convex $(n+2)$-gon
we associate a certain weight and consider the generating function of all
dissections of $(n+2)$-gon selected taken with that weight. One can show that
the reduced polynomial corresponding to the Coxeter element in the deformed
algebra is equal to that generating function. We show that certain
specializations of that reduced polynomial coincide, among others, with the
Grothendieck polynomials corresponding to the permutation $1\times
w_{0}^{(n-1)}\in\mathbb{S}_{n}$, the Lagrange inversion formula, as well as
give rise to combinatorial (i.e., positive expressions) multiparameters
deformations of Catalan and Fuss–Catalan, Motzkin, Riordan and Fine numbers,
Schröder numbers and Schröder trees. We expect (work in progress) a similar
connections between Schubert and Grothendieck polynomials associated with the
Richardson permutations $1^{k}\times w_{0}^{(n-k)}$, $k$-dissections of a
convex $(n+k+1)$-gon investigated in the present paper, and $k$-dimensional
Lagrange–Good inversion formula studied from combinatorial point of view,
e.g., in [22, 50].

##  1 Introduction

The Dunkl operators have been introduced in the later part of 80’s of the last
century by Charles Dunkl [35, 36] as a powerful mean to study of harmonic and
orthogonal polynomials related with finite Coxeter groups. In the present
paper we don’t need the definition of Dunkl operators for arbitrary (finite)
Coxeter groups, see, e.g., [35], but only for the special case of the
symmetric group ${\mathbb{S}}_{n}$.

######  Definition 1.1.

Let $P_{n}=\mathbb{C}[x_{1},\ldots,x_{n}]$ be the ring of polynomials in
variables $x_{1},\ldots,x_{n}$. The type $A_{n-1}$ (additive) rational Dunkl
operators $D_{1},\ldots,D_{n}$ are the differential-difference operators of
the following form

$\displaystyle D_{i}=\lambda{\partial\over\partial
x_{i}}+\sum_{j\not=i}{1-s_{ij}\over x_{i}-x_{j}},$ (1.1)

Here $s_{ij}$, $1\leq i<j\leq n$, denotes the exchange (or permutation)
operator, namely,

$\displaystyle
s_{ij}(f)(x_{1},\ldots,x_{i},\ldots,x_{j},\ldots,x_{n})=f(x_{1},\ldots,x_{j},\ldots,x_{i},\ldots,x_{n}),$

${\partial\over\partial x_{i}}$ stands for the derivative w.r.t. the variable
$x_{i}$, $\lambda\in\mathbb{C}$ is a parameter.

The key property of the Dunkl operators is the following result.

######  Theorem 1.2 (C. Dunkl [35]).

For any finite Coxeter group $(W,S)$, where $S=\\{s_{1},\ldots,s_{l}\\}$
denotes the set of simple reflections, the Dunkl operators $D_{i}:=D_{s_{i}}$
and $D_{j}:=D_{s_{j}}$ pairwise commute: $D_{i}D_{j}=D_{j}D_{i}$, $1\leq
i,j\leq l$.

Another fundamental property of the Dunkl operators which finds a wide variety
of applications in the theory of integrable systems, see, e.g., [56], is the
following statement: the operator

$\displaystyle\sum_{i=1}^{l}(D_{i})^{2}$

“essentially” coincides with the Hamiltonian of the rational Calogero–Moser
model related to the finite Coxeter group $(W,S)$.

######  Definition 1.3.

Truncated (additive) Dunkl operator (or the Dunkl operator at critical level),
denoted by ${\cal D}_{i}$, $i=1,\ldots,l$, is an operator of the form (1.1)
with parameter $\lambda=0$.

For example, the type $A_{n-1}$ rational truncated Dunkl operator has the
following form

$\displaystyle{\cal D}_{i}=\sum_{j\not=i}{1-s_{ij}\over x_{i}-x_{j}}.$

Clearly the truncated Dunkl operators generate a commutative algebra. The
important property of the truncated Dunkl operators is the following result
discovered and proved by C. Dunkl [36]; see also [8] for a more recent proof.

######  Theorem 1.4 (C. Dunkl [36], Yu. Bazlov [8]).

For any finite Coxeter group $(W,S)$ the algebra over $\mathbb{Q}$ generated
by the truncated Dunkl operators ${\cal D}_{1},\ldots,{\cal D}_{l}$ is
canonically isomorphic to the coinvariant algebra ${\cal{A}}_{W}$ of the
Coxeter group $(W,S)$.

Recall that for a finite crystallographic Coxeter group $(W,S)$ the
coinvariant algebra ${\cal{A}}_{W}$ is isomorphic to the cohomology ring
$H^{*}(G/B,\mathbb{Q})$ of the flag variety $G/B$, where $G$ stands for the
Lie group corresponding to the crystallographic Coxeter group $(W,S)$ we
started with.

######  Example 1.5.

In the case when $W={\mathbb{S}}_{n}$ is the symmetric group, Theorem 1.4
states that the algebra over $\mathbb{Q}$ generated by the truncated Dunkl
operators ${\cal D}_{i}=\sum\limits_{j\not=i}{1-s_{ij}\over x_{i}-x_{j}}$,
$i=1,\ldots,n$, is canonically isomorphic to the cohomology ring of the full
flag variety ${\cal F}l_{n}$ of type $A_{n-1}$

$\displaystyle\mathbb{Q}[{\cal D}_{1},\ldots,{\cal
D}_{n}]\cong\mathbb{Q}[x_{1},\ldots,x_{n}]/J_{n},$ (1.2)

where $J_{n}$ denotes the ideal generated by the elementary symmetric
polynomials $\\{e_{k}(X_{n}),\,1\leq k\leq n\\}$.

Recall that the elementary symmetric polynomials $e_{i}(X_{n})$,
$i=1,\ldots,n$, are defined through the generating function

$\displaystyle 1+\sum_{i=1}^{n}e_{i}(X_{n})t^{i}=\prod_{i=1}^{n}(1+tx_{i}),$

where we set $X_{n}:=(x_{1},\ldots,x_{n})$. It is well-known that in the case
$W=\mathbb{S}_{n}$, the isomorphism (1.2) can be defined over the ring of
integers $\mathbb{Z}$.

Theorem 1.4 by C. Dunkl has raised a number of natural questions:

  1. (A)

What is the algebra generated by the truncated

     * •

trigonometric,

     * •

elliptic,

     * •

super, matrix, …,

       1. (a)

additive Dunkl operators?

       2. (b)

Ruijsenaars–Schneider–Macdonald operators?

       3. (c)

Gaudin operators?

  2. (B)

Describe commutative subalgebra generated by the Jucys–Murphy elements in

     * •

the group ring of the symmetric group;

     * •

the Hecke algebra;

     * •

the Brauer algebra, ${\rm BMW}$ algebra, ….

  3. (C)

Does there exist an analogue of Theorem 1.4 for

     * •

classical and quantum equivariant cohomology and equivariant $K$-theory rings
of the partial flag varieties?

     * •

chomology and $K$-theory rings of affine flag varieties?

     * •

diagonal coinvariant algebras of finite Coxeter groups?

     * •

complex reflection groups?

The present paper is an extended introduction to a few items from Section 5 of
[72].

The main purpose of my paper “On some quadratic algebras, II” is to give some
partial answers on the above questions basically in the case of the symmetric
group ${\mathbb{S}}_{n}$.

The purpose of the present paper is to draw attention to an interesting class
of nonhomogeneous quadratic algebras closely connected (still mysteriously!)
with different branches of Mathematics such as classical and quantum Schubert
and Grothendieck calculi, low-dimensional topology, classical, basic and
elliptic hypergeometric functions, algebraic combinatorics and graph theory,
integrable systems, etc.

What we try to explain in [72] is that upon passing to a suitable
representation of the quadratic algebra in question, the subjects mentioned
above, are a manifestation of certain general properties of that quadratic
algebra.

From this point of view, we treat the commutative subalgebra generated (over a
universal Lazard ring ${\mathbb{L}}_{n}$ [88]) by the additive (resp.
multiplicative) truncated Dunkl elements in the algebra $3T_{n}(\beta)$, see
Definition 3.1, as universal cohomology (resp. universal $K$-theory) ring of
the complete flag variety ${\cal F}l_{n}$. The classical or quantum cohomology
(resp. the classical or quantum $K$-theory) rings of the flag variety ${\cal
F}l_{n}$ are certain quotients of that universal ring.

For example, in [74] we have computed relations among the (truncated) Dunkl
elements $\\{\theta_{i},\,i=1,\ldots,n\\}$ in the elliptic representation of
the algebra $3T_{n}(\beta=0)$. We expect that the commutative subalgebra
obtained is isomorphic to elliptic cohomology ring (not defined yet, but see
[48, 52]) of the flag variety ${\cal F}l_{n}$.

Another example from [72]. Consider the algebra $3T_{n}(\beta=0)$. One can
prove [72] the following identities in the algebra $3T_{n}(\beta=0)$:

  1. (A)

summation formula

$\displaystyle\sum_{j=1}^{n-1}\left(\prod_{b=j+1}^{n-1}u_{b,b+1}\right)u_{1,n}\left(\prod_{b=1}^{j-1}u_{b,b+1}\right)=\prod_{a=1}^{n-1}u_{a,a+1};$

  2. (B)

duality transformation formula, let $m\leq n$, then

$\displaystyle\sum_{j=m}^{n-1}\left(\prod_{b=j+1}^{n-1}u_{b,b+1}\right)\left[\prod_{a=1}^{m-1}u_{a,a+n-1}u_{a,a+n}\right]u_{m,m+n-1}\left(\prod_{b=m}^{j-1}u_{b,b+1}\right)$
$\displaystyle\qquad\quad{}+\sum_{j=2}^{m}\left[\prod_{a=j}^{m-1}u_{a,a+n-1}u_{a,a+n}\right]u_{m,n+m-1}\left(\prod_{b=m}^{n-1}u_{b,b+1}\right)u_{1,n}$
$\displaystyle\qquad{}=\sum_{j=1}^{m}\left[\prod_{a=1}^{m-j}u_{a,a+n}u_{a+1,a+n}\right]\left(\prod_{b=m}^{n-1}u_{b,b+1}\right)\left[\prod_{a=1}^{j-1}u_{a,a+n-1}u_{a,a+n}\right].$

One can check that upon passing to the elliptic representation of the algebra
$3T_{n}(\beta=0)$, see Section 3.1 or [74], for the definition of elliptic
representation, the above identities (A) and (B) finally end up
correspondingly, to be the summation formula and the $N=1$ case of the duality
transformation formula for multiple elliptic hypergeometric series (of type
$A_{n-1})$, see, e.g., [63] or Appendix A.6 for the explicit forms of the
latter. After passing to the so-called Fay representation [72], the identities
(A) and (B) become correspondingly to be the summation formula and duality
transformation formula for the Riemann theta functions of genus $g>0$ [72].
These formulas in the case $g\geq 2$ seems to be new.

Worthy to mention that the relation (A) above can be treated as a “non-
commutative analogue” of the well-known recurrence relation among the Catalan
numbers. The study of “descendent relations” in the quadratic algebras in
question was originally motivated by the author attempts to construct a
monomial basis in the algebra $3T_{n}^{(0)}$, and compute ${\rm
Hilb}(3T_{n}^{(0)},t)$ for $n\geq 6$. These problems are still widely open,
but gives rise the author to discovery of several interesting connections with

  * •

classical and quantum Schubert and Grothendieck calculi,

  * •

combinatorics of reduced decomposition of some special elements in the
symmetric group,

  * •

combinatorics of generalized Chan–Robbins–Yuen polytopes,

  * •

relations among the Dunkl and Gaudin elements,

  * •

computation of Tutte and chromatic polynomials of the weighted complete
multipartite graphs, etc.

A few words about the content of the present paper. Example 1.5 can be viewed
as an illustration of the main problems we are treated in Sections 2 and 3 of
the present paper, namely the following ones.

  * •

Let $\\{u_{ij},\,1\leq i,j\leq n\\}$ be a set of generators of a certain
algebra over a commutative ring $K$. The first problem we are interested in is
to describe “a natural set of relations” among the generators
$\\{u_{ij}\\}_{1\leq i,j\leq n}$ which implies the pair-wise commutativity of
dynamical Dunkl elements

$\displaystyle\theta_{i}=\theta_{i}^{(n)}=:\sum\limits_{j=1}^{n}u_{ij},\qquad
1\leq i\leq n.$

  * •

Should this be the case then we are interested in to describe the algebra
generated by “the integrals of motions”, i.e., to describe the quotient of the
algebra of polynomials $K[y_{1},\ldots,y_{n}]$ by the two-sided ideal
${\cal{J}}_{n}$ generated by non-zero polynomials $F(y_{1},\ldots,y_{n})$ such
that $F(\theta_{1},\ldots,\theta_{n})=0$ in the algebra over ring $K$
generated by the elements $\\{u_{ij}\\}_{1\leq i,j\leq n}$.

  * •

We are looking for a set of additional relations which imply that the
elementary symmetric polynomials $e_{k}(Y_{n})$, $1\leq k\leq n$, belong to
the set of integrals of motions. In other words, the value of elementary
symmetric polynomials $e_{k}(y_{1},\ldots,y_{n})$, $1\leq k\leq n$, on the
Dunkl elements $\theta_{1}^{(n)},\ldots,\theta_{n}^{(n)}$ do not depend on the
variables $\\{u_{ij},\,1\leq i\not=j\leq n\\}$. If so, one can defined
deformation of elementary symmetric polynomials, and make use of it and the
Jacobi–Trudi formula, to define deformed Schur functions, for example. We try
to realize this program in Sections 2 and 3.

In Section 2, see Definition 2.3, we introduce the so-called dynamical
classical Yang–Baxter algebra as “a natural quadratic algebra” in which the
Dunkl elements form a pair-wise commuting family. It is the study of the
algebra generated by the (truncated) Dunkl elements that is the main objective
of our investigation in [72] and the present paper. In Section 2.1 we describe
few representations of the dynamical classical Yang–Baxter algebra ${\rm
DCYB}_{n}$ related with

  * •

quantum cohomology $QH^{*}({\cal{F}}l_{n})$ of the complete flag variety
${\cal{F}}l_{n}$, cf. [41];

  * •

quantum equivariant cohomology $QH^{*}_{T^{n}\times
C^{*}}(T^{*}{\cal{F}}l_{n})$ of the cotangent bundle $T^{*}{\cal{F}}l_{n}$ to
the complete flag variety, cf. [54];

  * •

Dunkl–Gaudin and Dunkl–Uglov representations, cf. [108, 138].

In Section 3, see Definition 3.1, we introduce the algebra $3HT_{n}(\beta)$,
which seems to be the most general (noncommutative) deformation of the (even)
Orlik–Solomon algebra of type $A_{n-1}$, such that it’s still possible to
describe relations among the Dunkl elements, see Theorem 3.8. As an
application we describe explicitly a set of relations among the (additive)
Gaudin/Dunkl elements, cf. [108]. It should be stressed at this place that we
treat the Gaudin elements/operators (either additive or multiplicative) as
images of the universal Dunkl elements/operators (additive or multiplicative)
in the Gaudin representation of the algebra $3HT_{n}(0)$. There are several
other important representations of that algebra, for example, the
Calogero–Moser, Bruhat, Buchstaber–Felder–Veselov (elliptic), Fay trisecant
($\tau$-functions), adjoint, and so on, considered (among others) in [72].
Specific properties of a representation chosen666For example, in the cases of
either Calogero–Moser or Bruhat representations one has an additional
constraint, namely, $u_{ij}^{2}=0$ for all $i\not=j$. In the case of Gaudin
representation one has an additional constraint $u_{ij}^{2}=p_{ij}^{2}$, where
the (quantum) parameters $\\{p_{ij}={1\over x_{i}-x_{j}},\,i\not=j\\}$,
satisfy simultaneously the Arnold and Plücker relations, see Section 2, II.
Therefore, the (small) quantum cohomology ring of the type $A_{n-1}$ full flag
variety ${{\cal F}l}_{n}$ and the Bethe subalgebra(s) (i.e., the subalgebra
generated by Gaudin elements in the algebra $3HT_{n}(0)$) correspond to
different specializations of “quantum parameters” $\\{q_{ij}:=u_{ij}^{2}\\}$
of the universal cohomology ring (i.e., the subalgebra/ring in $3HT_{n}(0)$
generated by (universal) Dunkl elements). For more details and examples, see
Section 2.1 and [72]. (e.g., Gaudin representation) imply some additional
relations among the images of the universal Dunkl elements (e.g., Gaudin
elements) should to be unveiled.

We start Section 3 with definition of algebra $3T_{n}(\beta)$ and its “Hecke”
$3HT_{n}(\beta)$ and “elliptic” $3MT_{n}(\beta)$ quotients. In particular we
define an elliptic representation of the algebra $3T_{n}(0)$ [74], and show
how the well-known elliptic solutions of the quantum Yang–Baxter equation due
to A. Belavin and V. Drinfeld, see, e.g., [9], S. Shibukawa and K. Ueno [130],
and G. Felder and V. Pasquier [40], can be plug in to our construction, see
Section 3.1. At the end of Section 3.1.1 we point out on a mysterious (at
least for the author) appearance of the Euler numbers and “traces” of the
Brauer algebra in the equivariant Pieri rules hold for the algebra
$3TM_{n}(\beta,{\boldsymbol{q}},\psi)$ stated in Theorem 3.8.

In Section 3.2 we introduce a multiplicative analogue of the Dunkl elements
$\\{\Theta_{j}\in 3T_{n}(\beta)$, $1\leq j\leq n\\}$ and describe the
commutative subalgebra in the algebra $3T_{n}(\beta)$ generated by
multiplicative Dunkl elements [76]. The latter commutative subalgebra turns
out to be isomorphic to the quantum equivariant $K$-theory of the complete
flag variety ${\cal{F}}l_{n}$ [76].

In Section 3.3 we describe relations among the truncated Dunkl–Gaudin
elements. In this case the quantum parameters $q_{ij}=p_{ij}^{2}$, where
parameters $\\{p_{ij}=(z_{i}-z_{j})^{-1},\,1\leq i<j\leq n\\}$ satisfy the
both Arnold and Plücker relations. This observation has made it possible to
describe a set of additional rational relations among the Dunkl–Gaudin
elements, cf. [108].

In Section 3.4 we introduce an equivariant version of multiplicative Dunkl
elements, called shifted Dunkl elements in our paper, and describe (some)
relations among the latter. This result is a generalization of that obtained
in Section 3.1 and [76]. However we don’t know any geometric interpretation of
the commutative subalgebra generated by shifted Dunkl elements.

In Section 4.1 for any subgraph $\Gamma\subset K_{n}$ of the complete graph
$K_{n}$ we introduce777Independently the algebra $3T_{n}^{(0)}(\Gamma)$ has
been studied in [16], where the reader can find some examples and conjectures.
[70, 72], algebras $3T_{n}(\Gamma)$ and $3T_{n}^{(0)}(\Gamma)$ which can be
seen as analogues of algebras $3T_{n}$ and $3T_{n}^{(0)}$ correspondingly888To
avoid confusions, it must be emphasized that the defining relations for
algebras $3T_{n}(\Gamma)$ and $3T_{n}(\Gamma)^{(0)}$ may have more then three
terms..

We want to point out in the Introduction, cf. footnote 1, that an analog of
the algebras $3T_{n}$ and $3T_{n}^{(\beta)}$, $3HT_{n}$, etc. treated in the
present paper, can be defined for any (oriented or not) matroid $\cal{M}$. We
denote these algebras as $3T({\cal{M}})$ and $3T^{(\beta)}({\cal{M}})$. One
can show (A.K.) that the abelianization of the algebra
$3T^{(\beta)}({\cal{M}})$, denoted by ${3T^{(\beta)}({\cal{M}})}^{ab}$, is
isomorphic to the Gelfand–Varchenko algebra corresponding to a matroid
$\cal{M}$, whereas the algebra ${3T^{(\beta=0)}({\cal{M}})}^{ab}$ is
isomorphic to the (even) Orlik–Solomon algebra ${\rm OS}^{+}({\cal{M}})$ of a
matroid $\cal{M}$.999For a definition and basic properties of the
Orlik–Solomon algebra corresponding to a matroid, see, e.g., [49, 65]. We
consider and treat the algebras $3T({\cal{M}})$, $3HT({\cal{M}})$, …, as
equivariant noncommutative (or quantum) versions of the (even) Orlik–Solomon
algebras associated with matroid (including hyperplane, graphic, …
arrangements). However a meaning of a quantum deformation of the (even or odd)
Orlik–Solomon algebra suggested in the present paper, is missing, even for the
braid arrangement of type $A_{n}$. Generalizations of the Gelfand–Varchenko
algebra has been suggested and studied in [67, 72] and in the present paper
under the name quasi-associative Yang–Baxter algebra, see Section 5.

In the present paper we basically study the abelian quotient of the algebra
$3T_{n}^{(0)}(\Gamma)$, where graph $\Gamma$ has no loops and multiple edges,
since we expect some applications of our approach to the theory of chromatic
polynomials of planar graphs, in particular to the complete multipartite
graphs $K_{n_{1},\ldots,n_{r}}$ and the grid graphs $G_{m,n}$.101010See
http://reference.wolfram.com/language/ref/GridGraph.html for a definition of
grid graph $G_{m,n}$. Our main results hold for the complete multipartite,
cyclic and line graphs. In particular we compute their chromatic and Tutte
polynomials, see Proposition 4.19 and Theorem 4.24. As a byproduct we compute
the Tutte polynomial of the ${\boldsymbol{\ell}}$-weighted complete
multipartite graph $K_{n_{1},\ldots,n_{r}}^{({\boldsymbol{\ell}})}$ where
${\boldsymbol{\ell}}=\\{\ell_{ij}\\}_{1\leq i<j\leq r}$, is a collection of
weights, i.e., a set of non-negative integers.

More generally, for a set of variables $\\{\\{q_{ij}\\}_{1\leq i<j\leq
n},x,y\\}$ we define universal Tutte polynomial
$T_{n}(\\{q_{ij}\\},x,y)\in\mathbb{Z}[q_{ij}][x,y]$ such that for any
collection of non-negative integers $\\{m_{ij}\\}_{1\leq i<j\leq n}$ and a
subgraph $\Gamma\subset K_{n}^{({\boldsymbol{m}})}$ of the complete graph
$K_{n}$ with each edge $(i,j)$ comes with multiplicity $m_{ij}$, the
specialization

$\displaystyle q_{ij}\longrightarrow 0\quad\text{if~{}edge}\ \
(i,j)\notin\Gamma,\qquad
q_{ij}\longrightarrow[m_{ij}]_{y}:=\frac{y^{m_{ij}}-1}{y-1}\quad\text{if~{}edge}\
\ (i,j)\in\Gamma$

of the universal Tutte polynomial $T_{n}(\\{q_{ij}\\},x,y)$ is equal to the
Tutte polynomial of graph $\Gamma$ multiplied by the factor
$(t-1)^{\kappa(\Gamma)}$:

$\displaystyle(x-1)^{\kappa(\Gamma)}{\rm
Tutte}(\Gamma,x,y):=T_{n}(\\{q_{ij}\\},x,y)\bigg{|}_{q_{ij}=0\,\text{if}\,(i,j)\notin\Gamma\atop
q_{ij}={[m_{ij}]}_{y}\,\text{if}\,(i,j)\in\Gamma}.$

Here and after $\kappa(\Gamma)$ demotes the number of connected components of
a graph $\Gamma$. In other words, one can treat the universal Tutte polynomial
$T_{n}(\\{q_{ij}\\},x,y)$ as a “reproducing kernel” for the Tutte polynomials
of all (loop-less) graphs with the number of vertices not exceeded $n$.

We also state Conjecture 35 that for any loopless graph $\Gamma$ (possibly
with multiple edges) the algebra ${3T_{|\Gamma|}^{(0)}(\Gamma)}^{ab}$ is
isomorphic to the even Orlik–Solomon algebra ${\rm
OS}^{+}({\cal{A}}_{\Gamma})$ of the graphic arrangement associated with graph
$\Gamma$ in question111111For simple graphs, i.e., without loops and multiple
edges, this conjecture has been proved in [89]..

At the end we emphasize that the case of the complete graph $\Gamma=K_{n}$
reproduces the results of the present paper and those of [72], i.e., the case
of the full flag variety ${\cal F}l_{n}$. The case of the complete
multipartite graph $\Gamma=K_{n_{1},\ldots,n_{r}}$ reproduces the analogue of
results stated in the present paper for the case of full flag variety ${\cal
F}l_{n}$, to the case of the partial flag variety ${\cal
F}_{n_{1},\ldots,n_{r}}$, see [72] for details.

In Section 4.1.4 we sketch how to generalize our constructions and some of our
results to the case of the Lie algebras of classical types121212One can define
an analogue of the algebra $3T_{n}^{(0)}$ for the root system of $BC_{n}$ and
$C_{n}^{\vee}C_{n}$-types as well, but we are omitted these cases in the
present paper..

In Section 4.2 we briefly overview our results concerning yet another
interesting family of quadratic algebras, namely the six-term relations
algebras $6T_{n}$, $6T_{n}^{(0)}$ and related ones. These algebras also
contain a distinguished set of mutually commuting elements called Dunkl
elements $\\{\theta_{i},\,i=1,\ldots,n\\}$ given by
$\theta_{i}=\sum\limits_{j\not=i}r_{ij}$, see Definition 4.48.

In Section 4.2.2 we introduce and study the algebra $6T_{n}^{\bigstar}$ in
greater detail. In particular we introduce a “quantum deformation” of the
algebra generated by the curvature of $2$-forms of of the Hermitian linear
bundles over the flag variety ${\cal{F}}l_{n}$, cf. [118].

In Section 4.2.3 we state our results concerning the classical Yang–Baxter
algebra ${\rm CYB}_{n}$ and the $6$-term relation algebra $6T_{n}$. In
particular we give formulas for the Hilbert series of these algebras. These
formulas have been obtained independently in [7] The paper just mentioned,
contains a description of a basis in the algebra $6T_{n}$, and much more.

In Section 4.2.4 we introduce a super analog of the algebra $6T_{n}$, denoted
by $6T_{n,m}$, and compute its Hilbert series.

Finally, in Section 4.3 we introduce extended nil-three term relations algebra
${3\mathfrak{T}}_{n}$ and describe a subalgebra inside of it which is
isomorphic to the double affine Hecke algebra of type $A_{n-1}$, cf. [24].

In Section 5 we describe several combinatorial properties of some special
elements in the associative quasi-classical Yang–Baxter algebra131313The
algebra $\widehat{{\rm ACYB}}_{n}$ can be treated as “one-half” of the algebra
$3T_{n}(\beta)$. It appears that the basic relations among the Dunkl elements,
which do not mutually commute anymore, are still valid, see Lemma 5.3.,
denoted by $\widehat{{\rm ACYB}}_{n}$. The main results in that direction were
motivated and obtained as a by-product, in the process of the study of the the
structure of the algebra $3HT_{n}(\beta)$. More specifically, the main results
of Section 5 were obtained in the course of “hunting for descendant relations”
in the algebra mentioned, which is an important problem to be solved to
construct a basis in the nil-quotient algebra $3T_{n}^{(0)}$. This problem is
still widely-open.

The results of Section 5.1, see Proposition 5.4, items (1)–(5), are more or
less well-known among the specialists in the subject, while those of the item
(6) seem to be new. Namely, we show that the polynomial $Q_{n}(x_{ij}=t_{i})$
from [133, Exercise 6.C8(c)], essentially coincides with the
$\beta$-deformation [42] of the Lascoux–Schützenberger Grothendieck polynomial
[86] for some particular permutation. The results of Proposition 5.4(6), point
out on a deep connection between reduced forms of monomials in the algebra
$\widehat{{\rm ACYB}}_{n}$ and the Schubert and Grothendieck calculi. This
observation was the starting point for the study of some combinatorial
properties of certain specializations of the Schubert, the
$\beta$-Grothendieck [43] and the double $\beta$-Grothendieck polynomials in
Section 5.2. One of the main results of Section 5.2 can be stated as follows.

######  Theorem 1.6.

  1. $(1)$

Let $w\in\mathbb{S}_{n}$ be a permutation, consider the specialization
$x_{1}:=q$, $x_{i}=1$, $\forall\,i\geq 2$, of the $\beta$-Grothendieck
polynomial $\mathfrak{G}_{w}^{(\beta)}(X_{n})$. Then

$\displaystyle{\cal{R}}_{w}(q,\beta+1):=\mathfrak{G}_{w}^{(\beta)}(x_{1}=q,\,x_{i}=1,\,\forall\,i\geq
2)\in\mathbb{N}[q,1+\beta].$

In other words, the polynomial ${\cal{R}}_{w}(q,\beta)$ has non-negative
integer coefficients141414For a more general result see Appendix A.1,
Corollary A.7..

For late use we define polynomials

$\displaystyle\mathfrak{R}_{w}(q,\beta):=q^{1-w(1)}{\cal{R}}_{w}(q,\beta).$

  2. $(2)$

Let $w\in\mathbb{S}_{n}$ be a permutation, consider the specialization
$x_{i}:=q$, $y_{i}=t$, $\forall\,i\geq 1$, of the double $\beta$-Grothendieck
polynomial $\mathfrak{G}_{w}^{(\beta)}(X_{n},Y_{n})$. Then

$\displaystyle\mathfrak{G}_{w}^{(\beta-1)}(x_{i}:=q,\,y_{i}:=t,\,\forall\,i\geq
1)\in\mathbb{N}[q,t,\beta].$

  3. $(3)$

Let $w$ be a permutation, then

$\displaystyle\mathfrak{R}_{w}(1,\beta)=\mathfrak{R}_{1\times w}(0,\beta).$

Note that ${\cal{R}}_{w}(1,\beta)={\cal{R}}_{w^{-1}}(1,\beta)$, but
${\cal{R}}_{w}(t,\beta)\not={\cal{R}}_{w^{-1}}(t,\beta)$, in general.

For the reader convenience we collect some basic definitions and results
concerning the $\beta$-Grothendieck polynomials in Appendix A.1.

Let us observe that $\mathfrak{R}_{w}(1,1)=\mathfrak{S}_{w}(1)$, where
$\mathfrak{S}_{w}(1)$ denotes the specialization $x_{i}:=1$, $\forall\,i\geq
1$, of the Schubert polynomial $\mathfrak{S}_{w}(X_{n})$ corresponding to
permutation $w$. Therefore, $\mathfrak{R}_{w}(1,1)$ is equal to the number of
compatible sequences [13] (or pipe dreams, see, e.g., [129]) corresponding to
permutation $w$.

######  Problem 1.7.

Let $w\in\mathbb{S}_{n}$ be a permutation and $l:=\ell(w)$ be its length.
Denote by ${\rm CS}(w)=\\{{\boldsymbol{a}}=(a_{1}\leq a_{2}\leq\cdots\leq
a_{l})\in\mathbb{N}^{l}\\}$ the set of compatible sequences [13] corresponding
to permutation $w$.

  * •

Define statistics $r({\boldsymbol{a}})$ on the set of all compatible sequences
${\rm CS}_{n}:=\coprod\limits_{{w\in\mathbb{S}_{n}}}{\rm CS}(w)$ in a such way
that

$\displaystyle\sum_{{\boldsymbol{a}}\in{\rm
CS}(w)}q^{a_{1}}\beta^{r({\boldsymbol{a}})}={\cal{R}}_{w}(q,\beta).$

  * •

Find a geometric interpretation, and investigate combinatorial and algebra-
geometric properties of polynomials $\mathfrak{S}_{w}^{(\beta)}(X_{n})$, where
for a permutation $w\in\mathbb{S}_{n}$ we denoted by
$\mathfrak{S}_{w}^{(\beta)}(X_{n})$ the $\beta$-Schubert polynomial defined as
follows

$\displaystyle\mathfrak{S}_{w}^{(\beta)}(X_{n})=\sum_{{\boldsymbol{a}}\in{\rm
CS}(w)}\beta^{r({\boldsymbol{a}})}\prod_{i=1}^{l:=\ell(w)}x_{a_{i}}.$

We expect that polynomial $\mathfrak{S}_{w}^{(\beta)}(1)$ coincides with the
Hilbert polynomial of a certain graded commutative ring naturally associated
to permutation $w$.

######  Remark 1.8.

It should be mentioned that, in general, the principal specialization

$\displaystyle\mathfrak{G}_{w}^{(\beta-1)}\big{(}x_{i}:=q^{i-1},\,\forall\,i\geq
1\big{)}$

of the $(\beta-1)$-Grothendieck polynomial may have negative coefficients.

Our main objective in Section 5.2 is to study the polynomials
$\mathfrak{R}_{w}(q,\beta)$ for a special class of permutations in the
symmetric group $\mathbb{S}_{\infty}$. Namely, in Section 5.2 we study some
combinatorial properties of polynomials
$\mathfrak{R}_{\varpi_{\lambda,\phi}}(q,\beta)$ for the five parameters family
of vexillary permutations $\\{\varpi_{\lambda,\phi}\\}$ which have the shape
$\lambda:=\lambda_{n,p,b}=(p(n-i+1)+b$, $i=1,\ldots,n+1)$ and flag
$\phi:=\phi_{k,r}=(k+r(i-1),~{}i=1,\ldots,n+1)$.

This class of permutations is notable for many reasons, including that the
specialized value of the Schubert polynomial
$\mathfrak{S}_{\varpi_{\lambda,\phi}}(1)$ admits a nice product
formula151515One can prove a product formula for the principal specialization
$\mathfrak{S}_{\varpi_{\lambda,\phi}}(x_{i}:=q^{i-1},\,\forall\,i\geq 1)$ of
the corresponding Schubert polynomial. We don’t need a such formula in the
present paper., see Theorem 5.29. Moreover, we describe also some interesting
connections of polynomials $\mathfrak{R}_{\varpi_{\lambda,\phi}}(q,\beta)$
with plane partitions, the Fuss–Catalan numbers161616We define the
(generalized) Fuss–Catalan numbers to be ${\rm FC}_{n}^{(p)}(b):={1+b\over
1+b+(n-1)p}{np+b\choose n}$. Connection of the Fuss–Catalan numbers with the
$p$-ballot numbers ${\rm Bal}_{p}(m,n):={n-mp+1\over n+m+1}~{}{n+m+1\choose
m}$ and the Rothe numbers $R_{n}(a,b):={a\over a+bn}{a+bn\choose n}$ can be
described as follows  $\displaystyle{\rm FC}_{n}^{(p)}(b)=R_{n}(b+1,p)={\rm
Bal}_{p-1}(n,(n-1)p+b).$ and Fuss–Narayana polynomials, $k$-triangulations and
$k$-dissections of a convex polygon, as well as a connection with two families
of ${\rm ASM}$. For example, let $\lambda=(b^{n})$ and $\phi=(k^{n})$ be
rectangular shape partitions, then the polynomial
$\mathfrak{R}_{\varpi_{\lambda,\phi}}(q,\beta)$ defines a
$(q,\beta)$-deformation of the number of (ordinary) plane partitions171717Let
$\lambda$ be a partition. An ordinary plane partition (plane partition for
short)bounded by $d$ and shape $\lambda$ is a filling of the shape $\lambda$
by the numbers from the set $\\{0,1,\ldots,d\\}$ in such a way that the
numbers along columns and rows are weakly decreasing. A reverse plane
partition bounded by $d$ and shape $\lambda$ is a filling of the shape
$\lambda$ by the numbers from the set $\\{0,1,\ldots,d\\}$ in such a way that
the numbers along columns and rows are weakly increasing. sitting in the box
$b\times k\times n$. It seems an interesting problem to find an algebra-
geometric interpretation of polynomials $\mathfrak{R}_{w}(q,\beta)$ in the
general case.

######  Question 1.9.

Let $a$ and $b$ be mutually prime positive integers. Does there exist a family
of permutations $w_{a,b}\in{\mathbb{S}}_{ab(a+b)}$ such that the
specialization $x_{i}=1$, $\forall\,i$ of the Schubert polynomial
${\mathfrak{S}}_{w_{a,b}}$ is equal to the rational Catalan number $C_{a/b}$?
That is

$\displaystyle{\mathfrak{S}}_{w_{a,b}}(1)={1\over a+b}{a+b\choose a}.$

Many of the computations in Section 5.2 are based on the following
determinantal formula for $\beta$-Grothendieck polynomials corresponding to
grassmannian permutations, cf. [84].

######  Theorem 1.10 (see Comments 5.37(b)).

If $w=\sigma_{\lambda}$ is the grassmannian permutation with shape
$\lambda=(\lambda_{,}\ldots,\lambda_{n})$ and a unique descent at position
$n$, then181818The equality
$\displaystyle\mathfrak{G}_{\sigma_{\lambda}}^{(\beta)}(X_{n})={\operatorname{DET}\big{|}x_{i}^{\lambda_{j}+n-j}(1+\beta
x_{i})^{j-1}\big{|}_{1\leq i,j\leq n}\over\prod\limits_{1\leq i<j\leq
n}(x_{i}-x_{j})},$ has been proved independently in [107].

$\displaystyle({\rm
A})\quad\mathfrak{G}_{\sigma_{\lambda}}^{(\beta)}(X_{n})=\operatorname{DET}\big{|}h_{\lambda_{j}+i,j}^{(\beta)}(X_{n})\big{|}_{1\leq
i,j\leq n}={\operatorname{DET}\big{|}x_{i}^{\lambda_{j}+n-j}(1+\beta
x_{i})^{j-1}\big{|}_{1\leq i,j\leq n}\over\prod\limits_{1\leq i<j\leq
n}(x_{i}-x_{j})},$

where $X_{n}=(x_{i},x_{1},\ldots,x_{n})$, and for any set of variables $X$,

$\displaystyle h_{n,k}^{(\beta)}(X)=\sum_{a=0}^{k-1}~{}{k-1\choose
a}h_{n-k+a}(X)\beta^{a},$

and $h_{k}(X)$ denotes the complete symmetric polynomial of degree $k$ in the
variables from the set $X$.

$\displaystyle({\rm
B})\quad{\mathfrak{G}}_{\sigma_{\lambda}}(X,Y)={\operatorname{DET}\Big{|}\prod\limits_{a=1}^{\lambda_{j}+n-j}(x_{i}+y_{a}+\beta
x_{i}y_{a})(1+\beta x_{i})^{j-1}\Big{|}_{1\leq i,j\leq
n}\over\prod\limits_{1\leq i<j\leq n}(x_{i}-x_{j})}.$

In Sections 5.2.2 and 5.4.2 we study connections of Grothendieck polynomial
associated with the Richardson permutation $w_{k}^{(n)}=1^{k}\times
w_{0}^{(n-k)}$, $k$-dissections of a convex $(n+k+1)$-gon, generalized reduced
polynomial corresponding to a certain monomial in the algebra ${\widehat{{\rm
ACYB}}}_{n}$ and the Lagrange inversion formula. In the case of generalized
Richardson permutation $w_{n,p}^{(k)}$ corresponding to the $k$-shifted
dominant permutations $w^{(p,n)}$ associated with the Young diagram
$\lambda_{p,n}:=p(n-1,n-2,\ldots,1)$, namely, $w_{n,p}^{(k)}=1^{k}\times
w^{(p,n)}$, we treat only the case $k=1$, see also [39]. In the case $k\geq 2$
one comes to a task to count and find a lattice path type interpretation for
the number of $k$-pgulations of a convex $n$-gon that is the number of
partitioning of a convex $n$-gon on parts which are all equal to a convex
$(p+2)$-gon, by a (maximal) family of diagonals such that each diagonal has at
most $k$ internal intersections with the members of a family of diagonals
selected.

In Section 5.3 we give a partial answer on Question 6.C8(d) by R. Stanley
[133]. In particular, we relate the reduced polynomial corresponding to
monomial

$\displaystyle\bigl{(}x_{12}^{a_{2}}\cdots{x_{n-1,n}}^{a_{n}}\bigr{)}\prod_{j=2}^{n-2}\prod_{k=j+2}^{n}x_{jk},\qquad
a_{j}\in\mathbb{Z}_{\geq 0},\qquad\forall\,j,$

with the Ehrhart polynomial of the generalized Chan–Robbins–Yuen polytope, if
$a_{2}=\cdots=a_{n}=m+1$, cf. [101], with a $t$-deformation of the Kostant
partition function of type $A_{n-1}$ and the Ehrhart polynomials of some flow
polytopes, cf. [103].

In Section 5.4 we investigate certain specializations of the reduced
polynomials corresponding to monomials of the form

$\displaystyle x_{12}^{m_{1}}\cdots x_{n-1,n}^{m_{n}},\qquad
m_{j}\in\mathbb{Z}_{\geq 0},\qquad\forall\,j.$

First of all we observe that the corresponding specialized reduced polynomial
appears to be a piece-wise polynomial function of parameters
${\boldsymbol{m}}=(m_{1},\ldots,m_{n})\in(\mathbb{R}_{\geq 0})^{n}$, denoted
by $P_{{\boldsymbol{m}}}$. It is an interesting problem to compute the Laplace
transform of that piece-wise polynomial function. In the present paper we
compute the value of the function $P_{{\boldsymbol{m}}}$ in the dominant
chamber ${\cal{C}}_{n}=(m_{1}\geq m_{2}\geq\cdots\geq m_{n}\geq 0)$, and give
a combinatorial interpretation of the values of that function in points
$(n,m)$ and $(n,m,k)$, $n\geq m\geq k$.

For the reader convenience, in Appendices A.1–A.6 we collect some useful
auxiliary information about the subjects we are treated in the present paper.

Almost all results in Section 5 state that some two specific sets have the
same number of elements. Our proofs of these results are pure algebraic. It is
an interesting problem to find bijective proofs of results from Section 5
which generalize and extend remarkable bijective proofs presented in [103,
129, 135, 142] to the cases of

  * •

the $\beta$-Grothendieck polynomials,

  * •

the (small) Schröder numbers,

  * •

$k$-dissections of a convex $(n+k+1)$-gon,

  * •

special values of reduced polynomials.

We are planning to treat and present these bijections in separate
publication(s).

We expect that the reduced polynomials corresponding to the higher-order
powers of the Coxeter elements also admit an interesting combinatorial
interpretation(s). Some preliminary results in this direction are discussed in
Comments 5.67.

At the end of introduction I want to add a few remarks.

(a) After a suitable modification of the algebra $3HT_{n}$, see [75], and the
case $\beta\not=0$ in [72], one can compute the set of relations among the
(additive) Dunkl elements (defined in Section 2, equation (2.1)). In the case
$\beta=0$ and $q_{ij}=q_{i}\delta_{j-i,1}$, $1\leq i<j\leq n$, where
$\delta_{a,b}$ is the Kronecker delta symbol, the commutative algebra
generated by additive Dunkl elements (2.3) appears to be “almost” isomorphic
to the equivariant quantum cohomology ring of the flag variety ${\cal
F}l_{n}$, see [75] for details. Using the multiplicative version of Dunkl
elements, see Section 3.2, one can extend the results from [75] to the case of
equivariant quantum $K$-theory of the flag variety ${\cal F}l_{n}$, see [72].

(b) As it was pointed out previously, one can define an analogue of the
algebra $3T_{n}^{(0)}$ for any (oriented) matroid ${\cal{M}}_{n}$, and state a
conjecture which connects the Hilbert polynomial of the algebra
$3T_{n}^{(0)}(({\cal{M}}_{n})^{ab},t)$ and the chromatic polynomial of matroid
${\cal{M}}_{n}$. We expect that algebra
$3T_{n}^{(\beta=1)}({\cal{M}}_{n})^{ab}$ is isomorphic to the
Gelfand–Varchenko algebra associated with matroid $\cal{M}$. It is an
interesting problem to find a combinatorial meaning of the algebra
$3T_{n}^{(\beta)}({\cal{M}}_{n})$ for $\beta=0$ and $\beta\not=0$.

(c) Let $R$ be a (graded) ring (to be specified later) and
${\mathfrak{F}}_{n^{2}}$ be the free associative algebra over $R$ with the set
of generators $\\{u_{ij},\,1\leq i,j\leq n\\}$. In the subsequent text we will
distinguish the set of generators $\\{u_{ii}\\}_{1\leq i\leq n}$ from that
$\\{u_{ij}\\}_{1\leq i\not=j\leq n}$, and set

$\displaystyle x_{i}:=u_{ii},\qquad i=1,\ldots,n.$

A guiding idea to choose definitions and perform constructions in the present
paper is to impose a set of relations ${\cal{R}}_{n}$ among the generators
$\\{x_{i}\\}_{1\leq i\leq n}$ and that $\\{u_{ij}\\}_{1\leq i\not=j\leq n}$
which ensure the mutual commutativity of the following elements

$\displaystyle\theta_{i}^{(n)}:=\theta_{i}=x_{i}+\sum_{j\not=i}^{n}u_{ij},\qquad
i=1,\ldots,n,$

in the algebra ${\cal{F}}_{n^{2}}/{\cal{R}}_{n}$, as well as to have a good
chance to describe/compute

$\bullet$ “Integral of motions”, that is finding a big enough set of
algebraically independent polynomials (quite possibly that polynomials are
trigonometric or elliptic ones) $I_{\alpha}^{(n)}(y_{1},\ldots,y_{n})\in
R[Y_{n}]$ such that

$\displaystyle
I_{\alpha}^{(n)}\big{(}\theta_{1}^{(n)},\ldots,\theta_{n}^{(n)}\big{)}\in
R[X_{n}],\qquad\forall\,\alpha,$

in other words, the latter specialization of any integral of motion has to be
independent of the all generators $\\{u_{ij}\\}_{1\leq i\not=j\leq n}$.

$\bullet$ Give a presentation of the algebra ${\cal{I}}_{n}$ generated by the
integral of motions that is to find a set of defining relations among the
elements $\theta_{1},\ldots,\theta_{n}$, and describe a $R$-basis
$\big{\\{}m_{\alpha}^{(n)}\big{\\}}$ in the algebra ${\cal{I}}_{n}$.

$\bullet$ Generalized Littlewood–Richardson and Murnaghan–Nakayama problems.
Given an integral of motion $I_{\beta}^{(m)}(Y_{m})$ and an integer $n\geq m$,
find an explicit positive (if possible) expression in the quotient algebra
${\cal{F}}_{n^{2}}/{\cal{R}}_{n}$ of the element

$\displaystyle
I_{\beta}^{(m)}\big{(}\theta_{1}^{(n)},\ldots,\theta_{m}^{(n)}\big{)}.$

For example in the case of the 3-term relations algebra $3T_{n}^{(0)}$ (as
well as its equivariant, quantum, etc. versions) the generalized
Littlewood–Richardson problem is to find a positive expression in the algebra
$3T_{n}^{(0)}$ for the element
${\mathfrak{S}}_{w}\big{(}\theta_{1}^{(n)},\ldots,\theta_{m}^{(n)}\big{)}$,
where ${\mathfrak{S}}_{w}(Y_{n})$ stands for the Schubert polynomial
corresponding to a permutation $w\in\mathbb{S}_{n}$.

Generalized Murnaghan–Nakayama problem consists in finding a combinatorial
expression in the algebra $3T_{n}^{(0)}$ for the element
$\sum\limits_{i=1}^{m}(\theta_{i}^{(n)})^{k}$.

Partial results concerning these problems have been obtained as far as we
aware in [45, 70, 72, 73, 104, 117].

$\bullet$ “Partition functions”. Assume that the (graded) algebra
${\cal{I}}_{n}$ generated over $R$ by the elements
$\theta_{1},\ldots,\theta_{n}$ has finite dimension/rank, and the (non zero)
maximal degree component ${\cal{I}}_{\max}^{(n)}$ of that algebra has
dimension/rank one and generated by an element $\omega$. For any element
$g\in{\cal{F}}_{n^{2}}$ let us denote by $\operatorname{Res}_{\omega}(g)$ an
element in $R$ such that

$\displaystyle\overline{g}=\operatorname{Res}_{\omega}(g)\omega,$

where we denote by $\overline{g}$ the image of element $g$ in the component
${\cal{I}}_{\max}^{(n)}$.

We define partition function associated with the algebra ${\cal{I}}_{n}$ as
follows

$\displaystyle{\cal{Z}}({\cal{I}}_{n})=\operatorname{Res}_{w}\bigg{(}\exp\bigg{(}\sum_{\alpha}q_{\alpha}m_{\alpha}^{(n)}\bigg{)}\bigg{)},$

where $\\{q_{\alpha}\\}$ is a set of parameters which is consistent in one-to-
one correspondence with a basis $\big{\\{}m_{\alpha}^{(n)}\big{\\}}$ chosen.

We are interesting in to find a closed formula for the partition function
${\cal{Z}}({\cal{I}}_{n})$ as well as that for a small partition function

$\displaystyle{\cal{Z}}^{(0)}({\cal{I}}_{n}):=\operatorname{Res}_{\omega}\bigg{(}\exp\bigg{(}\sum_{1\leq
i,j\leq n}\lambda_{ij}u_{ij}\bigg{)}\bigg{)},$

where $\\{\lambda_{ij}\\}_{1\leq i,j\leq n}$ stands for a set of parameters.
One can show [68] that the partition function ${\cal{Z}}({\cal{I}}_{n})$
associated with algebra $3T_{n}^{{\boldsymbol{q}}}$ satisfies the famous
Witten–Dijkraaf–Verlinde–Verlinde equations.

As a preliminary steps to perform our guiding idea we

  1. (i)

investigate properties of the abelianization of the algebra
${\cal{F}}_{n^{2}}/{\cal{R}}_{n}$. Some unexpected connections with the theory
of hyperplane arrangements and graph theory are discovered;

  2. (ii)

investigate a variety of descendent relations coming from the defining
relations. Some polynomials with interesting combinatorial properties are
naturally appear.

To keep the size of the present paper reasonable, several new results are
presented as exercises.

We conclude Introduction by a short historical remark. As far as we aware, the
commutative version of $3$-term relations which provided the framework for a
definition of the FK algebra ${\cal{E}}_{n}$ [45] and a plethora of its
generalizations, have been frequently used implicitly in the theory of
elliptic functions and related topics, starting at least from the middle of
the 19th century, see, e.g., [141] for references, and up to now, and for sure
will be used for ever. The key point is that the Kronecker sigma function

$\displaystyle\sigma_{z}(w):={\frac{\sigma(z-w)\theta^{\prime}(0)}{\sigma(z)\sigma(-w)}},$

where $\sigma(z)$ denotes the Weierstrass sigma function, satisfies the
quadratic three terms addition formula or functional equation discovered, as
far as we aware, by K. Weierstrass. In fact this functional equation is really
equivalent191919We refer the reader to a nice written paper by Tom H.
Koornwinder [79] for more historical information. to the famous Jacobi–Riemann
three term relation of degree four between the Riemann theta functions
$\theta(x)$. In the rational degeneration of theta functions, the three term
relation between Kronecker sigma functions turns to the famous three term
Jacobi identity which can be treated as an associative analogue of the Jacobi
identity in the theory of Lie algebras.

To our best knowledge, in an abstract form that is as a set of defining
relations in a certain algebra, an anticommutative version of three term
relations had been appeared in a remarkable paper by V.I. Arnold [3]. Nowadays
these relations are known as Arnold relations. These relations and its various
generalizations play fundamental role in the theory of arrangements, see,
e.g., [113], in topology, combinatorics and many other branches of
Mathematics.

In commutative set up abstract form of $3$-term relations has been invented by
O. Mathieu [96]. In the context of the braided Hopf algebras (of type A)
$3$-term relations like algebras (as some examples of the Nichols algebras)
have appeared in papers by A. Milinski and H.-J. Schneider (2000), N.
Andruskiewitsch (2002), S. Madjid (2004), I. Heckenberger (2005) and many
others202020We refer the reader to the site
https://en.wikipedia.org/wiki/Nichols_algebra for basic definitions and
results concerning Nichols’ algebras and references on vast literature treated
different aspects of the theory of Nichols’ algebras and braided Hopf
algebras..

It is well-known that the Nichols algebra associated with the symmetric group
${\mathbb{S}}_{n}$ and trivial conjugacy class is a quotient of the algebra
$FK_{n}$. It is still an open problem to prove (or disprove) that these two
algebras are isomorphic.

##  2 Dunkl elements

Having in mind to fulfill conditions suggested by our guiding line mentioned
in Introduction as far as it could be done till now, we are led to introduce
the following algebras212121Surprisingly enough, in many cases to find
relations among the elements $\theta_{1},\ldots,\theta_{n}$ there is no need
to require that the elements $\\{\theta_{i}\\}_{1\leq i\leq n}$ are pairwise
commute..

######  Definition 2.1 (additive Dunkl elements).

The (additive) Dunkl elements $\theta_{i}$, $i=1,\dots,n$, in the algebra
${\cal F}_{n}$ are defined to be

$\displaystyle\theta_{i}=x_{i}+\sum_{j=1\atop j\not=i}^{n}u_{ij}.$ (2.1)

We are interested in to find “natural relations” among the generators
$\\{u_{ij}\\}_{1\leq i,j\leq n}$ such that the Dunkl elements (2.1) are pair-
wise commute. One of the natural conditions which is the commonly accepted in
the theory of integrable systems, is

  * •

locality conditions:

$\displaystyle(a)\quad[x_{i},x_{j}]=0\qquad\text{if}\ \ i\not=j,$
$\displaystyle(b)\quad u_{ij}u_{kl}=u_{kl}u_{ij}\qquad\text{if}\ \ i\not=j,\ \
k\not=l\ \ \text{and}\ \ \\{i,j\\}\cap\\{k,l\\}=\varnothing.$ (2.2)

######  Lemma 2.2.

Assume that elements $\\{u_{ij}\\}$ satisfy the locality condition (2.1). If
$i\not=j$, then

$\displaystyle[\theta_{i},\theta_{j}]=\biggl{[}x_{i}+\sum_{k\not=i,j}u_{ik},u_{ij}+u_{ji}\biggr{]}+\biggl{[}u_{ij},\sum_{k=1}^{n}x_{k}\biggr{]}+\sum_{k\not=i,j}w_{ijk},$

where

$\displaystyle
w_{ijk}=[u_{ij},u_{ik}+u_{jk}]+[u_{ik},u_{jk}]+[x_{i},u_{jk}]+[u_{ik},x_{j}]+[x_{k},u_{ij}].$
(2.3)

Therefore in order to ensure that the Dunkl elements form a pair-wise
commuting family, it’s natural to assume that the following conditions hold

  * •

unitarity:

$\displaystyle[u_{ij}+u_{ji},u_{kl}]=0=[u_{ij}+u_{ji},x_{k}]\qquad\text{for
all distinct}\ \ i,\,j,\,k,\,l,$ (2.4)

i.e., the elements $u_{ij}+u_{ji}$ are central.

  * •

“conservation laws”:

$\displaystyle\left[\sum_{k=1}^{n}x_{k},u_{ij}\right]=0\qquad\text{for all}\ \
i,\,j,$ (2.5)

i.e., the element $E:=\sum\limits_{k=1}^{n}x_{k}$ is central,

  * •

unitary dynamical classical Yang–Baxter relations:

$\displaystyle[u_{ij},u_{ik}+u_{jk}]+[u_{ik},u_{jk}]+[x_{i},u_{jk}]+[u_{ik},x_{j}]+[x_{k},u_{ij}]=0,$
(2.6)

if $i$, $j$, $k$ are pair-wise distinct.

######  Definition 2.3 (dynamical six term relations algebra $6DT_{n}$).

We denote by $6DT_{n}$ (and frequently will use also notation ${\rm
DCYB}_{n}$) the quotient of the algebra ${\cal F}_{n}$ by the two-sided ideal
generated by relations (2.2)–(2.6).

Clearly, the Dunkl elements (2.1) generate a commutative subalgebra inside of
the algebra $6DT_{n}$, and the sum
$\sum\limits_{i=1}^{n}\theta_{i}=\sum\limits_{i=1}^{n}x_{i}$ belongs to the
center of the algebra $6DT_{n}$.

######  Remark 2.4.

Occasionally we will call the Dunkl elements of the form (2.1) by dynamical
Dunkl elements to distinguish the latter from truncated Dunkl elements,
corresponding to the case $x_{i}=0$, $\forall\,i$.

###  2.1 Some representations of the algebra $\boldsymbol{6DT_{n}}$

####  2.1.1 Dynamical Dunkl elements and equivariant quantum cohomology

(I) ( cf. [41]). Given a set $q_{1},\ldots,q_{n-1}$ of mutually commuting
parameters, define

$\displaystyle q_{ij}=\prod_{a=i}^{j-1}q_{a}\qquad\text{if}\quad i<j,$

and set $q_{ij}=q_{ji}$ in the case $i>j$. Clearly, that if $i<j<k$, then
$q_{ij}q_{jk}=q_{ik}$.

Let $z_{1},\ldots,z_{n}$ be a set of (mutually commuting) variables. Denote by
$P_{n}:=\mathbb{Z}[z_{1},\ldots,z_{n}]$ the corresponding ring of polynomials.
We consider the variable $z_{i}$, $i=1,\ldots,n$, also as the operator acting
on the ring of polynomials $P_{n}$ by multiplication on the variable $z_{i}$.

Let $s_{ij}\in\mathbb{S}_{n}$ be the transposition that swaps the letters $i$
and $j$ and fixes the all other letters $k\not=i,j$. We consider the
transposition $s_{ij}$ also as the operator which acts on the ring $P_{n}$ by
interchanging $z_{i}$ and $z_{j}$, and fixes all other variables. We denote by

$\displaystyle\partial_{ij}={1-s_{ij}\over
z_{i}-z_{j}},\qquad\partial_{i}:=\partial_{i,i+1},$

the divided difference operators corresponding to the transposition $s_{ij}$
and the simple transposition $s_{i}:=s_{i,i+1}$ correspondingly. Finally we
define operator (cf. [41])

$\displaystyle\partial_{(ij)}:=\partial_{i}\cdots\partial_{j-1}\partial_{j}\partial_{j-1}\cdots\partial_{i}\qquad\text{if}\
\ i<j.$

The operators $\partial_{(ij)}$, $1\leq i<j\leq n$, satisfy (among other
things) the following set of relations (cf. [41])

  * •

$[z_{j},\partial_{(ik)}]=0$ if $j\notin[i,k]$,
$\Big{[}\partial_{(ij)},\sum\limits_{a=i}^{j}z_{a}\Big{]}=0$,

  * •

$[\partial_{(ij)},\partial_{(kl)}]=\delta_{jk}[z_{j},\partial_{(il)}]+\delta_{il}[\partial_{(kj)},z_{i}]$
if $i<j$, $k<l$.

Therefore, if we set $u_{ij}=q_{ij}\partial_{(ij)}$ if $i<j$, and
$u_{ij}=-u_{ji}$ if $i>j$, then for a triple $i<j<k$ we will have

$\displaystyle[u_{ij},u_{ik}+u_{jk}]+[u_{ik},u_{jk}]+[z_{i},u_{jk}]+[u_{ik},z_{j}]+[z_{k},u_{jk}]$
$\displaystyle\qquad{}=q_{ij}q_{jk}[\partial_{(ij)},\partial_{(jk)}]+q_{ik}[\partial_{(ik)},z_{j}]=0.$

Thus the elements $\\{z_{i},\,i=1,\ldots,n\\}$ and $\\{u_{ij},\,1\leq i<j\leq
n\\}$ define a representation of the algebra ${\rm DCYB}_{n}$, and therefore
the Dunkl elements

$\displaystyle\theta_{i}:=z_{i}+\sum_{j\not=i}u_{ij}=z_{i}-\sum_{j<i}q_{ji}\partial_{(ji)}+\sum_{j>i}q_{ij}\partial_{(ij)}$

form a pairwise commuting family of operators acting on the ring of
polynomials

$\displaystyle\mathbb{Z}[q_{1},\ldots,q_{n-1}][z_{1},\ldots,z_{n}],$

cf. [41]. This representation has been used in [41] to construct the small
quantum cohomology ring of the complete flag variety of type $A_{n-1}$.

(II) Consider degenerate affine Hecke algebra ${\mathfrak{H}}_{n}$ generated
by the central element $h$, the elements of the symmetric group
${\mathbb{S}}_{n}$, and the mutually commuting elements $y_{1},\ldots,y_{n}$,
subject to relations

$\displaystyle s_{i}y_{i}-y_{i+1}s_{i}=h,\quad 1\leq i<n,\qquad
s_{i}y_{j}=y_{j}s_{i},\quad j\not=i,i+1,$

where $s_{i}$ stand for the simple transposition that swaps only indices $i$
and $i+1$. For $i<j$, let $s_{ij}=s_{i}\cdots s_{j-1}s_{j}s_{j-1}\cdots s_{i}$
denotes the permutation that swaps only indices $i$ and $j$. It is an easy
exercise to show that

  * •

$[y_{j},s_{ik}]=h[s_{ij},s_{jk}]$ if $i<j<k$,

  * •

$y_{i}s_{ik}-s_{ik}y_{k}=h+hs_{ik}\sum\limits_{i<j<k}s_{jk}$ if $i<k$.

Finally, consider a set of mutually commuting parameters $\\{p_{ij},\,1\leq
i\not=j\leq n,\,p_{ij}+p_{ji}=0\\}$, subject to the constraints

$\displaystyle p_{ij}p_{jk}=p_{ik}p_{ij}+p_{jk}p_{ik}+hp_{ik},\qquad i<j<k.$

######  Comments 2.5.

If parameters $\\{p_{ij}\\}$ are invertible, and satisfy relations

$\displaystyle p_{ij}p_{jk}=p_{ik}p_{ij}+p_{jk}p_{ik}+\beta p_{ik},\qquad
i<j<k,$

then one can rewrite the above displayed relations in the following form

$\displaystyle 1+{\beta\over p_{ik}}=\left(1+{\beta\over
p_{ij}}\right)\left(1+{\beta\over p_{jk}}\right),\qquad 1\leq i<j<k\leq n.$

Therefore there exist parameters $\\{q_{1},\ldots,q_{n}\\}$ such that
$1+\beta/p_{ij}=q_{i}/q_{j}$, $1\leq i<j\leq n$. In other words,
$p_{ij}={\beta q_{j}\over q_{j}-q_{j}}$, $1\leq i<j\leq n$. However in
general, there are many other types of solutions, for example, solutions
related to the Heaviside function222222See
https://en.wikipedia.org/wiki/Heaviside_step_function. $H(x)$, namely,
$p_{ij}=H(x_{i}-x_{j})$, $x_{i}\in\mathbb{R}$, $\forall\,i$, and its discrete
analogue, see Example (III) below. In the both cases $\beta=-1$; see also
Comments 2.12 for other examples.

To continue presentation of Example (II), define elements
$u_{ij}=p_{ij}s_{ij}$, $1\leq i\not=j\leq n$.

######  Lemma 2.6 (dynamical classical Yang–Baxter relations).

$\displaystyle[u_{ij},u_{ik}+u_{jk}]+[u_{ik},u_{jk}]+[u_{ik},y_{j}]=0,\qquad
1<i<j<k\leq n.$ (2.7)

Indeed,

$\displaystyle
u_{ij}u_{jk}=u_{ik}u_{ij}+u_{jk}u_{ik}+hp_{ik}s_{ij}s_{jk},\qquad
u_{jk}u_{ij}=u_{ij}u_{ik}+u_{ik}u_{jk}+hp_{ik}s_{jk}s_{ij},$

and moreover, $[y_{j},u_{ik}]=hp_{ik}[s_{ij},s_{jk}]$.

Therefore, the elements

$\displaystyle\theta_{i}=y_{i}-h\sum_{j<i}u_{ij}+h\sum_{i<j}u_{ij},\qquad
i=1,\ldots,n,$

form a mutually commuting set of elements in the algebra
$\mathbb{Z}[\\{p_{ij}\\}]\otimes_{\mathbb{Z}}{\mathfrak{H}}_{n}$.

######  Theorem 2.7.

Define matrix $M_{n}=(m_{i,j})_{1\leq i,j\leq n}$ as follows

$\displaystyle m_{i,j}(u;z_{1},\ldots,z_{n})=\begin{cases}u-z_{i}&\text{if
$i=j$},\\\ -h-p_{ij}&\text{if $i<j$},\\\ p_{ij}&\text{if $i>j$}.\end{cases}$

Then

$\displaystyle\operatorname{DET}\big{|}M_{n}(u;\theta_{1},\ldots,\theta_{n})\big{|}=\prod_{j=1}^{n}(u-y_{j}).$

Moreover, let us set
$q_{ij}:=h^{2}(p_{ij}+p_{ij}^{2})=h^{2}q_{i}q_{j}(q_{i}-q_{j})^{-2}$, $i<j$,
then

$\displaystyle
e_{k}(\theta_{1},\ldots,\theta_{n})=e_{k}^{({\boldsymbol{q}})}(y_{1},\ldots,y_{n}),\qquad
1\leq k\leq n,$

where $e_{k}(x_{1},\ldots,x_{n})$ and
$e_{k}^{({\boldsymbol{q}})}(x_{1},\ldots,x_{n})$ denote correspondingly the
classical and multiparameter quantum [45] elementary polynomials232323For the
reader convenience we remind [45] a definition of the quantum elementary
polynomial $e_{k}^{\boldsymbol{q}}(x_{1},\ldots,x_{n})$. Let
${\boldsymbol{q}}:=\\{q_{ij}\\}_{1\leq i<j\leq n}$ be a collection of “quantum
parameters”, then  $\displaystyle
e_{k}^{\boldsymbol{q}}(x_{1},\ldots,x_{n})=\sum_{\ell}\sum_{1\leq
i_{1}<\cdots<i_{\ell}\leq n\atop
j_{1}>i_{1},\ldots,j_{\ell}>i_{\ell}}e_{k-2\ell}(X_{\overline{I\cup
J}})\prod_{a=1}^{\ell}q_{i_{a},j_{a}},$ where $I=(i_{1},\ldots,i_{\ell})$,
$J=(j_{1},\ldots,j_{\ell})$ should be distinct elements of the set
$\\{1,\ldots,n\\}$, and $X_{\overline{I\cup J}}$ denotes set of variables
$x_{a}$ for which the subscript $a$ is neither one of $i_{m}$ nor one of the
$j_{m}$..

Let’s stress that the elements $y_{i}$ and $\theta_{j}$ do not commute in the
algebra ${\mathfrak{H}}_{n}$, but the symmetric functions of
$y_{1},\ldots,y_{n}$, i.e., the center of the algebra ${\mathfrak{H}}_{n}$,
do.

A few remarks in order. First of all, $u_{ij}^{2}=p_{ij}^{2}$ are central
elements. Secondly, in the case $h=0$ and $y_{i}=0$, $\forall\,i$, the
equality

$\displaystyle\operatorname{DET}\big{|}M_{n}(u;x_{1},\ldots,x_{n})\big{|}=u^{n}$

describes the set of polynomial relations among the Dunkl–Gaudin elements
(with the following choice of parameters $p_{ij}=(q_{i}-q_{j})^{-1}$ are
taken). And our final remark is that according to [54, Section 8], the
quotient ring

$\displaystyle{\cal{H}}_{n}^{{\boldsymbol{q}}}:=\mathbb{Q}[y_{1},\ldots,y_{n}]^{{\mathbb{S}}_{n}}\otimes\mathbb{Q}[\theta_{1},\dots,\theta_{n}]\otimes\mathbb{Q}[h]\Big{/}\bigg{\langle}M_{n}(u;\theta_{1},\ldots,\theta_{n})=\prod_{j=1}^{n}(u-y_{j})\bigg{\rangle}$

is isomorphic to the quantum equivariant cohomology ring of the cotangent
bundle $T^{*}{\cal{F}}l_{n}$ of the complete flag variety of type $A_{n-1}$,
namely,

$\displaystyle{\cal{H}}_{n}^{{\boldsymbol{q}}}\cong
QH^{*}_{T^{n}\times\mathbb{C}^{*}}(T^{*}{\cal{F}}l_{n})$

with the following choice of quantum parameters: $Q_{i}:=hq_{i+1}/q_{i}$,
$i=1,\ldots,n-1$.

On the other hand, in [75] we computed the so-called multiparameter
deformation of the equivariant cohomology ring of the complete flag variety of
type $A_{n-1}$.

A deformation defined in [75] depends on parameters $\\{q_{ij},\,1\leq i<j\leq
n\\}$ without any constraints are imposed. For the special choice of
parameters $\displaystyle q_{ij}:=h^{2}{q_{i}~{}q_{j}\over(q_{i}-q_{j})^{2}}$
the multiparameter deformation of the equivariant cohomology ring of the type
$A_{n-1}$ complete flag variety ${\cal{F}}l_{n}$ constructed in [75], is
isomorphic to the ring ${\cal{H}}_{n}^{{\boldsymbol{q}}}$.

######  Comments 2.8.

Let us fix a set of independent parameters $\\{q_{1},\ldots,q_{n}\\}$ and
define new parameters

$\displaystyle\left\\{q_{ij}:=hp_{ij}(p_{ij}+h)=h^{2}{q_{i}q_{j}\over(q_{i}-q_{j})^{2}}\right\\},\quad
1\leq i<j\leq n,\\!\qquad\text{where}\quad p_{ij}={q_{j}\over
q_{i}-q_{j}},\quad i<j.$

We set $\deg(q_{ij})=2$, $\deg(p_{ij})=1$, $\deg(h)=1$.

The new parameters $\\{q_{ij}\\}_{1\leq i<j\leq n}$, do not free anymore, but
satisfy rather complicated algebraic relations. We display some of these
relations soon, having in mind a question: is there some intrinsic meaning of
the algebraic variety defined by the set of defining relations among the
“quantum parameters” $\\{q_{ij}\\}$?

Let us denote by ${{\cal{A}}}_{n,h}$ the quotient ring of the ring of
polynomials $\mathbb{Q}[h][x_{ij},\,1\leq i<j\leq n]$ modulo the ideal
generating by polynomials $f(x_{ij})$ such that the specialization
$x_{ij}=q_{ij}$ of a polynomial $f(x_{ij})$, namely $f(q_{ij})$, is equal to
zero. The algebra ${\cal{A}}_{n,h}$ has a natural filtration, and we denote by
${\cal{A}}_{n}=\operatorname{gr}{\cal{A}}_{n,h}$ the corresponding associated
graded algebra.

To describe (a part of) relations among the parameters $\\{q_{ij}\\}$ let us
observe that parameters $\\{p_{ij}\\}$ and $\\{q_{ij}\\}$ are related by the
following identity

$\displaystyle
q_{ij}q_{jk}-q_{ik}(q_{ij}+q_{jk})+h^{2}q_{ik}=2p_{ij}p_{ik}p_{jk}(p_{ik}+h)\qquad\text{if}\quad
i<j<k.$

Using this identity we can find the following relations among parameters in
question

$\displaystyle
q_{ij}^{2}q_{jk}^{2}+q_{ij}^{2}q_{ik}^{2}+h^{4}q_{ik}^{2}q_{jk}^{2}-2q_{ij}q_{ik}q_{jk}(q_{ij}+q_{jk}+q_{ik})$
$\displaystyle\qquad{}-2h^{2}q_{ik}(q_{ij}q_{jk}+q_{ij}q_{ik}+q_{jk}q_{ik})=8hq_{ij}q_{ik}q_{jk}p_{ik},$
(2.8)

if $1\leq i<j<k\leq n$.

Finally, we come to a relation of degree $8$ among the “quantum parameters”
$\\{q_{ij}\\}$

$\displaystyle\big{(}\text{l.h.s.\ of
\eqref{eq:xdef}}\big{)}^{2}=64h^{2}q_{ij}^{2}q_{ik}^{3}q_{jk}^{2},\qquad 1\leq
i<j<k\leq n.$

There are also higher degree relations among the parameters $\\{q_{ij}\\}$
some of whose in degree $16$ follow from the deformed Plücker relation between
parameters $\\{p_{ij}\\}$:

$\displaystyle{1\over p_{ik}p_{jl}}={1\over p_{ij}p_{kl}}+{1\over
p_{il}p_{jk}}+{h\over p_{ij}p_{jk}p_{kl}},\qquad i<j<k<l.$

However, we don’t know how to describe the algebra ${{\cal{A}}}_{n,h}$
generated by quantum parameters $\\{q_{ij}\\}_{1\leq i<j\leq n}$ even for
$n=4$.

The algebra ${\cal{A}}_{n}=\operatorname{gr}({\cal{A}}_{n,h})$ is isomorphic
to the quotient algebra of $\mathbb{Q}[x_{ij},\,1\leq i<j\leq n]$ modulo the
ideal generated by the set of relations between “quantum parameters”

$\displaystyle\left\\{\overline{q}_{ij}:=\left({1\over
z_{i}-z_{j}}\right)^{2}\right\\}_{1\leq i<j\leq n},$

which correspond to the Dunkl–Gaudin elements $\\{\theta_{i}\\}_{1\leq i\leq
n}$, see Section 3.2 below for details. In this case the parameters
$\\{\overline{q}_{ij}\\}$ satisfy the following relations

$\displaystyle\overline{q}_{ij}^{2}\overline{q}_{jk}^{2}+\overline{q}_{ij}^{2}\overline{q}_{ik}^{2}+\overline{q}_{jk}^{2}\overline{q}_{ik}^{2}=2\overline{q}_{ij}\overline{q}_{ik}\overline{q}_{jk}(\overline{q}_{ij}+\overline{q}_{jk}+\overline{q}_{jk})$

which correspond to the relations (2.8) in the special case $h=0$. One can
find a set of relations in degrees $6$, $7$ and $8$, namely for a given pair-
wise distinct integers $1\leq i,j,k,l\leq n$, one has

  * •

one relation in degree $6$

$\displaystyle\overline{q}_{ij}^{2}\overline{q}_{ik}^{2}\overline{q}_{il}^{2}+\overline{q}_{ij}^{2}\overline{q}_{jk}^{2}\overline{q}_{jl}^{2}+\overline{q}_{ik}^{2}\overline{q}_{jk}^{2}\overline{q}_{kl}^{2}+\overline{q}_{il}^{2}\overline{q}_{jl}^{2}\overline{q}_{kl}^{2}$
$\displaystyle\qquad{}-2\overline{q}_{ij}\overline{q}_{ik}\overline{q}_{il}\overline{q}_{jk}\overline{q}_{jl}\overline{q}_{kl}\left({{{\overline{q}_{ij}}\over{\overline{q}_{kl}}}}+{{{\overline{q}_{kl}}\over{\overline{q}_{ij}}}}+{{{\overline{q}_{ik}}\over{\overline{q}_{jl}}}}+{{{\overline{q}_{jl}}\over{\overline{q}_{ik}}}}+{{{\overline{q}_{il}}\over{\overline{q}_{jk}}}}+{{{\overline{q}_{jk}}\over{\overline{q}_{il}}}}\right)$
$\displaystyle\qquad{}+8\overline{q}_{ij}\overline{q}_{ik}\overline{q}_{il}\overline{q}_{jk}\overline{q}_{jl}\overline{q}_{kl}=0;$

  * •

three relations in degree $7$

$\displaystyle\overline{q}_{ik}\bigl{(}\overline{q}_{ij}\overline{q}_{il}\overline{q}_{kl}-\overline{q}_{ij}\overline{q}_{il}\overline{q}_{jk}+\overline{q}_{ij}\overline{q}_{jk}\overline{q}_{kl}-\overline{q}_{il}\overline{q}_{jk}\overline{q}_{kl}\bigr{)}^{2}$
$\displaystyle\qquad{}=8\overline{q}_{ij}^{2}\overline{q}_{ik}^{2}\overline{q}_{jk}\overline{q}_{kl}\bigl{(}\overline{q}_{jk}+\overline{q}_{jl}+\overline{q}_{kl}\bigr{)}-4\overline{q}_{ij}^{2}\overline{q}_{il}^{2}\overline{q}_{jl}\bigl{(}\overline{q}_{jk}^{2}+\overline{q}_{kl}^{2}\bigr{)},$

  * •

one relation in degree $8$

$\displaystyle\overline{q}_{ij}^{2}\overline{q}_{il}^{2}\overline{q}_{jk}^{2}\overline{q}_{kl}^{2}+\overline{q}_{ij}^{2}\overline{q}_{ik}^{2}\overline{q}_{jl}^{2}\overline{q}_{kl}^{2}+\overline{q}_{ik}^{2}\overline{q}_{il}^{2}\overline{q}_{jk}^{2}\overline{q}_{jl}^{2}=2\overline{q}_{ij}\overline{q}_{ik}\overline{q}_{il}\overline{q}_{jk}\overline{q}_{jl}\overline{q}_{kl}\bigl{(}\overline{q}_{ij}\overline{q}_{kl}+\overline{q}_{ik}\overline{q}_{jl}+\overline{q}_{il}\overline{q}_{jk}\bigr{)},$

However we don’t know does the list of relations displayed above, contains the
all independent relations among the elements $\\{\overline{q}_{ij}\\}_{1\leq
i<j\leq n}$ in degrees $6$, $7$ and $8$, even for $n=4$. In degrees $\geq 9$
and $n\geq 5$ some independent relations should appear.

Notice that the parameters $\big{\\{}p_{ij}={hq_{j}\over
q_{i}-q_{j}},\,i<j\big{\\}}$ satisfy the so-called Gelfand–Varchenko
relations, see, e.g., [67]

$\displaystyle p_{ij}p_{jk}=p_{ik}p_{ij}+p_{jk}p_{ik}+hp_{ik},\qquad i<j<k,$

whereas parameters $\big{\\{}{\overline{p}}_{ij}={1\over
q_{i}-q_{j}},\,i<j\big{\\}}$ satisfy the so-called Arnold relations

$\displaystyle{\overline{p}}_{ij}{\overline{p}}_{jk}={\overline{p}}_{ik}{\overline{p}}_{ij}+{\overline{p}}_{jk}{\overline{p}}_{ik},\qquad
i<j<k.$

######  Project 2.9.

Find Hilbert series ${\rm Hilb}({\cal{A}}_{n},t)$ for $n\geq 4$.242424This is
a particular case of more general problem we are interested in. Namely, let
$\\{f_{\alpha}\in\mathbb{R}[x_{1},\ldots,x_{n}]\\}_{1\leq\alpha\leq N}$ be a
collection of linear forms, and $k\geq 2$ be an integer. Denote by
$I(\\{f_{\alpha}\\})$ the ideal in the ring of polynomials
$\mathbb{R}[z_{1},\ldots,z_{N}]$ generated by polynomials
$\Phi(z_{1},\ldots,z_{N})$ such that
$\displaystyle\Phi\big{(}f_{1}^{-k},\ldots,f_{N}^{-k}\big{)}=0.$ Compute the
Hilbert series (polynomial?) of the quotient algebra
$\mathbb{R}[z_{1},\ldots,z_{N}]/I(\\{f_{\alpha}\\})$.

For example, ${\rm Hilb}({\cal{A}}_{3},t)={(1+t)(1+t^{2})\over(1-t)^{2}}$.

Finally, if we set $q_{i}:=\exp(hz_{i})$ and take the limit $\lim\limits_{h\to
0}\frac{h^{2}q_{i}q_{j}}{(q_{i}-q_{j})^{2}}$, as a result we obtain the
Dunkl–Gaudin parameter ${\overline{q}}_{ij}=\frac{1}{(z_{i}-z_{j})^{2}}$.

(III) Consider the following representation of the degenerate affine Hecke
algebra ${\mathfrak{H}}_{n}$ on the ring of polynomials
$P_{n}=\mathbb{Q}[x_{1},\ldots,x_{n}]$:

  * •

the symmetric group ${\mathbb{S}}_{n}$ acts on $P_{n}$ by means of operators

$\displaystyle\overline{s}_{i}=1+(x_{i+1}-x_{i}-h)\partial_{i},\qquad
i=1,\ldots,n-1,$

  * •

$y_{i}$ acts on the ring $P_{n}$ by multiplication on the variable $x_{i}$:
$y_{i}(f(x))=x_{i}f(x)$, $f\in P_{n}$. Clearly,

$\displaystyle
y_{i}\overline{s_{i}}-y_{i+1}\overline{s_{i}}=h\qquad\text{and}\qquad
y_{i}({\overline{s}}_{i}-1)=({\overline{s}}_{i}-1)y_{i+1}+x_{i+1}-x_{i}-h.$

In the subsequent discussion we will identify the operator of multiplication
by the variable $x_{i}$, namely the operator $y_{i}$, with $x_{i}$.

This time define $u_{ij}=p_{ij}(\overline{s}_{i}-1)$, if $i<j$ and set
$u_{ij}=-u_{ji}$ if $i>j$, where parameters $\\{p_{ij}\\}$ satisfy the same
conditions as in the previous example.

######  Lemma 2.10.

The elements $\\{u_{ij},\,1\leq i<j\leq n\\}$, satisfy the dynamical classical
Yang–Baxter relations displayed in Lemma 2.6, equation (2.7).

Therefore, the Dunkl elements

$\displaystyle\overline{\theta}_{i}:=x_{i}+\sum_{j\atop j\not=i}u_{ij},\qquad
i=1,\ldots,n,$

form a commutative set of elements.

######  Theorem 2.11 ([54]).

Define matrix $\overline{M}_{n}=(\overline{m}_{ij})_{1\leq i,j\leq n}$ as
follows

$\displaystyle\overline{m}_{i,j}(u;z_{1},\ldots,z_{n})=\begin{cases}u-z_{i}+\sum\limits_{j\not=i}hp_{ij}&\text{if
$i=j$},\\\ -h-p_{ij}&\text{if $i<j$},\\\ p_{ij}&\text{if $i>j$}.\end{cases}$

Then

$\displaystyle\operatorname{DET}\big{|}\overline{M}_{n}(u;\overline{\theta}_{1},\ldots,\overline{\theta}_{n})\big{|}=\prod_{j=1}^{n}(u-x_{j}).$

######  Comments 2.12.

Let us list a few more representations of the dynamical classical Yang–Baxter
relations.

  * •

Trigonometric Calogero–Moser representation. Let $i<j$, define

$\displaystyle u_{ij}={x_{j}\over
x_{i}-x_{j}}(s_{ij}-\epsilon),\qquad\epsilon=0\ \text{or}\ 1,$ $\displaystyle
s_{ij}(x_{i})=x_{j},\qquad s_{ij}(x_{j})=x_{i},\qquad
s_{ij}(x_{k})=x_{k},\qquad\forall\,k\not=i,j.$

  * •

Mixed representation:

$\displaystyle
u_{ij}=\left({\lambda_{j}\over\lambda_{i}-\lambda_{j}}-{x_{j}\over
x_{i}-x_{j}}\right)(s_{ij}-\epsilon),\qquad\epsilon=0\ \text{or}\ 1,\qquad
s_{ij}(\lambda_{k})=\lambda_{k},\qquad\forall\,k.$

We set $u_{ij}=-u_{ji}$, if $i>j$. In all cases we define Dunkl elements to be
$\theta_{i}=\sum\limits_{j\not=i}u_{ij}$.

Note that operators

$\displaystyle
r_{ij}=\left({\lambda_{i}+\lambda_{j}\over\lambda_{i}-\lambda_{j}}-{x_{i}+x_{j}\over
x_{i}-x_{j}}\right)s_{ij}$

satisfy the three term relations: $r_{ij}r_{jk}=r_{ik}r_{ij}+r_{jk}r_{ik}$,
and $r_{jk}r_{ij}=r_{ij}r_{jk}+r_{ik}r_{jk}$, and thus satisfy the classical
Yang–Baxter relations.

####  2.1.2 Step functions and the Dunkl–Uglov representations
of the degenerate affine Hecke algebras [138]

Consider step functions $\eta^{\pm}\colon\mathbb{R}\longrightarrow\\{0,1\\}$

$\displaystyle\text{(Heaviside
function)}\qquad\eta^{+}(x)=\begin{cases}1&\text{if $x\geq 0$},\\\ 0&\text{if
$x<0$},\end{cases}\qquad\eta^{-}(x)=\begin{cases}1&\text{if $x>0$},\\\
0&\text{if $x\leq 0$}.\end{cases}$

For any two real numbers $x_{i}$ and $x_{j}$ set
$\eta_{ij}^{\pm}=\eta^{\pm}(x_{i}-x_{j})$.

######  Lemma 2.13.

The functions $\eta_{ij}$ satisfy the following relations

$\displaystyle\eta_{ij}^{\pm}+\eta_{ji}^{\pm}=1+\delta_{x_{i},x_{j}},\qquad(\eta_{ij}^{\pm})^{2}=\eta_{ij}^{\pm},$
$\displaystyle\eta_{ij}^{\pm}\eta_{jk}^{\pm}=\eta_{ik}^{\pm}\eta_{ij}^{\pm}+\eta_{jk}^{\pm}\eta_{ik}^{\pm}-\eta_{ik}^{\pm},$

where $\delta_{x,y}$ denotes the Kronecker delta function.

To introduce the Dunkl–Uglov operators [138] we need a few more definitions
and notation. To start with, denote by $\Delta_{i}^{\pm}$ the finite
difference operators:
$\Delta_{i}^{\pm}(f)(x_{1},\ldots,x_{n})=f(\ldots,x_{i}\pm 1,\ldots)$. Let as
before, $\\{s_{ij},\,1\leq i\not=j\leq n,\,s_{ij}=s_{ji}\\}$, denotes the set
of transpositions in the symmetric group ${\mathbb{S}}_{n}$. Recall that
$s_{ij}(x_{i})=x_{j}$, $s_{ij}(x_{k})=x_{k}$, $\forall\,k\not=i,j$. Finally
define Dunkl–Uglov operators
$d_{i}^{\pm}\colon\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$ to be

$\displaystyle
d_{i}^{\pm}=\Delta_{i}^{\pm}+\sum_{j<i}\delta_{x_{i},x_{j}}-\sum_{j<i}\eta_{ji}^{\pm}s_{ij}+\sum_{j>i}\eta_{ij}^{\pm}s_{ij}.$

To simplify notation, set $u_{ij}^{\pm}:=\eta_{ij}^{\pm}s_{ij}$ if $i<j$, and
${\widetilde{\Delta}}_{i}^{\pm}=\Delta_{i}^{\pm}+\sum\limits_{j<i}\delta_{x_{i},x_{j}}$.

######  Lemma 2.14.

The operators $\\{u_{ij}^{\pm},\,1\leq i<j\leq n\\}$ satisfy the following
relations

$\displaystyle\big{[}u_{ij}^{\pm},u_{ik}^{\pm}+u_{jk}^{\pm}\big{]}+\big{[}u_{ik}^{\pm},u_{jk}^{\pm}\big{]}+\bigg{[}u_{ik}^{\pm},\sum_{j<i}\delta_{x_{i},x_{j}}\bigg{]}=0\qquad\text{if}\
\ i<j<k.$

From now on we assume that $x_{i}\in\mathbb{Z}$, $\forall\,i$, that is, we
will work with the restriction of the all operators defined at beginning of
Example 2.28(c), to the subset $\mathbb{Z}^{n}\subset\mathbb{R}^{n}$. It is
easy to see that under the assumptions $x_{i}\in\mathbb{Z}$, $\forall\,i$, we
will have

$\displaystyle\Delta_{j}^{\pm}\eta_{ij}^{\pm}=(\eta_{ij}^{\pm}\mp\delta_{x_{i},x_{j}})\Delta_{i}^{\pm}.$
(2.9)

Moreover, using relations (2.12), (2.13) one can prove that

######  Lemma 2.15.

  * •

$[u_{ij}^{\pm},{\widetilde{\Delta}}_{i}^{\pm}+{\widetilde{\Delta}}_{j}^{\pm}]=0$,

  * •

$[u_{ik}^{\pm},{\widetilde{\Delta}_{j}}^{\pm}]=\big{[}u_{ik}^{\pm},\sum\limits_{j<i}\delta_{x_{i},x_{j}}\big{]}$,
$i<j<k$.

######  Corollary 2.16.

  * •

The operators $\\{u_{ij}^{\pm},\,1\leq i<j<k\leq n\\}$, and
${\widetilde{\Delta}}_{i}^{\pm}$, $i=1,\ldots,n$ satisfy the dynamical
classical Yang–Baxter relations

$\displaystyle\big{[}u_{ij}^{\pm},u_{ik}^{\pm}+u_{jk}^{\pm}\big{]}+\big{[}u_{ik}^{\pm},u_{jk}^{\pm}\big{]}+\big{[}u_{ik}^{\pm},{\widetilde{\Delta}}_{j}\big{]}=0\qquad\text{if}\
\ i<j<k.$

  * •

The operators $\\{s_{i}:=s_{i,i+1},\,1\leq
i<n,\,and\,{\widetilde{\Delta}}_{j}^{\pm},\,1\leq j\leq n\\}$ give rise to two
representations of the degenerate affine Hecke algebra ${\mathfrak{H}}_{n}$.
In particular, the Dunkl–Uglov operators are mutually commute:
$[d_{i}^{\pm},d_{j}^{\pm}]=0$ [138].

####  2.1.3 Extended Kohno–Drinfeld algebra and Yangian Dunkl–Gaudin elements

######  Definition 2.17.

Extended Kohno–Drinfeld algebra is an associative algebra over
$\mathbb{Q}[\beta]$ generated by the elements $\\{z_{1},\ldots,z_{n}\\}$ and
$\\{y_{ij}\\}_{1\leq i\not=j\leq n}$ subject to the set of relations

  1. (i)

The elements $\\{y_{ij}\\{_{1\leq i\not=j\leq n}$ satisfy the Kohno–Drinfeld
relations

     * •

$y_{ij}=y_{ji}$, $[y_{ij},y_{kl}]=0$ if $i$, $j$, $k$, $l$ are distinct,

     * •

$[y_{ij},y_{ik}+y_{jk}]=0=[y_{ij}+y_{ik},y_{jk}]$ if $i<j<k$.

  2. (ii)

The elements $z_{1},\ldots,z_{n}$ generate the free associative algebra
${\cal{F}}_{n}$.

  3. (iii)

Crossing relations:

     * •

$[z_{i},y_{jk}]=0$ if $i\not=j,k$, $[z_{i},z_{j}]=\beta[y_{ij},z_{i}]$ if
$i\not=j$.

To define the (Yangian) Dunkl–Gaudin elements, cf. [54], let us consider a set
of elements $\\{p_{ij}\\}_{1\leq i\not=j\leq n}$ subject to relations

  * •

$p_{ij}+p_{ji}=\beta$, $[p_{ij},y_{kl}]=0=[p_{ij},z_{k}]$ for all $i$, $j$,
$k$,

  * •

$p_{ij}p_{jk}=p_{ik}(p_{jk}-p_{ji})$ if $i<j<k$.

Let us set $u_{ij}=p_{ij}y_{ij}$, $i\not=j$, and define the (Yangian)
Dunkl–Gaudin elements as follows

$\displaystyle\theta_{i}=z_{i}+\sum_{j\not=i}u_{ij},\qquad i=1,\ldots,n.$

######  Proposition 2.18 (cf. [54, Lemma 3.5]).

The elements $\theta_{1},\ldots,\theta_{n}$ form a mutually commuting family.

Indeed, let $i<j$, then

$\displaystyle[\theta_{i},\theta_{j}]=[z_{i},z_{j}]+\beta[z_{i},y_{ij}]+p_{ij}[y_{ij},z_{i}+z_{j}]$
$\displaystyle\hphantom{[\theta_{i},\theta_{j}]=}{}+\sum_{k\not=i,j}\big{(}p_{ik}p_{jk}\big{[}y_{ij}+y_{ik},y_{jk}\big{]}+p_{ik}p_{ji}\big{[}y_{ij},y_{ik}+y_{jk}\big{]}\big{)}=0.$

A representation of the extended Kohno–Drinfeld algebra has been constructed
in [54], namely one can take

$\displaystyle y_{ij}:=T_{ij}^{(1)}T_{ji}^{(1)}-T_{jj}^{(1)}=y_{ji},\qquad
z_{i}:=\beta
T_{ii}^{(2)}-\frac{\beta}{2}T_{ii}^{(1)}\big{(}T_{ii}^{(1)}-1\big{)},$
$\displaystyle p_{ij}:=\frac{\beta q_{j}}{q_{i}-q_{j}},\qquad i\not=j,$

where $q_{1},\ldots,q_{n}$ stands for a set of mutually commuting quantum
parameters, and $\big{\\{}T_{ij}^{(s)}\big{\\}}_{1\leq i,j\leq n\atop
s\in\mathbb{Z}_{\geq 0}}$ denotes the set of generators of the Yangian
$Y({\mathfrak{gl}}_{n})$, see, e.g., [106].

A proof that the elements $\\{z_{i}\\}_{1\leq i\leq n}$ and
$\\{y_{ij}\\}_{1\leq i\not=j\leq n}$ satisfy the extended Kohno–Drinfeld
algebra relations is based on the following relations, see, e.g., [54, Section
3],

$\displaystyle\big{[}T_{ij}^{(1)},T_{kl}^{(s)}\big{]}=\delta_{il}T_{kj}^{(s)}-\delta_{jk}T_{il}^{(s)},\qquad
i,j,k,l=1,\ldots,n,\qquad s\in\mathbb{Z}_{\geq 0}.$

###  2.2 “Compatible” Dunkl elements, Manin matrices and algebras
related with weighted complete graphs $\boldsymbol{rK_{n}}$

Let us consider a collection of generators $\\{u_{ij}^{(\alpha)},\,1\leq
i,j\leq n,\,\alpha=1,\ldots,r\\}$, subject to the following relations

  * •

either the unitarity (the case of sign “${+}$”) or the symmetry relations (the
case of sign “${-}$”)252525More generally one can impose the $q$-symmetry
conditions  $\displaystyle u_{ij}+qu_{ji}=0,\qquad 1\leq i<j\leq n$ and ask
about relations among the local Dunkl elements to ensure the commutativity of
the global ones. As one might expect, the matrix
$Q:=\big{(}\theta_{j}^{(a)}\big{)}_{1\leq a\leq r\atop 1\leq j\leq n}$
composed from the local Dunkl elements should be a $q$-Manin matrix. See,
e.g., [25], or https://en.wikipedia.org/wiki/Manin.matrix for a definition and
basic properties of the latter.

$\displaystyle u_{ij}^{(\alpha)}\pm
u_{ji}^{(\alpha)}=0,\qquad\forall\,\alpha,i,j,$ (2.10)

  * •

local $3$-term relations:

$\displaystyle
u_{ij}^{(\alpha)}u_{jk}^{(\alpha)}+u_{jk}^{(\alpha)}u_{ki}^{\alpha)}+u_{ki}^{(\alpha)}u_{ij}^{(\alpha)}=0,\qquad
i,j,k\ \ \text{are distinct},\quad 1\leq\alpha\leq r.$ (2.11)

We define global 3-term relations algebra $3T_{n,r}^{(\pm)}$ as “compatible
product” of the local 3-term relations algebras. Namely, we require that the
elements

$\displaystyle
U_{ij}^{({\boldsymbol{\lambda}})}:=\sum_{\alpha=1}^{r}\lambda_{\alpha}u_{ij}^{(\alpha)},\qquad
1\leq i,j\leq n,$

satisfy the global 3-term relations

$\displaystyle
U_{ij}^{(\boldsymbol{\lambda})}U_{jk}^{(\boldsymbol{\lambda})}+U_{jk}^{(\boldsymbol{\lambda})}U_{ki}^{(\boldsymbol{\lambda})}+U_{ki}^{(\boldsymbol{\lambda})}U_{ij}^{(\boldsymbol{\lambda})}=0$

for all values of parameters $\\{\lambda_{i}\in\mathbb{R},\,1\leq\alpha\leq
r\\}$.

It is easy to check that our request is equivalent to a validity of the
following sets of relations among the generators
$\big{\\{}u_{ij}^{(\alpha)}\big{\\}}$

  1. (a)

local $3$-term relations:
$u_{ij}^{(\alpha)}u_{jk}^{\alpha)}+u_{jk}^{(\alpha)}u_{ki}^{(\alpha)}+u_{ki}^{\alpha)}u_{ij}^{(\alpha)}=0$,

  2. (b)

$6$-term crossing relations:

$\displaystyle
u_{ij}^{(\alpha)}u_{jk}^{(\beta)}+u_{ij}^{(\beta)}u_{jk}^{(\alpha)}+u_{k,i}^{(\alpha)}u_{ij}^{(\beta)}u_{ki}^{(\alpha)}+u_{jk}^{(\alpha)}u_{ki}^{(\beta)}+u_{jk}^{(\beta)}u_{ki}^{(\alpha)}=0,$

$i$, $j$, $k$ are distinct, $\alpha\not=\beta$.

Now let us consider local Dunkl elements

$\displaystyle\theta_{i}^{(\alpha)}:=\sum_{j\neq i}u_{ij}^{(\alpha)},\qquad
j=1,\ldots,n,\quad\alpha=1,\ldots,r.$

It follows from the local 3-term relations (2.11) that for a fixed
$\alpha\in[1,r]$ the local Dunkl elements
$\big{\\{}\theta_{i}^{(\alpha)}\big{\\}}_{1\leq i\leq n\atop 1\leq\alpha\leq
r}$ either mutually commute (the sign “$+$”), or pairwise anticommute (the
sign “$-$”). Similarly, the global 3-term relations imply that the global
Dunkl elements

$\displaystyle\theta_{i}^{(\lambda)}:=\lambda_{1}\theta_{i}^{(1)}+\cdots+\lambda_{r}\theta_{i}^{(r)}=\sum_{j\not=i}U_{ij}^{(\lambda)},\qquad
i=1,\ldots,n,$

also either mutually commute (the case “$+$”) or pairwise anticommute (the
case “$-$”).

Now we are looking for a set of relations among the local Dunkl elements which
is a consequence of the commutativity (anticommutativity) of the global Dunkl
elements. It is quite clear that if $i<j$, then

$\displaystyle\big{[}\theta_{i}^{(a)},\theta_{j}^{(b)}\big{]}_{\pm}=\sum_{a=1}^{r}\lambda_{a}^{2}\big{[}\theta_{i}^{(a)},\theta_{j}^{(a)}\big{]}_{\pm}+\sum_{1\leq
a<b\leq
r}\lambda_{a}\lambda_{b}\big{(}\big{[}\theta_{i}^{(a)},\theta_{j}^{(b)}\big{]}_{\pm}+\big{[}\theta_{i}^{(b)},\theta_{j}^{(a)}\big{]}_{\pm}\big{)},$

and the commutativity (or anticommutativity) of the global Dunkl elements for
all $(\lambda_{1},\ldots,\lambda_{r})\in\mathbb{R}^{r}$ is equivalent to the
following set of relations

  * •

$[\theta_{i}^{(a)},\theta_{j}^{(a)}]_{\pm}=0$,

  * •

$[\theta_{i}^{(a)},\theta_{j}^{(b)}]_{\pm}+[\theta_{i}^{(b)},\theta_{j}^{(a)}]_{\pm}=0$,
$a<b$ and $i<j$, where by definition we set $[a,b]_{\pm}:=ab\mp ba$.

In other words, the matrix
$\varTheta_{n}:=\big{(}\theta_{i}^{(a)}\big{)}_{1\leq a\leq r\atop 1\leq i\leq
n}$ should be either a Manin matrix (the case “$+$”), or its super analogue
(the case “$-$”). Clearly enough that a similar construction can be applied to
the algebras studied in Section 2, I–III, and thus it produces some
interesting examples of the Manin matrices. It is an interesting problem to
describe the algebra generated by the local Dunkl elements
$\big{\\{}\theta_{i}^{(a)}\big{\\}}_{1\leq a\leq r\atop 1\leq i\leq n}$ and a
commutative subalgebra generated by the global Dunkl elements inside the
former. It is also an interesting question whether or not the coefficients
$C_{1},\ldots,C_{n}$ of the column characteristic polynomial
$\operatorname{Det}^{\rm
col}|\varTheta_{n}-tI_{n}|=\sum\limits_{k=0}^{n}C_{k}t^{n-k}$ of the Manin
matrix $\varTheta_{n}$ generate a commutative subalgebra? For a definition of
the column determinant of a matrix, see, e.g., [25].

However a close look at this problem and the question posed needs an
additional treatment and has been omitted from the content of the present
paper.

Here we are looking for a “natural conditions” to be imposed on the set of
generators $\\{u_{ij}^{\alpha}\\}_{1\leq\alpha\leq r\atop 1\leq i,j\leq n}$ in
order to ensure that the local Dunkl elements satisfy the commutativity (or
anticommutativity) relations:

$\displaystyle\big{[}\theta_{i}^{(\alpha)},\theta_{j}^{(\beta)}\big{]}_{\pm}=0,\qquad\text{for
all}\ \ 1\leq i<j\leq n,\qquad 1\leq\alpha,\beta\leq r.$

The “natural conditions” we have in mind are

  * •

locality relations:

$\displaystyle\big{[}u_{ij}^{(\alpha)},u_{kl}^{(\beta)}\big{]}_{\pm}=0,$
(2.12)

  * •

twisted classical Yang–Baxter relations:

$\displaystyle\big{[}u_{ij}^{(\alpha)},u_{jk}^{(\beta)}\big{]}_{\pm}+\big{[}u_{ik}^{(\alpha)},u_{ji}^{(\beta)}\big{]}_{\pm}+\big{[}u_{ik}^{(\alpha)},u_{jk}^{(\beta)}\big{]}_{\pm}=0,$
(2.13)

if $i$, $j$, $k$, $l$ are distinct and $1\leq\alpha,\beta\leq r$.

Finally we define a multiple analogue of the three term relations algebra,
denoted by $3T^{\pm}(rK_{n})$, to be the quotient of the global $3$-term
relations algebra $3T_{n,r}^{\pm}$ modulo the two-sided ideal generated by the
left hand sides of relations (2.12), (2.13) and that of the following
relations

  * •

$\big{(}u_{ij}^{(\alpha)}\big{)}^{2}=0$,
$\big{[}u_{ij}^{(\alpha)},u_{ij}^{(\beta)}\big{]}_{\pm}=0$, for all $i\not=j$,
$\alpha\not=\beta$.

The outputs of this construction are

  * •

commutative (or anticommutative) quadratic algebra $3T^{(\pm)}(rK_{n})$
generated by the elements $\big{\\{}u_{ij}^{(\alpha)}\big{\\}}_{1\leq i<j\leq
n\atop\alpha=1,\ldots,r}$,

  * •

a family of $nr$ either mutually commuting (the case “$+$”), or pair-wise
anticommuting (the case “$-$”) local Dunkl elements
$\big{\\{}\theta_{i}^{(\alpha)}\big{\\}}_{i=1,\ldots,n\atop\alpha=1,\ldots,r}$.

We expect that the subalgebra generated by local Dunkl elements in the algebra
$3T^{+}(rK_{n})$ is closely related (isomorphic for $r=2$) with the
coinvariant algebra of the diagonal action of the symmetric group
${\mathbb{S}}_{n}$ on the ring of polynomials
$\mathbb{Q}\big{[}X_{n}^{(1)},\ldots,X_{n}^{(r)}\big{]}$, where $X_{n}^{(j)}$
stands for the set of variables
$\big{\\{}x_{1}^{(j)},\ldots,x_{n}^{(j)}\big{\\}}$. The algebra
$3T^{-}(2K_{n})^{\rm anti}$ has been studied in [72] and [12]. In the present
paper we state only our old conjecture.

######  Conjecture 2.19 (A.N. Kirillov, 2000).

$\displaystyle{\rm Hilb}\big{(}3T^{-}(3K_{n})^{\rm
anti},t\big{)}=(1+t)^{n}(1+nt)^{n-2},$

where for any algebra $A$ we denote by $A^{\rm anti}$ the quotient of algebra
$A$ by the two-sided ideal generated by the set of anticommutators
$\\{ab+ba\,|\,(a,b)\in A\times A\\}$.

According to observation of M. Haiman [55], the number $2^{n}(n+1)^{n-2}$ is
thought of as being equal to the dimension of the space of triple coinvariants
of the symmetric group $\mathbb{S}_{n}$.

###  2.3 Miscellany

####  2.3.1 Non-unitary dynamical classical Yang–Baxter algebra
$\boldsymbol{{\rm DCYB}_{n}}$

Let $\widetilde{{\cal A}_{n}}$ be the quotient of the algebra
${\mathfrak{F}}_{n}$ by the two-sided ideal generated by the relations (2.2),
(2.5) and (2.6). Consider elements

$\displaystyle\theta_{i}=x_{i}+\sum_{a\not=i}u_{ia}\qquad\text{and}\qquad{\bar{\theta_{j}}}=-x_{j}+\sum_{b\not=j}u_{bj},\qquad
1\leq i<j\leq n.$

Clearly, if $i<j$, then

$\displaystyle[\theta_{i},{\bar{\theta}_{j}}]+[x_{i},x_{j}]=\left[\sum_{k=1}^{n}x_{k},u_{ij}\right]+\sum_{k\not=i,j}w_{ikj},$

where the elements $w_{ijk}$, $i<j$, have been defined in Lemma 2.2, equation
(2.3).

Therefore the elements $\theta_{i}$ and ${\bar{\theta}_{j}}$ commute in the
algebra ${\widetilde{A}_{n}}$.

In the case when $x_{i}=0$ for all $i=1,\ldots,n$, the relations

$\displaystyle w_{ijk}:=[u_{ij},u_{ik}+u_{jk}]+[u_{ik},u_{jk}]=0\qquad\text{if
$i$, $j$, $k$ are all distinct},$

are well-known as the non-unitary classical Yang–Baxter relations. Note that
for a given triple of pair-wise distinct $(i,j,k)$ one has in fact 6
relations. These six relations imply that $[\theta_{i},{\bar{\theta_{j}}}]=0$.
However, in general,

$\displaystyle[\theta_{i},\theta_{j}]=\biggl{[}\sum_{k\not=i,j}u_{ik},u_{ij}+u_{ji}\biggr{]}\not=0.$

Dynamical classical Yang–Baxter algebra ${\rm DCYB}_{n}$. In order to ensure
the commutativity relations among the Dunkl elements (2.1), i.e.,
$[\theta_{i},\theta_{j}]=0$ for all $i$, $j$, let us remark that if $i\not=j$,
then

$\displaystyle[\theta_{,}\theta_{j}]=[x_{i}+u_{ij},x_{j}+u_{ji}]+[x_{i}+x_{j},u_{ij}]+\left[u_{ij},\sum_{k=1}^{n}x_{k}\right]$
$\displaystyle\hphantom{[\theta_{,}\theta_{j}]=}{}+\sum_{k=1\atop
k\not=i,j}^{n}[u_{ij}+u_{ik},u_{jk}]+[u_{ik},u_{ji}]+[x_{i},u_{jk}]+[u_{ik},x_{j}]+[x_{k},u_{ij}].$

######  Definition 2.20.

Define dynamical non-unitary classical Yang–Baxter algebra ${\rm DNUCYB}_{n}$
to be the quotient of the free associative algebra
$\mathbb{Q}\langle\\{x_{i},\,1\leq i\leq n\\},\,\\{u_{ij}\\}_{1\leq
i\not=j\leq n}\rangle$ by the two-sided ideal generated by the following set
of relations

  * •

zero curvature conditions:

$\displaystyle[x_{i}+u_{ij},x_{j}+u_{ji}]=0,\qquad 1\leq i\not=j\leq n,$
(2.14)

  * •

conservation laws conditions:

$\displaystyle\left[u_{ij},\sum_{k=1}^{n}x_{k}\right]=0\qquad\text{for all}\ \
i\not=j,k.$

  * •

crossing relations:

$\displaystyle[x_{i}+x_{j},u_{ij}]=0,\qquad i\not=j.$

  * •

twisted dynamical classical Yang–Baxter relations:

$\displaystyle[u_{ij}+u_{ik},u_{jk}]+[u_{ik},u_{ji}]+[x_{i},u_{jk}]+[u_{ik},x_{j}]+[x_{k},u_{ij}]=0,$

$i$, $j$, $k$ are distinct.

It is easy to see that the twisted classical Yang–Baxter relations

$\displaystyle[u_{ij}+u_{ik},u_{jk}]+[u_{ik},u_{ji}]=0,\qquad i,j,k\ \
\text{are distinct},$ (2.15)

for a fixed triple of distinct indices $i$, $j$, $k$ contain in fact $3$
different relations whereas the non-unitary classical Yang–Baxter relations

$\displaystyle[u_{ij}+u_{ik},u_{jk}]+[u_{ij},u_{ik}],\qquad i,j,k\ \ \text{are
distinct},$

contain $6$ different relations for a fixed triple of distinct indices $i$,
$j$, $k$.

######  Definition 2.21.

  * •

Define dynamical classical Yang–Baxter algebra ${\rm DCYB}_{n}$ to be the
quotient of the algebra ${\rm DNUCYB}_{n}$ by the two-sided ideal generated by
the elements

$\displaystyle\sum_{k\not=i,j}[u_{ik},u_{ij}+u_{ji}]\qquad\text{for all}\ \
i\not=j.$

  * •

Define classical Yang–Baxter algebra ${\rm CYB}_{n}$ to be the quotient of the
dynamical classical Yang–Baxter algebra ${\rm DCYB}_{n}$ by the set of
relations

$\displaystyle x_{i}=0\qquad\text{for}\ \ i=1,\dots,n.$

######  Example 2.22.

Define

$\displaystyle
p_{ij}(z_{1},\ldots,z_{n})=\begin{cases}\dfrac{z_{i}}{z_{i}-z_{j}}&\text{if
$1\leq i<j\leq n$},\vspace{1mm}\\\ -\dfrac{z_{j}}{z_{j}-z_{i}}&\text{if $n\geq
i>j\geq 1$}.\end{cases}$

Clearly, $p_{ij}+p_{ji}=1$. Now define operators $u_{ij}=p_{ij}s_{ij}$, and
the truncated Dunkl operators to be $\theta_{i}=\sum\limits_{j\not=i}u_{ij}$,
$i=1,\ldots,n$. All these operators act on the field of rational functions
$\mathbb{Q}(z_{1},\ldots,z_{n})$; the operator $s_{ij}=s_{ji}$ acts as the
exchange operator, namely, $s_{ij}(z_{i})=z_{j}$, $s_{ij}(z_{k})=z_{k}$,
$\forall\,k\not=i,j$, $s_{ij}(z_{j})=z_{i}$.

Note that this time one has

$\displaystyle p_{12}p_{23}=p_{13}p_{12}+p_{23}p_{13}-p_{13}.$

It is easy to see that the operators $\\{u_{ij},\,1\leq i\not=j\leq n\\}$
satisfy relations (3.1), and therefore, satisfy the twisted classical
Yang–Baxter relations (2.13). As a corollary we obtain that the truncated
Dunkl operators $\\{\theta_{i},\,i=1,\ldots,n\\}$ are pair-wise commute. Now
consider the Dunkl operator $D_{i}=\partial_{{z_{i}}}+h\theta_{i}$,
$i=1,\ldots,n$, where $h$ is a parameter. Clearly that
$[\partial_{{z_{i}}}+\partial_{{z_{j}}},u_{ij}]=0$, and therefore
$[D_{i},D_{j}]=0$, $\forall\,i,j$. It easy to see that

$\displaystyle
s_{i,i+1}D_{i}-D_{i+1}s_{i,i+1}=h,\qquad[D_{i},s_{j,j+1}]=0\qquad\text{if}\ \
j\not=i,i+1.$

In such a manner we come to the well-known representation of the degenerate
affine Hecke algebra ${\mathfrak{H}}_{n}$.

####  2.3.2 Dunkl and Knizhnik–Zamolodchikov elements

Assume that $\forall\,i$, $x_{i}=0$, and generators $\\{u_{ij},\,1\leq i<j\leq
n\\}$ satisfy the locality conditions (2.2) and the classical Yang–Baxter
relations

$\displaystyle[u_{ij},u_{ik}+u_{jk}]+[u_{ik},u_{jk}]=0\qquad\text{if}\ \ 1\leq
i<j<k\leq n.$

Let $y,z,t_{1},\ldots,t_{n}$ be parameters, consider the rational function

$\displaystyle F_{\rm CYB}(z;{\boldsymbol{t}}):=F_{\rm
CYB}(z;t_{1},\ldots,t_{n})=\sum_{1\leq i<j\leq
n}{(t_{i}-t_{j})u_{ij}\over(z-t_{i})(z-t_{j})}.$

Then

$\displaystyle[F_{\rm CYB}(z;{\boldsymbol{t}}),F_{\rm
CYB}(y;{\boldsymbol{t}})]=0\qquad\text{and}\qquad\operatorname{Res}_{z=t_{i}}F_{\rm
CYB}(z;{\boldsymbol{t}})=\theta_{i}.$

Now assume that a set of generators $\\{c_{ij},\,1\leq i\not=j\leq n\\}$
satisfy the locality and symmetry (i.e., $c_{ij}=c_{ji}$) conditions, and the
Kohno–Drinfeld relations:

$\displaystyle[c_{ij},c_{kl}]=0\qquad\text{if}\ \
\\{i,j\\}\cap\\{k,l\\}={\varnothing},$
$\displaystyle[c_{ij},c_{jk}+c_{ik}]=0=[c_{ij}+c_{ik},c_{jk}],\qquad i<j<k.$

Let $y,z,t_{1},\ldots,t_{n}$ be parameters, consider the rational function

$\displaystyle F_{\rm KD}(z;{\boldsymbol{t}}):=F_{\rm
KD}(z;t_{1},\ldots,t_{n})=\sum_{1\leq i\not=j\leq
n}{c_{ij}\over(z-t_{i})(t_{i}-t_{j})}=\sum_{1\leq i<j\leq
n}{c_{ij}\over(z-t_{i})(z-t_{j})}.$

Then

$\displaystyle[F_{\rm KD}(z;{\boldsymbol{t}}),F_{\rm
KD}(y;{\boldsymbol{t}})]=0\qquad\text{and}\qquad\operatorname{Res}_{z=t_{i}}F_{\rm
KD}(z;{\boldsymbol{t}})={\rm KZ}_{i},$

where

$\displaystyle{\rm KZ}_{i}=\sum_{j=1\atop j\not=i}^{n}{c_{ij}\over
t_{i}-t_{j}}$

denotes the truncated Knizhnik–Zamolodchikov element.

####  2.3.3 Dunkl and Gaudin operators

(a) Rational Dunkl operators. Consider the quotient of the algebra ${\rm
DCYB}_{n}$, see Definition 2.3, by the two-sided ideal generated by elements

$\displaystyle\\{[x_{i}+x_{j},u_{ij}]\\}\qquad\text{and}\qquad\\{[x_{k},u_{ij}],\,k\not=i,j\\}.$

Clearly the Dunkl elements (2.1) mutually commute. Now let us consider the so-
called Calogero–Moser representation of the algebra ${\rm DCYB}_{n}$ on the
ring of polynomials $R_{n}:=\mathbb{R}[z_{1},\ldots,z_{n}]$ given by

$\displaystyle x_{i}(p(z))=\lambda{\partial p(z)\over\partial z_{i}},\qquad
u_{ij}(p(z))={1\over z_{i}-z_{j}}(1-s_{ij})p(z),\qquad p(z)\in R_{n}.$

The symmetric group ${\mathbb{S}}_{n}$ acts on the ring $R_{n}$ by means of
transpositions $s_{ij}\in{\mathbb{S}}_{n}$: $s_{ij}(z_{i})=z_{j}$,
$s_{ij}(z_{j})=z_{i}$, $s_{ij}(z_{k})=z_{k}$ if $k\not=i,j$.

In the Calogero–Moser representation the Dunkl elements $\theta_{i}$ becomes
the rational Dunkl operators [35], see Definition 1.1. Moreover, one has
$[x_{k},u_{ij}]=0$ if$k\not=i,j$, and

$\displaystyle x_{i}u_{ij}=u_{ij}x_{j}+{1\over
z_{i}-z_{j}}(x_{i}-x_{j}-u_{ij}),\qquad x_{j}u_{ij}=u_{ij}x_{i}-{1\over
z_{i}-z_{j}}(x_{i}-x_{j}-u_{ij}).$

(b) Gaudin operators. The Dunkl–Gaudin representation of the algebra ${\rm
DCYB}_{n}$ is defined on the field of rational functions
$K_{n}:=\mathbb{R}(q_{1},\ldots,q_{n})$ and given by

$\displaystyle x_{i}(f(q)):=\lambda{\partial f(q)\over\partial q_{i}},\qquad
u_{ij}={s_{ij}\over q_{i}-q_{j}},\qquad f(q)\in K_{n},$

but this time we assume that $w(q_{i})=q_{i}$, $\forall\,i\in[1,n]$ and for
all $w\in{\mathbb{S}}_{n}$. In the Dunkl–Gaudin representation the Dunkl
elements becomes the rational Gaudin operators, see, e.g., [108]. Moreover,
one has $[x_{k},u_{ij}]=0$, if $k\not=i,j$, and

$\displaystyle x_{i}u_{ij}=u_{ij}x_{j}-{u_{ij}\over q_{i}-q_{j}},\qquad
x_{j}u_{ij}=u_{ij}x_{i}+{u_{ij}\over q_{i}-q_{j}}.$

######  Comments 2.23.

It is easy to check that if $f\in\mathbb{R}[z_{1},\ldots,z_{n}]$, and
$x_{i}:={\frac{\partial}{\partial z_{i}}}$, then the following commutation
relations are true

$\displaystyle x_{i}f=fx_{i}+\frac{\partial}{\partial_{z_{i}}}(f),\qquad
u_{ij}f=s_{ij}(f)u_{ij}+\partial_{z_{i},z_{j}}(f).$

Using these relations it easy to check that in the both cases
$({\boldsymbol{a}})$ and $({\bf b})$ the elementary symmetric polynomials
$e_{k}(x_{1},\ldots,x_{n})$ commute with the all generators
$\\{u_{ij}\\}_{1\leq i,j\leq n}$, and therefore commute with the all Dunkl
elements $\\{\theta_{i}\\}_{1\leq i\leq n}$. Let us stress that
$[\theta_{i},x_{k}]\not=0$ for all $1\leq i,k\leq n$.

######  Project 2.24.

Describe a commutative algebra generated by the Dunkl elements
$\\{\theta_{i}\\}_{1\leq i\leq n}$ and the elementary symmetric polynomials
$\\{e_{k}(x_{1},\ldots,x_{n})\\}_{1\leq k\leq n}$.

####  2.3.4 Representation of the algebra $\boldsymbol{3T_{n}}$ on the free
algebra $\boldsymbol{\mathbb{Z}\langle t_{1},\ldots,t_{n}\rangle}$

Let ${\mathcal{F}}_{n}=\mathbb{Z}\langle t_{1},\ldots,t_{n}\rangle$ be free
associative algebra over the ring of integers $\mathbb{Z}$, equipped with the
action of the symmetric group $\mathbb{S}_{n}$: $s_{ij}(t_{i})=t_{j}$,
$s_{ij}(t_{k})=t_{k}$, $\forall\,k\not=i,j$.

Define the action of $u_{ij}\in 3T_{n}$ on the set of generators of the
algebra $\mathcal{F}_{n}$ as follows

$\displaystyle u_{ij}(t_{k})=\delta_{i,k}t_{i}t_{j}-\delta_{j,k}t_{j}t_{i}.$

The action of generator $u_{ij}$ on the whole algebra $\mathcal{F}_{n}$ is
defined by linearity and the twisted Leibniz rule:

$\displaystyle u_{ij}(1)=0,\qquad u_{ij}(a+b)=u_{ij}(a)+u_{ij}(b),\qquad
u_{ij}(ab)=u_{ij}(a)b+s_{ij}(a)u_{ij}(b).$

It is easy to see from (2.14) that

$\displaystyle s_{ij}u_{jk}=u_{ik}s_{ij},\qquad
s_{ij}u_{kl}=u_{kl}s_{ij}\qquad\text{if}\ \
\\{i,j\\}\cap\\{k,l\\}=\varnothing,\qquad u_{ij}+u_{ji}=0.$

Now let us consider operator

$\displaystyle u_{ijk}:=u_{ij}u_{jk}-u_{jk}u_{ik}-u_{ik}u_{ij},\qquad 1\leq
i<j<k\leq n.$

######  Lemma 2.25.

$\displaystyle u_{ijk}(ab)=u_{ijk}(a)b+s_{ij}s_{jk}(a)u_{ijk}(b),\qquad
a,b\in\mathcal{F}_{n}.$

######  Lemma 2.26.

$\displaystyle u_{ijk}(a)=0\qquad\forall\,a\in\mathcal{F}_{n}.$

Indeed,

$\displaystyle
u_{ijk}(t_{i})=-u_{jk}(u_{ij}(t_{i}))-u_{ik}(u_{ij}(t_{i}))=-t_{i}u_{jk}(t_{k})-u_{ik}(t_{i})t_{j}=t_{i}(t_{k}t_{j})-(t_{i}t_{k})t_{j}=0,$
$\displaystyle
u_{ijk}(t_{k})=u_{ij}(u_{jk}(t_{k}))-u_{jk}(u_{ik}(t_{k}))=-u_{ij}(t_{k}t_{j})+u_{jk}(t_{k}t_{i})=t_{k}(u_{ij}(t_{j})+u_{jk}(t_{k})t_{i}=0,$
$\displaystyle
u_{ijk}(t_{j})=u_{ij}(u_{jk}(t_{j}))-u_{ik}(u_{ij}(t_{j}))=-u_{ij}(t_{j})t_{k}-t_{j}u_{ik}(t_{i})=(t_{j}t_{i})t_{k}-t_{j}(t_{i}t_{k})=0.$

Therefore Lemma 2.26 follows from Lemma 2.25.

Let $\mathcal{F}_{n}^{\bullet}$ be the quotient of the free algebra
$\mathcal{F}_{n}$ by the two-sided ideal generated by elements
$t_{i}^{2}t_{j}-t_{j}t_{i}^{2}$, $1\leq i\not=j\leq n$. Since
$u_{i,j}^{2}(t_{i})=t_{i}t_{j}^{2}-t_{j}^{2}t_{i}$, one can define a
representation of the algebra $3T_{n}^{(0)}$ on that
$\mathcal{F}_{n}^{\bullet}$. One can also define a representation of the
algebra $3T_{n}^{(0)}$ on that $\mathcal{F}_{n}^{(0)}$, where
$\mathcal{F}_{n}^{(0)}$ denotes the quotient of the algebra $\mathcal{F}_{n}$
by the two-sided ideal generated by elements $\\{t_{i}^{2},\,1\leq i\leq
n\\}$. Note that
$(u_{i,k}u_{j,k}u_{i,j})(t_{k})=[t_{i}t_{j}t_{i},t_{k}]\not=0$ in the algebra
$\mathcal{F}_{n}^{(0)}$, but the elements $u_{i,j}u_{i,k}u_{j,k}u_{i,j}$,
$1\leq i<j<k\leq n$, which belong to the kernel of the Calogero–Moser
representation [72], act trivially both on the algebras
$\mathcal{F}_{n}^{(0)}$ and that $\mathcal{F}_{n}^{\bullet}$.

Note finally that the algebra $\mathcal{F}_{n}^{(0)}$ is Koszul and has
Hilbert series ${\rm Hilb}\big{(}\mathcal{F}_{n}^{(0)},t\big{)}={1+t\over
1-(n-1)t}$, whereas the algebra $\mathcal{F}_{n}^{\bullet}$ is not Koszul for
$n\geq 3$, and

$\displaystyle{\rm
Hilb}(\mathcal{F}_{n}^{\bullet},t)={1\over(1-t)(1-(n-1)t)(1-t^{2})^{n-1}}.$

In Appendix A.5 we apply the representation introduced in this section to the
study of relations in the subalgebra $Z_{n}^{(0)}$ of the algebra
$3T_{n}^{(0)}$ generated by the elements $u_{1,n},\ldots,u_{n-1,n}$. To
distinguish the generators $\\{u_{ij}\\}$ of the algebra $3T_{n}^{(0)}$ from
the introduced in this section operators $u_{ij}$ acting on it, in Appendix
A.5 we will use for the latter notation $\nabla_{ij}:=u_{ij}$.

####  2.3.5 Kernel of Bruhat representation

Bruhat representations, classical and quantum, of algebras $3T_{n}^{(0)}$ and
$3QT_{n}$ can be seen as a connecting link between commutative subalgebras
generating by either additive or multiplicative Dunkl elements in these
algebras, and classical and quantum Schubert and Grothendieck calculi.

$(\bf Ia)$ Bruhat representation of algebra $3T_{n}^{(0)}$, cf. [45]. Define
action of $u_{i,j}\in 3T_{n}^{(0)}$ on the group ring of the symmetric group
$\mathbb{Z}[{\mathbb{S}}_{n}]$ as follows: let $w\in{\mathbb{S}}_{n}$, then

$\displaystyle u_{i,j}w=\begin{cases}ws_{ij}&\text{if \
$l(ws_{ij})=l(w)+1$},\\\ 0&\text{otherwise}.\end{cases}$

Let us remind that $s_{ij}\in{\mathbb{S}}_{n}$ denotes the transposition that
interchanges $i$ and $j$ and fixes each $k\not=i,j$; for each permutation
$u\in{\mathbb{S}}_{n}$, $l(u)$ denotes its length.

$(\bf Ib)$ Quantum Bruhat representation of algebra $3QT_{n}$, cf. [45]. Let
us remind that algebra $3QT_{n}$ is the quotient of the 3-term relations
algebra $3T_{n}$ by the two-sided ideal generated by the elements

$\displaystyle\\{u_{ij}^{2},|j-i|\geq
2\\}\bigcup\\{u_{i,i+1}^{2}=q_{i},~{}i=1,\ldots,n-1\\}.$

Define the $\mathbb{Z}[q]-$linear action of $u_{i,j}\in 3QT_{n}$, $i<j$, on
the extended group ring of the symmetric group
$\mathbb{Z}[q][{\mathbb{S}}_{n}]$ as follows: let $w\in{\mathbb{S}}_{n}$, and
$q_{ij}=q_{i}q_{i+1}\cdots q_{j-1}$, $i<j$, then

$\displaystyle u_{i,j}w=\begin{cases}ws_{ij}&\text{if \
$l(ws_{ij})=l(w)+1$},\\\ q_{ij}ws_{ij}&\text{if \
$l(ws_{ij})=l(w)-l(s_{ij})$},\\\ 0&\text{otherwise}.\end{cases}$

Let us remind, see, e.g., [92], that in general one has

$\displaystyle l(ws_{ij})=\begin{cases}l(w)-2e_{ij}-1&\text{if}\ \
w(i)>w(j),\\\ l(w)+2~{}e_{ij}+1&\text{if}\ \ w(i)<w(j).\end{cases}$

Here $e_{ij}(w)$ denotes the number of $k$ such that $i<k<j$ and $w(k)$ lies
between $w(i)$ and $w(j)$. In particular, $l(ws_{ij})=l(w)+1$ iff
$e_{ij}(w)=0$ and $w(i)<w(j)$; $l(ws_{ij})=l(w)-l(s_{ij})=l(w)-2(j-i)+1$ iff
$w(i)>w(j)$ and $e_{ij}=j-i-1$ is the maximal possible.

$({\bf II})$ Kernel of the Bruhat representation. It is not difficult to see
that the following elements of degree three and four belong to the kernel of
the Bruhat representation:

$\displaystyle({\bf IIa})\quad u_{i,j}u_{i,k}u_{i,j}\qquad\text{and}\qquad
u_{i,k}u_{j,k}u_{i,k}\qquad\text{if}\ \ 1\leq i<j<k\leq n;$
$\displaystyle({\bf IIb})\quad u_{i,k}u_{i,l}u_{j,l}\qquad\text{and}\qquad
u_{j,l}u_{i,l}u_{i,k};$ $\displaystyle({\bf IIc})\quad
u_{il}u_{ik}u_{jl}u_{il},\qquad u_{il}u_{ij}u_{kl}u_{il},\qquad
u_{ik}u_{il}u_{jk}u_{ik},$ $\displaystyle\hphantom{({\bf
IIc})\quad}u_{ij}u_{ik}u_{il}u_{ij},\qquad
u_{ik}u_{il}u_{ij}u_{ik}\qquad\text{if}\ \ 1\leq i<j<k<l\leq n.$

This observation motivates the following definition.

######  Definition 2.27.

The reduced 3-term relation algebra $3T_{n}^{\rm red}$ is defined to be the
quotient of the algebra $3T_{n}^{(0)}$ by the two-sided ideal generated by the
elements displayed in IIa–IIc above.

######  Example 2.28.

$\displaystyle{\rm Hilb}\big{(}3T_{3}^{\rm
red},t\big{)}=(1,3,4,1),\qquad\dim\big{(}3T_{3}^{\rm red}\big{)}=9,$
$\displaystyle{\rm Hilb}\big{(}3T_{4}^{\rm
red},t\big{)}=(1,6,19,32,19,6,1),\qquad\dim\big{(}3T_{4}^{\rm red}\big{)}=84,$
$\displaystyle{\rm Hilb}\big{(}3T_{5}^{\rm
red},t\big{)}=(1,10,55,190,383,370,227,102,34,8,1),\qquad\dim\big{(}3T_{5}^{\rm
red}\big{)}=1374.$

We expect that $\dim(3T_{n}^{red})_{{n\choose 2}-1}=2(n-1)$ if $n\geq 3$.

######  Theorem 2.29.

  1. $1.$

The algebra $3T_{n}^{\rm red}$ is finite-dimensional, and its Hilbert
polynomial has degree ${n\choose 2}$.

  2. $2.$

The maximal degree ${n\choose 2}$ component of the algebra $3T_{n}^{\rm red}$
has dimension one and generated by any element which is equal to the product
$($in any order$)$ of all generators of the algebra $3T_{n}^{\rm red}$.

  3. $3.$

The subalgebra in $3T_{n}^{\rm red}$ generated by the elements
$\\{u_{i,i+1},\,i=1,\ldots,n-1\\}$ is canonically isomorphic to the nil-
Coxeter algebra ${\rm NC}_{n}$. In particular, its Hilbert polynomial is equal
to $[n]_{t}!:=\prod\limits_{j=1}^{n}{(1-t^{j})\over 1-t}$, and the element
$\prod\limits_{j=1}^{n-1}\prod\limits_{a=j}^{1}u_{a,a+1}$ of degree ${n\choose
2}$ generates the maximal degree component of the algebra $3T_{n}^{\rm red}$.

  4. $4.$

The subalgebra over $\mathbb{Z}$ generated by the Dunkl elements
$\\{\theta_{1},\ldots,\theta_{n}\\}$ in the algebra $3T_{n}^{\rm red}$ is
canonically isomorphic to the cohomology ring $H^{*}({\cal
F}l_{n},\mathbb{Z})$ of the type $A$ flag variety ${\cal F}l_{n}$.

A definition of the nil-Coxeter algebra ${\rm NC}_{n}$ one can find in Section
4.1.1. It is known, see [8] or Section 4.1.1, that the subalgebra generated by
the elements $\\{u_{i,i+1},\,i=1,\ldots,n-1\\}$ in the whole algebra
$3T_{n}^{(0)}$ is canonically isomorphic to the nil-Coxeter algebra ${\rm
NC}_{n}$ as well.

We expect that the kernel of the Bruhat representation of the algebra
$3T_{n}^{(0)}$ is generated by all monomials of the form
$u_{i_{1},j_{1}}\cdots u_{i_{k},j_{k}}$ such that the sequence of
transpositions $t_{i_{1},j_{1}},\ldots,t_{i_{k},j_{k}}$ does not correspond to
a path in the Bruhat graph of the symmetric group ${\mathbb{S}}_{n}$. For
example if $1\leq i<j<k<l\leq n$, the elements $u_{i,k}u_{i,l}u_{j,l}$ and
$u_{j,l}u_{i,l}u_{i,k}$ do belong to the kernel of the Bruhat representation.

######  Problem 2.30.

  1. $1.$

The image of the Bruhat representation of the algebra $3T_{n}^{(0)}$ defines a
subalgebra

$\displaystyle\operatorname{Im}\big{(}3T_{n}^{(0)}\big{)}\subset\operatorname{End}_{\mathbb{Q}}(\mathbb{Q}[{\mathbb{S}}_{n}]).$

Does this image isomorphic to the algebra $3T_{n}^{\rm red}$? Compute Hilbert
polynomials of algebras $\operatorname{Im}\big{(}3T_{n}^{(0)}\big{)}$ and
$3T_{n}^{\rm red}$.

  2. $2.$

Describe the image$($s$)$ of the affine nil-Coxeter algebra ${\widetilde{{\rm
NC}}}_{n}$, see Section 4.1.1, in the algebras $3T_{n}^{\rm red}$ and
$\operatorname{End}_{\mathbb{Q}}(\mathbb{Q}[{\mathbb{S}}_{n}])$.

####  2.3.6 The Fulton universal ring [47], multiparameter quantum cohomology
of flag varieties [45] and the full Kostant–Toda lattice [29, 80]

Let $X_{n}=(x_{1},\ldots,x_{n})$ be be a set of variables, and

$\displaystyle{\boldsymbol{g}}:={\boldsymbol{g}}^{(n)}=\\{g_{a}[b]\,|\,a\geq
1,\,b\geq 1,\,a+b\leq n\\}$

be a set of parameters; we put $\deg(x_{i})=1$ and $\deg(g_{a}[b])=b+1$, and
set $g_{k}[0]:=x_{k}$, $k=1,\ldots,n$. For a subset $S\subset[1,n]$ we denote
by $X_{S}$ the set of variables $\\{x_{i}\,|\,i\in S\\}$.

Let $t$ be an auxiliary variable, denote by $M=(m_{ij})_{1\leq i,j\leq n}$ the
matrix of size $n$ by $n$ with the following elements:

$\displaystyle m_{i,j}=\begin{cases}x_{i}+t&\text{if \ $i=j$},\\\
g_{i}[j-i]&\text{if \ $j>i$},\\\ -1&\text{if \ $i-j=1$},\\\ 0&\text{if \
$i-j>1$}.\end{cases}$

Let $P_{n}(X_{n},t)=\det|M|$.

######  Definition 2.31.

The Fulton universal ring ${\cal R}_{n-1}$ is defined to be the
quotient262626If $P(t,X_{n})=\sum\limits_{k\geq 1}f_{k}(X_{n})t^{k}$,
$f_{k}(X_{n})\in\mathbb{Q}[Xn]$ is a polynomial, we denote by $\langle
P(t,X_{n})\rangle$ the ideal in the polynomial ring $\mathbb{Q}[X_{n}]$
generated by the coefficients $\\{f_{1},f_{2},\ldots\\}$.

$\displaystyle{\cal
R}_{n-1}=\mathbb{Z}\big{[}{\boldsymbol{g}}^{(n)}\big{]}[x_{1},\ldots,x_{n}]/\langle
P_{n}(X_{n},t)-t^{n}\rangle.$

######  Lemma 2.32.

Let $P_{n}(X_{n},t)=\sum\limits_{k=0}^{n}c_{k}(n)t^{n-k}$, $c_{0}(n)=1$. Then

$\displaystyle
c_{k}(n):=c_{k}\big{(}n;X_{n},{\boldsymbol{g}}^{(n)}\big{)}=\sum_{{1\leq
i_{1}<i_{2}<\cdots<i_{s}<n\atop j_{1}\geq 1,\ldots,j_{s}\geq 1}\atop
m:=\sum(j_{a}+1)\leq
n}\prod_{a=1}^{s}g_{i_{a}}[j_{a}]e_{k-m}\big{(}X_{[1,n]{\setminus}\bigcup\limits_{a=1}^{s}[i_{a},i_{a}+j_{a}]}\big{)},$
(2.16)

where in the summation we assume additionally that the sets
$[i_{a},i_{a}+j_{a}]:=\\{i_{a},i_{a}+1,\ldots,i_{a}+j_{a}\\}$, $a=1,\ldots,s$,
are pair-wise disjoint.

It is clear that ${\cal
R}_{n-1}=\mathbb{Z}[{\boldsymbol{g}}^{(n)}][x_{1},\ldots,x_{n}]/\langle
c_{n}(1),\ldots,c_{n}(n)\rangle$. One can easily see that the coefficients
$c_{k}(n)$ and $g_{m}[k]$ satisfy the following recurrence relations [47]:

$\displaystyle
c_{k}(n)=c_{k}(n-1)+\sum_{a=0}^{k-1}g_{n-a}[a]c_{k-a-1}(n-a-1),\qquad
c_{0}(n)=1,$ $\displaystyle
g_{m}[k]=c_{k+1}(m+k)-c_{k+1}(m+k-1)-\sum_{a=0}^{k-1}g_{m+k-a}[a]c_{k-a}(m+k-a),$
$\displaystyle g_{m}[0]:=x_{m}.$

On the other hand, let $\\{q_{ij}\\}_{1\leq i<j\leq n}$ be a set of (quantum)
parameters, and $e_{k}^{({\boldsymbol{q}})}(X_{n})$ be the multiparameter
quantum elementary polynomial introduced in [45]. We are interested in to
describe a set of relations between the parameters $\\{g_{i}[j]\\}_{i\geq
1,j\geq 1\atop i+j\leq n}$ and the quantum parameters $\\{q_{ij}\\}_{1\leq
i<j\leq n}$ which implies that

$\displaystyle c_{k}(n)=e_{k}^{({\boldsymbol{q}})}(X_{n})\qquad\text{for}\quad
k=1,\ldots,n.$

To start with, let us recall the recurrence relations among the quantum
elementary polynomials, cf. [117]. To do so, consider the generating function

$\displaystyle E_{n}\big{(}X_{n};\\{q_{ij}\\}_{1\leq i<j\leq
n}\big{)}=\sum_{k=0}^{n}e_{k}^{({\boldsymbol{q}})}(X_{n})t^{n-k}.$

######  Lemma 2.33 ([41, 117]).

One has

$\displaystyle E_{n}\big{(}X_{n};\\{q_{ij}\\}_{1\leq i<j\leq
n}\big{)}=(t+x_{n})E_{n-1}\big{(}X_{n-1};\\{q_{ij}\\}_{1\leq i<j\leq
n-1}\big{)}$ $\displaystyle\hphantom{E_{n}\big{(}X_{n};\\{q_{ij}\\}_{1\leq
i<j\leq
n}\big{)}=}{}+\sum_{j=1}^{n-1}q_{jn}E_{n-2}\big{(}X_{[1,n-1]{\setminus}\\{j\\}};\\{q_{a,b}\\}_{1\leq
a<b\leq n-1\atop a\not=j,b\not=j}\big{)}.$

######  Proposition 2.34.

Parameters $\\{g_{a}[b]\\}$ can be expressed polynomially in terms of quantum
parameters $\\{q_{ij}\\}$ and variables $x_{1},\ldots,x_{n}$, in a such way
that

$\displaystyle c_{k}(n)=e_{k}^{({\boldsymbol{q}})}(X_{n}),\qquad\forall\,k,n.$

Moreover,

  * •

$g_{a}[b]=\sum\limits_{k=1}^{a}q_{k,a+b}\prod\limits_{j=a+1}^{a+b-1}(x_{j}-x_{k})+\text{lower
degree polynomials in $x_{1},\ldots,x_{n}$}$,

  * •

the quantum parameters $\\{q_{ij}\\}$ can be presented as rational functions
in terms of variables $x_{1},\ldots,x_{n}$ and polynomially in terms of
parameters $\\{g_{a}[b]\\}$ such that the equality
$c_{k}(n)=e_{k}^{({\boldsymbol{q}})}(X_{n})$ holds for all $k$, $n$.

In other words, the transformation

$\displaystyle\\{q_{ij}\\}_{1\leq i<j\leq
n}\longleftrightarrow\\{g_{a}[b]\\}_{a+b\leq n\atop a\geq 1,\,b\geq 1}$

defines a “birational transformation” between the algebra
$\mathbb{Z}[{\boldsymbol{g}}^{(n)}][X_{n}]/\langle
P_{n}(X_{n},t)-t^{n}\rangle$ and multiparameter quantum deformation of the
algebra $H^{*}({\cal{F}}l_{n},\mathbb{Z})$.

######  Example 2.35.

Clearly,

$\displaystyle g_{n-1}[1]=\sum_{j=1}^{n-1}q_{j,n},\quad n\geq
2\qquad\text{and}\qquad g_{n-2}[2]=\sum_{j=1}^{n-2}q_{jn}(x_{n-1}-x_{j}),\quad
n\geq 3.$

Moreover

$\displaystyle
g_{1}[3]=q_{14}\big{(}(x_{2}-x_{1})(x_{3}-x_{1})+q_{23}-q_{12}\big{)}+q_{24}\big{(}q_{13}-q_{12}\big{)},$
$\displaystyle
g_{2}[3]=q_{15}\big{(}(x_{3}-x_{1})(x_{4}-x_{1})+q_{24}+q_{34}-q_{12}-q_{13}\big{)}$
$\displaystyle\hphantom{g_{1}[3]=}{}+q_{25}\big{(}(x_{3}-x_{2})(x_{4}-x_{2})+q_{14}+q_{34}-q_{12}-q_{23}\big{)}+q_{35}\big{(}q_{14}+q_{24}-q_{13}-q_{23}\big{)}.$

######  Comments 2.36.

The full Kostant–Toda lattice (FKTL for short) has been introduced in the end
of $70^{\prime}s$ of the last century by B. Kostant and since that time has
been extensively studied both in Mathematical and Physical literature. We
refer the reader to the original paper by B. Kostant [29, 80] for the
definition of the ${\rm FKTL}$ and its basic properties. In the present paper
we just want to point out on a connection of the Fulton universal ring and
hence the multiparameter deformation of the cohomology ring of complete flag
varieties, and polynomial integral of motion of the FKTL. Namely,

Polynomials $c_{k}(n;X_{n},{\boldsymbol{g}}^{(n)})$ defined by (2.16) coincide
with the polynomial integrals of motion of the FKTL.

It seems an interesting task to clarify a meaning of the ${\rm FKTL}$ rational
integrals of motion in the context of the universal Schubert calculus [47] and
the algebra $3HT_{n}(0)$, as well as any meaning of universal Schubert or
Grothendieck polynomials in the context of the Toda or full Kostant–Toda
lattices.

##  3 Algebra $\boldsymbol{3HT_{n}}$

Consider the twisted classical Yang–Baxter relation

$\displaystyle[u_{ij}+u_{ik},u_{jk}]+[u_{ik},u_{ji}]=0,$

where $i$, $j$, $k$ are distinct. Having in mind applications of the Dunkl
elements to combinatorics and algebraic geometry, we split the above relation
into two relations

$\displaystyle u_{ij}u_{jk}=u_{jk}u_{ik}-u_{ik}u_{ji}\qquad\text{and}\qquad
u_{jk}u_{ij}=u_{ik}u_{jk}-u_{ji}u_{ik}$ (3.1)

and impose the following unitarity constraints

$\displaystyle u_{ij}+u_{ji}=\beta,$

where $\beta$ is a central element. Summarizing, we come to the following
definition.

######  Definition 3.1.

Define algebra $3T_{n}(\beta)$ to be the quotient of the free associative
algebra

$\displaystyle\mathbb{Z}[\beta]\langle u_{ij},\,1\leq i<j\leq n\rangle$

by the set of relations

  * •

locality: $u_{ij}u_{kl}=u_{kl}u_{ij}$ if $\\{i,j\\}\cap\\{k,l\\}=\varnothing$,

  * •

$3$-term relations: $u_{ij}u_{jk}=u_{ik}u_{ij}+u_{jk}u_{ik}-\beta u_{ik}$, and
$u_{jk}u_{ij}=u_{ij}u_{ik}+u_{ik}u_{jk}-\beta u_{ik}$ if $1\leq i<j<k\leq n$.