However, note that $\widehat{G}_{C}(P_{2}^{\scriptscriptstyle\vee})$ (see
Figure 37(d)) consists of a single cycle fixed orientation-wise by the
involution, so the only cycle binomial is $0$. Consequently,

$R_{P^{\prime{\scriptscriptstyle\vee}}}^{\mathrm{rt}}\cong
k[U_{1\bar{2}},U_{12},U_{\bar{3}4},U_{\bar{2}3},U_{\bar{2}5},U_{4\bar{5}}]/(U_{\bar{3}4}U_{\bar{2}3}-U_{4\bar{5}}{\bar{2}5}),$

and closing the notch gives

$\displaystyle R_{{P^{\scriptscriptstyle\vee}}}^{\mathrm{rt}}$
$\displaystyle\cong
k[U_{1\bar{2}},U_{12},U_{\bar{3}4},U_{\bar{2}3},U_{\bar{2}5},U_{4\bar{5}}]/(U_{\bar{3}4}U_{\bar{2}3}-U_{4\bar{5}}{\bar{2}5},U_{\bar{2}5}-U_{\bar{1}5})$
$\displaystyle\cong
k[U_{1\bar{2}},U_{12},U_{\bar{3}4},U_{\bar{2}3},U_{\bar{1}4}]/(U_{\bar{3}4}U_{\bar{2}3}-U_{\bar{1}4}U_{\bar{2}5}).$

Then the Hilbert series is

$\mathrm{Hilb}(R_{{P^{\scriptscriptstyle\vee}}}^{\mathrm{rt}},\bm{x})=\dfrac{1-x_{2}^{-1}x_{4}}{(1-x_{1}^{-1}x_{4})(1-x_{3}^{-1}x_{4})(1-x_{2}^{-1}x_{3})(1-x_{1}x_{2}^{-1})(1-x_{1}x_{2})}.$

Applying Proposition 6.14 gives

$\Psi^{\scriptscriptstyle\vee}_{{P^{\scriptscriptstyle\vee}}}(\bm{x})=\dfrac{x_{4}-x_{2}}{(x_{4}-x_{1})(x_{4}-x_{3})(x_{3}-x_{2})(x_{1}-x_{2})(x_{1}+x_{2})}.$

On the other hand,

$\Psi^{\scriptscriptstyle\vee}_{{P^{\scriptscriptstyle\vee}}}(\bm{x})=\sum_{w\in\mathcal{L}(P)}w\left(\dfrac{1}{(x_{1}-x_{2})(x_{2}-x_{3})(x_{3}-x_{4})2x_{4}}\right)\\\
=\dfrac{x_{4}-x_{2}}{(x_{4}-x_{3})(x_{3}-x_{2})(x_{4}-x_{1})(x_{1}-x_{2})(x_{1}+x_{2})},$

agreeing with the computation via the Hilbert series.

With the regular sequence in hand, one can apply Proposition 6.14 to obtain
the following corollary.

######  Corollary 15.20.

Suppose $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) is a strongly planar signed
poset. Then

$\Psi_{P}(\bm{x})=\dfrac{\prod_{\rho}\mathrm{sgn}(\min(\rho))x_{\min(\rho)}-\mathrm{sgn}(\max(\rho))x_{\max(\rho)}}{\prod_{(i,j)\in\Sigma_{P}}\mathrm{sgn}(i)x_{i}-\mathrm{sgn}(j)x_{j}},$

where $\rho$ runs over the regions of $\widehat{G}(P)$ and $(i,j)$ runs over
the edges of the Hasse diagram.

##  Chapter The Weight Cone Semigroup

Associated to the weight cone, $K_{P}^{\mathrm{wt}}$, is a semigroup, called
the _weight cone semigroup_ , defined by the intersection of
$K_{P}^{\mathrm{wt}}$ with the coweight lattice. In this chapter, only
$\Phi_{B_{n}}$-signed posets will be considered. Section 16 discusses the
generators of the semigroup, Section 17 gives a generating set for the toric
ideal. Finally, Section 18 considers the question of computing
$\Phi_{P}(\bm{x})$ and

$\sum_{w\in\mathcal{L}(P)}q^{\mathrm{maj}(w)}.$

Theorem 18.10 characterizes signed posets which are so-called initial complete
intersections, for which the aforementioned sums can be readily computed, and
Theorem 18.21 explains how the initial complete intersections are constructed.

Before beginning, recall some definitions and facts about the weight cone:

  * •

an ideal $I$ is said to be _extensible_ if there is a (nonempty)
$J\subset\pm[n]$ such that $I\cup J$ and $I\cup-J$ are ideals (see Definition
12.4)

  * •

the extreme rays of $K_{P}^{\mathrm{wt}}$ correspond to the connected,
nonextensible ideals of $P$ (see Proposition 12.5)

  * •

the elements of the weight cone semigroup are the $P$-partitions (see Section
9)

  * •

when considering the ideals of a signed poset, it suffices to look at
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$ instead of
$\widehat{G}_{B}(P)$. To that end, $\widehat{G}(P)$ in this chapter denotes
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$, even though signed posets are
$P\subset\Phi_{B_{n}}$. (see Definition 9.5)

  * •

the _signed support_ of a $P$-partition $f$ is an ideal

$\mathop{\mathrm{supp}_{\pm}}(f)=\\{\mathrm{sgn}(f_{i})i\colon
i\in[n],|f_{i}|\geq 1\\}$

(see Definition 12.8).

  * •

the set of connected ideals is denoted $J_{\mathrm{conn}}(P)$.

###  16 Generators of the Semigroup

Lemma 12.7 has a consequence that is key to the discussion of the weight cone
semigroup, which is stated now.

######  Proposition 16.1.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset and $f$ is a $P$-partition.
Then there are ideals $J_{1}\supset J_{2}\supset\cdots\supset J_{k}$ such that
$f=\chi_{J_{1}}+\chi_{J_{2}}+\cdots\chi_{J_{k}}$.

The $J_{i}$ are precisely those from Lemma 12.7.

######  Proposition 16.2.

The semigroup $K_{P}^{\mathrm{wt}}\cap L^{\mathrm{cowt}}_{B}$ is generated by
the connected ideals of $P$, though not necessarily minimally.

###### Proof.

Suppose $f$ is a $P$-partition. Then there are ideals
$I_{1}\subseteq\cdots\subseteq I_{k}$ such that

$f=\sum_{j}\chi_{I_{j}}.$

Let $I_{j}^{(i)}$ be the connected components of $I_{j}$. Then $f$ is in the
semigroup generated by the $I_{j}^{(i)}$, which lies in the semigroup
generated by the connected ideals. ∎

It is simple to see that this is not necessarily a minimal generating set.
Suppose $P=\\{e_{1}+e_{2},+e_{2}\\}$. Then
$J_{\mathrm{conn}}(P)=\\{\\{1\\},\\{2\\},\\{-1,2\\}\\}$, but $\\{1\\}$ and
$\\{-1,2\\}$ suffice to generate the semigroup.

###  17 The Toric Ideal

To discuss the toric ideal of the weight cone semigroup, one needs a few
additional definitions involving ideals of a signed poset.

######  Definition 17.1.

Suppose $J$ is an ideal of a signed poset. The _support_ of $J$ is the set
$\mathop{\mathrm{supp}}J=\\{i\colon\pm i\in J\\}\subset[n]$. Two ideals, $J$
and $K$, _intersect nontrivially_ if
$\mathop{\mathrm{supp}}J\cap\mathop{\mathrm{supp}}K\neq\emptyset$ and neither
$J\subset K$ nor $K\subset J$. Two ideals which do not intersect nontrivially
will be said to _intersect trivially_.

If $J$ and $K$ are ideals, let $J+K$ denote the $P$-partition
$\chi_{J}+\chi_{K}$. In the sequel, it will be pairs of nontrivially
intersecting connected ideals that are paramount. Denote the set of such pairs
by $\Pi(P)$.

With this setup one can now give a presentation for the toric ideal of the
weight cone semigroup. The result closely resembles that of [21, Theorem 1.2
], save that one must account for the fact that the union of two ideals of a
$\Phi_{B_{n}}$-poset is not necessarily an ideal, since the union of two
isotropic order ideals of $\widehat{G}(P)$ is not necessarily isotropic.

######  Definition 17.2.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset. Let
$S^{\mathrm{wt}}_{P}=k[U_{J_{1}},\ldots,U_{J_{k}}]$, where the $J_{i}$ are the
connected ideals of $P$. Define a map

$\displaystyle\varphi\colon S^{\mathrm{wt}}_{P}$ $\displaystyle\to
k[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ $\displaystyle U_{J}$
$\displaystyle\mapsto\bm{x}^{\chi_{J}}.$

Suppose $J_{1},J_{2}$ are two ideals which intersect nontrivially. Define

$\mathrm{syz}(U_{J_{1}},U_{J_{2}})=U_{J_{1}}U_{J_{2}}-\prod U_{J^{(i)}}\prod
U_{K^{(j)}},$

where the $J^{(i)}$ are the connected components of $J_{1}\cap J_{2}$ and the
$K^{(j)}$ are the connected components of
$\mathop{\mathrm{supp}_{\pm}}(J_{1}+J_{2})$.

######  Theorem 17.3.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset. One has an exact sequence

$0\to I^{\mathrm{wt}}_{P}\to S^{\mathrm{wt}}_{P}\to R_{P}^{\mathrm{wt}}\to 0,$

with

$I^{\mathrm{wt}}_{P}=(\mathrm{syz}(U_{J_{1}},U_{J_{2}})),$

where $\\{J_{1},J_{2}\\}$ runs over $\Pi(P)$, the set of pairs of nontrivially
intersecting connected ideals.

Theorem 17.3 is the generalization of the type A result [21, Theorem 1.2].

Consider the signed poset in Figure 44. The connected ideals are

$\\{1\\},\\{-3\\},\\{-4\\},\\{1,2\\},\\{1,4\\},\\{-2,-3\\},\\{1,2,4\\},\\{1,2,3\\},\\{1,2,3,4\\},$

so one has
$S^{\mathrm{wt}}_{P}=k[U_{1},U_{\bar{3}},U_{\bar{4}},U_{12},U_{14},U_{\bar{2}\bar{3}},U_{124},U_{123},U_{1234}]$.
Then

$I^{\mathrm{wt}}_{P}=(U_{\bar{3}}U_{123}-U_{12},U_{\bar{3}}U_{1234}-U_{124},U_{\bar{4}}U_{14}-U_{1},U_{\bar{4}}U_{124}-U_{12},U_{\bar{4}}U_{1234}-U_{123},\\\
U_{12}U_{14}-U_{1}U_{124},U_{12}U_{\bar{2}\bar{3}}-U_{1}U_{\bar{3}},U_{\bar{2}\bar{3}}U_{124}-U_{14}U_{\bar{3}},U_{124}U_{123}-U_{12}U_{1234})$

$1$$2$$4$$3$$-1$$-4$$-2$$-3$ Figure 44:
$P=\\{+e_{1},+e_{1}-e_{2},+e_{1}-e_{3},+e_{1}-e_{4},+e_{2}-e_{3}\\}$

The proof of Theorem 17.3 requires a number of lemmas and a few facts from the
theory of Gröbner bases. Section 17.1 discusses Gröbner bases and Section 17.2
contains the lemmas and proof of the theorem.

####  17.1 Term Orders and Gröbner Bases

In this section, we pause for a moment to review a few facts about terms
orders and Gröbner bases before the proof of Theorem 17.3 in the next section.
This material is standard and can be found in many places, including [18, 20,
58].

######  Definition 17.4.

Suppose $S=k[x_{1},\ldots,x_{n}]$ is a polynomial ring. A _term order_ , $<$,
is a total order on the monomials of $S$ such that

  1. (a)

there are no infinite descending chains

  2. (b)

$1=\bm{x}^{0}$ is the minimal element

  3. (c)

if $\bm{x}^{\alpha}\leq\bm{x}^{\beta}$ and $\bm{x}^{\gamma}$ is any monomial,
then $\bm{x}^{\alpha}\bm{x}^{\gamma}\leq\bm{x}^{\beta}\bm{x}^{\gamma}$.

Suppose $f=a_{1}\bm{x}^{\alpha_{1}}+\cdots+a_{k}\bm{x}^{\alpha_{k}}$ is a
polynomial in $R$. The _multidegree_ of $f$ is $\alpha_{\deg f}$ which is
maximal among $\alpha_{1},\ldots,\alpha_{k}$ with respect to the term order
$<$. The _initial term_ of $f$ is then $a_{\deg f}\bm{x}^{\alpha_{\deg f}}$,
denoted $\mathrm{in}_{<}f$.

######  Definition 17.5.

Given an ideal $I\subset R$, the _initial ideal_ with respect to $<$ is

$\mathrm{in}_{<}I=(\mathrm{in}_{<}f\colon f\in I).$

An ideal and its initial ideal are connected by the notion of a Gröbner basis.
A set $\mathcal{G}=\\{g_{1},\ldots,g_{m}\\}\subset I$ is a _Gröbner basis_ of
$I$ if

$\mathrm{in}_{<}I=(\mathrm{in}_{<}g_{1},\ldots,\mathrm{in}_{<}g_{m}).$

A priori, it is not clear that an arbitrary ideal in a polynomial ring should
have a finite Gröbner basis. However, this is a consequence of Dickson’s Lemma
(see Theorem 1.9, Corollary 1.10 and page 20 of [20]). The following facts
will underpin the proof of Theorem 17.3.

######  Theorem 17.6 (Macaulay [38]).

Suppose $S$ is a polynomial ring over the field $K$, there is an ideal
$I\subset S$ and a term order $<$. The monomials of $S$ which do not belong to
$\mathrm{in}_{<}I$ form a $k$-basis of $S/I$.

######  Corollary 17.7.

Suppose $I$ is a homogeneous ideal in a polynomial ring $S$ and $<$ is a term
order. Then

$\mathrm{Hilb}_{S/\mathrm{in}_{<}I}(\bm{x})=\mathrm{Hilb}_{S/I}(\bm{x}).$

######  Proposition 17.8.

If $I$ is an ideal in a polynomial ring $S$ and $\mathcal{G}$ is a Gröbner
basis of $I$, then $\mathcal{G}$ generates $I$.

As an aside, Gordan observed in [29] that this last proposition, when combined
with the existence of finite Gröbner bases gives the Hilbert Basis Theorem,
though it is common to rely on the Hilbert Basis Theorem to obtain the
existence of finite Gröbner bases.

####  17.2 The Proof of Theorem 17.3

The basic idea of the proof of Theorem 17.3 is to show that the generating set
given in the theorem is a Gröbner basis for $I^{\mathrm{wt}}_{P}$ in a
particular term order. This is accomplished by showing that the rings
$S^{\mathrm{wt}}_{P}/I^{\mathrm{wt}}_{P}$ and
$S^{\mathrm{wt}}_{P}/I^{\mathrm{in}}_{P}$ share the same Hilbert series. (See
Lemma 17.11 for the definition of $I^{\mathrm{in}}_{P}$.)

######  Lemma 17.9.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset. A $P$-partition $f$ can be
written uniquely as the sum of trivially intersecting connected ideals.

Lemma 17.9 generalizes [21, Proposition 2.5(ii)], which has the same statement
when $P$ is a poset.

###### Proof.

From Proposition 16.1, any $P$-partition can be written as
$f=\sum\chi_{I_{k}}$ with ideals $I_{1}\subset I_{2}\subset\cdots\subset
I_{\ell}$. Thus, the lemma is a matter of transforming this collection of
ideals into a set of trivially intersecting connected ideals. Proceed by
induction on $|f|=|f_{1}|+\cdots+|f_{n}|$. The base case $|f|=0$ is trivial.

Suppose $|f|\geq 1$. Let $J=\mathop{\mathrm{supp}_{\pm}}(f)$. Recall that the
signed support is an ideal and $J=I_{k}$. Let $J^{(1)},J^{(2)},\cdots,J^{(c)}$
be the connected components of $J$. Suppose $c>1$. Consider $f|_{J^{(i)}}$.
One has

$f|_{J^{(i)}}=\sum_{i=1}^{k}I_{k}|_{J^{(i)}},$

so $f|_{J^{(i)}}$ is a $J^{(i)}$-partition. One then has that
$|J^{(i)}|<|J|\leq|f|$, so, by induction, $f|_{J^{(i)}}$ can be written
uniquely as a sum of trivially intersecting connected ideals. Consequently,
$f$ can be written uniquely as a sum of trivially intersecting connected
ideals.

In the other case suppose $c=1$. Then $J=\mathop{\mathrm{supp}_{\pm}}(f)$ is
connected. Let $\hat{f}=\sum_{i=i}^{k-1}I_{i}$. Then $f=\chi_{J}+\hat{f}$ and
one has that $\hat{f}$ is a (possibly zero) $P$-partition with
$|\hat{f}|<|f|$. Then, by induction, $\hat{f}$ can be written uniquely as a
sum of trivially intersecting connected ideals. Further, each of these ideals
is contained in $\chi_{J}=I_{k}$, so $f$ can be written uniquely as a sum of
trivially intersecting connected order ideals. ∎

######  Definition 17.10.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset. Let

$I_{P}^{\mathrm{in}}=(U_{J_{1}}U_{J_{2}})$

with $\\{J_{1},J_{2}\\}$ running over $\Pi(P)$.

######  Lemma 17.11.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset. Then $R_{P}^{\mathrm{wt}}$
and $S^{\mathrm{wt}}_{P}/I_{P}^{\mathrm{in}}$ have the same
${\mathbb{Z}}^{n}$-graded Hilbert series, namely

$\sum_{f\in\mathcal{A}(P)}\bm{x}^{f}.$

###### Proof.

Since the $P$-partitions are the elements of the semigroup,
$R_{P}^{\mathrm{wt}}$ has $\sum_{f\in\mathcal{A}(P)}\bm{x}^{f}$ as its
${\mathbb{Z}}^{n}$-graded Hilbert series. The monomials killed by
$I_{P}^{\mathrm{in}}$ are precisely those mapped to $P$-partitions expressed
as a sum of (at least) two nontrivially intersecting ideals. By Lemma 17.9,
every $P$-partition can be written uniquely as a sum of nontrivially
intersecting connected order ideals, i.e. each $P$-partition corresponds to a
unique monomial surviving in $S^{\mathrm{wt}}_{P}/I_{P}^{\mathrm{in}}$. ∎

The notation $I_{P}^{\mathrm{in}}$ is not a coincidence, as Lemma 17.13 will
show. One defines a term order for $S^{\mathrm{wt}}_{P}=k[U_{J}]$ that
specializes to the term order given by Féray and Reiner in [21]. First, recall
their term order. Place a total order $<$ on the $U_{J}$ such that
$U_{J}<U_{K}$ whenever $|J|<|K|$. (This amounts to choosing a linear extension
of the poset of nonempty connected order ideals ordered by inclusion.) Suppose
$U^{J}=U_{J_{1}}\cdots U_{J_{r}}$ and $U^{K}=U_{K_{1}}\cdots U_{K_{s}}$ are
two monomials and assume without loss of generality that $r<s$ and the
$U_{J_{i}}$ and $U_{K_{\ell}}$ are ordered by $<$. Find the first $i$ such
that $J_{i}\neq K_{i}$. If $J_{i}<K_{i}$, then $U^{J}<U^{K}$, and if
$K_{i}<J_{i}$, then $U^{J}<U^{K}$. If no such $i$ exists, then $U^{J}$ divides
$U^{K}$, so $U^{J}<U^{K}$.

In type A, the initial term of $\mathrm{syz}(U_{J},U_{K})$ is always
$U_{J}U_{K}$ with respect to this term order. However, in type B, this is not
always the case. Consider the poset in Figure 44.
$S^{\mathrm{wt}}_{P}=k[U_{1},U_{\bar{3}},U_{\bar{4}},U_{12},U_{14},U_{\bar{2}\bar{3}},U_{124},U_{123},U_{1234}]$.
Order the ideals as the corresponding variables appear from left to right.
Then, using the term order from [21], the leading term of
$\mathrm{syz}(U_{123},U_{\bar{3}})=U_{123}U_{\bar{3}}-U_{12}$ is $U_{12}$ and
not $U_{123}U_{\bar{3}}$.

To resolve this issue, define a new term order.

######  Definition 17.12.

Let $w=(|J|)_{J\in J_{\text{conn}}(P)}$ be a weight vector and define a term
order as follows. Consider monomials $U^{\alpha}$ and $U^{\beta}$. If $\langle
w,\alpha\rangle>\langle w,\beta\rangle$, then $U^{\alpha}>U^{\beta}$. If
$\langle w,\alpha\rangle=\langle w,\beta\rangle$, break the tie using the term
order from [21] described above. Denote this new term order $\preceq$.

In the example from Figure 44, the weight vector is $(1,1,1,2,2,2,3,3,4)$, and
$U_{123}U_{\bar{3}}$ has weight four while $U_{12}$ has weight two.
Consequently, using the $\preceq$ order, the initial term of
$\mathrm{syz}(U_{123},U_{\bar{3}})$ is $U_{123}U_{\bar{3}}$.

######  Lemma 17.13.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset. Suppose $J_{1}$ and $J_{2}$
are connected ideals that intersect nontrivially. Then
$\mathrm{in}_{\preceq}(\mathrm{syz}(U_{J_{1}},U_{J_{2}}))=U_{J_{1}}U_{J_{2}}$.

###### Proof.

There are two cases to consider. First, suppose
$\mathop{\mathrm{supp}}(J_{1}+J_{2})=\mathop{\mathrm{supp}}(J_{1})\cup\mathop{\mathrm{supp}}(J_{2})$.
In other words, no cancellation occurs in $J_{1}+J_{2}$. Then

$|J_{1}|+|J_{2}|=|\mathop{\mathrm{supp}_{\pm}}(J_{1}+J_{2})|+|J_{1}\cap
J_{2}|.$

In this case, the weight vector $w$ has the same inner product with the
exponent vectors of both monomials of $\mathrm{syz}(U_{J_{1}},U_{J_{2}})$.
Using the term order $>$ from [21] to break the tie, one will always have that
$U_{J_{1}}U_{J_{2}}>\prod U_{J^{i}}\prod U_{K^{(j)}}$, since any connected
component of $J_{1}\cap J_{2}$ is a (proper) subset of $J_{1}$ and $J_{2}$.
Therefore,
$\mathrm{in}_{\preceq}(\mathrm{syz}(U_{J_{1}},U_{J_{2}}))=U_{J_{1}}U_{J_{2}}$.

In the other case, cancellation occurs in $J_{1}+J_{2}$. In this case,

$|J_{1}|+|J_{2}|>|\mathop{\mathrm{supp}_{\pm}}(J_{1}+J_{2})|+|J_{1}\cap
J_{2}|,$

so
$\mathrm{in}_{\preceq}(\mathrm{syz}(U_{J_{1}},U_{J_{2}}))=U_{J_{1}}U_{J_{2}}$.
∎

The proof of Theorem 17.3 now proceeds much as the proof does for type A given
in [21].

###### Proof of Theorem 17.3.

For simplicity, let $K=\ker(\varphi\colon S\to k[x_{1}^{\pm
1},\ldots,x_{n}^{\pm 1}])$ and, as before, let
$I_{P}^{\mathrm{in}}=(U_{J_{1}}U_{J_{2}})$, where $\\{J_{1},J_{2}\\}$ runs
over $\Pi(P)$. Observe that by definition, $I_{P}\subset K$. One then has
$\mathrm{in}_{\preceq}(I_{P})\subset\mathrm{in}_{\preceq}(K)$. On the other
hand, recall from Lemma 17.13 that
$U_{J_{1}}U_{J_{2}}=\mathrm{in}_{\preceq}(\mathrm{syz}(U_{J_{1}},U_{J_{2}}))$,
so $I_{P}^{\mathrm{in}}\subset\mathrm{in}_{\preceq}(I_{P})$. Therefore, one
has surjections

$S/I_{P}^{\mathrm{in}}\to S/\mathrm{in}_{\preceq}(I_{P})\to
S/\mathrm{in}_{\preceq}(K).$

By the definition of $\varphi$, the ideals $K$ and $I_{P}$ are homogeneous in
the ${\mathbb{Z}}^{n}$-grading, so $S/K$ and $S/I_{P}$ each share Hilbert
series with $S/\mathrm{in}_{\preceq}(K)$ and $S/\mathrm{in}_{\preceq}(I_{P})$,
respectively, by Corollary 17.7. Furthermore, one knows from Lemma 17.11 that
$S/K$ and $S/I_{P}^{\mathrm{in}}$ share the same ${\mathbb{Z}}^{n}$-Hilbert
series. Hence, one has

$\mathrm{Hilb}(S/I_{P}^{\mathrm{in}},\bm{x})=\mathrm{Hilb}(S/I,\bm{x})=\mathrm{Hilb}(S/K,\bm{x})=\mathrm{Hilb}(S/\mathrm{in}_{\preceq}(I),\bm{x})=\mathrm{Hilb}(S/\mathrm{in}_{\preceq}(K),\bm{x}).$

Consequently, the surjections must be isomorphisms and
$S/I_{P}^{\mathrm{in}}\cong S/\mathrm{in}_{\preceq}(I_{P})\cong
S/\mathrm{in}_{\preceq}(K)$, meaning
$I_{P}^{\mathrm{in}}=\mathrm{in}_{\preceq}(I_{P})=\mathrm{in}_{\preceq}(K)$.
One then has that the $\mathrm{syz}(U_{J_{1}},U_{J_{2}})$ form a Gröbner basis
for both $I_{P}$ and $K$, meaning $I_{P}=K$, since an ideal is always
generated by a Gröbner basis. ∎

###  18 Complete intersections and the sum over linear extensions

In [47], Reiner proved the following result about the $P$-partition generating
function for signed posets. Recall the definition of _major index_ of a signed
permutation $w$ from Definition 7.7:

$\mathrm{maj}(w)=\sum_{i\in\mathrm{Des}(w)}i.$

######  Proposition 18.1.

Suppose $P$ is a signed poset. Then

$\sum_{f\in\mathcal{A}(P)}q^{|f|}=\dfrac{\sum_{w\in\mathcal{L}(P)}q^{\mathrm{maj}(w)}}{(1-q)(1-q^{2})\cdots(1-q^{n})}.$

This parallels the result in type A due to Stanley (see [54] or [52]). Féray
and Reiner observed (again for type A) that the left hand side is the Hilbert
series of $R_{P}^{\mathrm{wt}}$ with the grading $\deg\bm{x}^{f}=|f|$.
However, this is _not_ the case when $P$ is a signed poset—$\deg x^{f}=|f|$ is
not a grading of $R_{P}^{\mathrm{wt}}$, never mind a specialization of the
${\mathbb{Z}}^{n}$ grading! Additionally, while Theorem 17.3 gives a
presentation of $R_{P}^{\mathrm{wt}}$ as
$S^{\mathrm{wt}}_{P}/I^{\mathrm{wt}}_{P}$, the toric ideal
$I^{\mathrm{wt}}_{P}$ is not homogeneous in this ${{\mathbb{N}}}$-grading.

All is not lost, however, as, using the same logic as in Lemma 17.11, one
observes that when $S^{\mathrm{wt}}_{P}$ is graded by $\deg U_{J}=|J|$, one
has

$\mathrm{Hilb}(S^{\mathrm{wt}}_{P}/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P},q)=\sum_{f\in\mathcal{A}(P)}q^{|f|}.$

Additionally, if one grades $S^{\mathrm{wt}}_{P}$ by $\deg U_{J}=1$, one has

$\mathrm{Hilb}(S^{\mathrm{wt}}_{P}/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P},t)=\sum_{f\in\mathcal{A}(P)}t^{\nu(f)},$

where $\nu(f)$ is the number of ideals used in the unique expression of the
$P$-partition $f$ as a sum of nontrivially intersecting connected order
ideals.

This section is concerned with computing these sums in the case where
$S^{\mathrm{wt}}_{P}/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}{P}$ is
a complete intersection. Section 18.1 will reduce the case of signed posets
which are not full-dimensional to the work of Féray and Reiner in [21].
Section 18.2 will define the notion of _initial complete intersection_ and
characterize in two ways the signed posets which are initial complete
intersections:

  * •

they are the signed posets avoiding a certain list of induced subposets
(Theorem 18.10);

  * •

they are the signed posets constructed using certain moves (Theorem 18.21).

By shifting one’s focus to
$S/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$ from
$S/I^{\mathrm{wt}}_{P}$, one sees immediate benefit in that one now has a
minimal generating set for the ideal of interest.

######  Proposition 18.2.

Suppose $P$ is a signed poset. Then
$\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$ is minimally generated by
$U_{J}U_{K}$ where $\\{J,K\\}$ runs over all nontrivially intersecting pairs
of connected order ideals of $P$.

###### Proof.

Since the $\mathrm{syz}(U_{J},U_{K})$ form a Gröbner basis of $I$, their
initial terms, the $U_{J}U_{K}$, certainly generate
$\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$. A generating set of a
monomial ideal is a minimal generating set precisely when none of the
monomials divides another. (See [36, Proposition 1.1.6], for instance.) Since
the $U_{J}U_{K}$ are all distinct and all quadratic, none can divide any of
the others. ∎

####  18.1 Reducing to connected $\widehat{G}(P)$

As with the root cone, one is able to reduce discussion of the weight cone
semigroup to a more convenient case: that where $\widehat{G}(P)$ is connected.
First, one uses a similar logic to the biconnected component reduction of
Proposition 15.2.

######  Definition 18.3.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset. Let a _signed component_ of
$\widehat{G}(P)$ be

  1. (a)

a connected component $P_{1}$ of $\widehat{G}(P)$ such that if $i\in P_{1}$,
then $-i\in P_{1}$, or

  2. (b)

a pair of connected components $P_{1},P_{2}$ such that $P_{1}=-P_{2}$.

Observe that each signed component of $\widehat{G}(P)$ is the Fischer poset of
some smaller signed poset, call them $P_{1},\ldots,P_{k}$. One then has the
following.

######  Proposition 18.4.

Let $P$ be a signed poset and let $P_{1},\ldots,P_{k}$ be the signed posets
corresponding to its signed components. Then

$R_{P}^{\mathrm{wt}}\cong R_{P_{1}}^{\mathrm{wt}}\otimes\cdots\otimes
R_{P_{k}}^{\mathrm{wt}}.$

###### Proof.

Begin by observing that each connected order ideal of $P$ lies entirely in a
signed component. Therefore,

$S^{\mathrm{wt}}_{P}\cong S^{\mathrm{wt}}_{P_{1}}\otimes\cdots\otimes
S^{\mathrm{wt}}_{P_{k}},$

and

$I^{\mathrm{wt}}_{P}\cong I^{\mathrm{wt}}_{P_{1}}\oplus\cdots\oplus
I^{\mathrm{wt}}_{P_{k}}.$

Consequently, one has

$R_{P}^{\mathrm{wt}}=R_{P_{1}}^{\mathrm{wt}}\otimes\cdots\otimes
R_{P_{k}}^{\mathrm{wt}},$

as desired. ∎

As a result of Proposition 18.4 it suffices to consider signed posets having
only one signed component. This assumption will hold for the remainder of the
chapter.

As in Section 15.1, one may assume that each signed poset under consideration
is not contained in $\text{span}\\{e_{i}\colon i\in A\\}$ for any
$A\subsetneq[n]$. We explain here why it suffices to consider signed posets
$P$ for which $K_{P}^{\mathrm{rt}}$ is full-dimensional and
$K_{P}^{\mathrm{wt}}$ is pointed.

Since it was assumed that $\widehat{G}(P)$ has only one signed component, if
$K_{P}^{\mathrm{wt}}$ is not pointed, one must have that $\widehat{G}(P)$
consists of two isotropic connected components, $P_{1}=-P_{2}$. Without loss
of generality, one may assume $P_{1}=[n]$, meaning both $[n]$ and $-[n]$ are
both ideals.

######  Proposition 18.5.

Suppose $P$ is a signed poset such that $K_{P}^{\mathrm{wt}}$ is not pointed.
Then there is a poset $Q$ on $[n]$ such that

$R_{P}^{\mathrm{wt}}\cong R_{Q}^{\mathrm{wt}}[(x_{1}x_{2}\cdots x_{n})^{-1}],$

the localization of $R_{Q}^{\mathrm{wt}}$ at the multiplicatively closed set
$\\{(x_{1}\cdots x_{n})^{k}\colon k\geq 0\\}$.

###### Proof.

One may assume that $\widehat{G}(P)$ consists of two connected components
$K_{1}$ and $K_{2}$ such that $K_{1}=-K_{2}$ and $[n]$ are the vertices of
$K_{1}$. Let $Q$ be the poset whose Hasse diagram coincides with $K_{1}$. One
defines a map

$\varphi\colon R_{P}^{\mathrm{wt}}\to R_{Q}^{\mathrm{wt}}[(x_{1}x_{2}\cdots
x_{n})^{-1}]$

as follows. Given any $f\in\mathcal{A}(P)$, the $P$-partition $f$ can be
written uniquely as a sum of nontrivially intersecting connected ideals:
$f=J_{1}+\cdots+J_{k}$. Without loss of generality, one may assume
$J_{1},\ldots,J_{i}\subset[n]$ and $J_{i+1},\ldots,J_{k}\subset-[n]$. Then
define

$\varphi(\bm{x}^{f})=\dfrac{\bm{x}^{J_{1}}}{1}\cdots\dfrac{\bm{x}^{J_{i}}}{1}\cdot\dfrac{\bm{x}^{([n]\smallsetminus-
J_{i+1})}}{x_{1}\cdots x_{n}}\cdots\dfrac{\bm{x}^{([n]\smallsetminus-
J_{k})}}{x_{1}\cdots x_{n}}.$

Since $[n]$ and $-[n]$ are both ideals of $P$, for any ideal $J\subset-[n]$,
one has that $[n]\smallsetminus-J$ is also an ideal of $P$, meaning it is an
ideal of $Q$. (It may not be a _connected_ ideal of $Q$, but it is an
$Q$-partition, which is all that is needed.)

Certainly $\varphi$ is an injection. To complete the proof, one must now show
that $\varphi$ is surjective. A prototypical monomial of
$R_{Q}^{\mathrm{wt}}[(x_{1}x_{2}\cdots x_{n})^{-1}]$ is

$\dfrac{\bm{x}^{f}}{(x_{1}\cdots x_{n})^{k}},$

with $k$ minimal (i.e. having cancelled as many powers of $x_{1}\cdots x_{n}$
from the numerator as possible). Since $f$ is an $Q$-partition, it is also a
$P$-partition. Then $\bm{x}^{f}(x_{1}\cdots x_{n})^{-k}\in
R_{P}^{\mathrm{wt}}$ and, since $\varphi$ is a ring homomorphism,

$\varphi(\bm{x}^{f}(x_{1}\cdots x-n)^{-k})=\dfrac{\bm{x}^{f}}{(x_{1}\cdots
x_{n})^{k}},$

meaning $\varphi$ is surjective, completing the proof. ∎

####  18.2 Characterizing the initial complete intersections

######  Proposition 18.6.

Suppose $P$ is a signed poset. One has that

$0\to(U_{J}U_{K})\stackrel{{\scriptstyle\varphi}}{{\to}}S^{\mathrm{wt}}_{P}\to
S^{\mathrm{wt}}_{P}/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}\to 0,$
(10)

where $\\{J,K\\}$ runs over the set $\Pi(P)$ of all pairs of nontrivially
intersecting connected ideals, is a complete intersection presentation if and
only if no connected order ideal of $P$ intersects two or more other connected
ideals nontrivially.

Proposition 18.6 is a special case of an easy fact about Stanley-Reisner
rings.

######  Definition 18.7.

Let $V$ be a finite set and suppose $\Delta$ is a _simplicial complex_ on $V$,
i.e. a collection of subsets of $V$ such that if $F\in\Delta$ and
$F^{\prime}\subset F$, then $F^{\prime}\in\Delta$. Let $S$ be a polynomial
ring whose variables are indexed by elements of $V$. The _Stanley-Reisner
ideal_ of $\Delta$ is the ideal

$I_{\Delta}=\left(\prod_{v\in F}x_{v}\colon F\notin\Delta\right).$

The quotient ring $S/I$ is then called a _Stanley-Reisner ring_.

######  Proposition 18.8.

Suppose $I$ is a Stanley-Reisner ideal. The Stanley-Reisner ring $S/I$ is a
complete intersection if and only if

$I=(x_{11}\cdots x_{1n_{1}},\ldots,x_{m1}\cdots x_{mn_{m}}),$

i.e. $I$ is generated by a collection of square-free monomials with pairwise
disjoint support.

See [17, Theorem 4.1.1 ] for proof of a more general statement.

Proposition 18.6 follows from the observation that $I^{\mathrm{wt}}_{P}$ is
the Stanley-Reisner ideal of the simplicial complex on $J_{\mathrm{conn}}(P)$
defined by $F\subset J_{\mathrm{conn}}(P)$ if all ideals in $F$ pairwise
intersect trivially.

This characterization of posets such that
$S/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$ is a complete
intersection allows one to characterize such posets as those avoiding a list
of induced subposets, similar to that of [21, Theorem 10.5].

######  Definition 18.9.

Suppose $P$ is a signed poset such that

$0\to(U_{J}U_{K})\stackrel{{\scriptstyle\varphi}}{{\to}}S^{\mathrm{wt}}_{P}\to
S^{\mathrm{wt}}_{P}/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}\to 0$

is a complete intersection presentation. Then $P$ is said to be an _initial
complete intersection_.

######  Theorem 18.10.

Suppose $P$ is a signed poset. $P$ is an initial complete intersection if and
only if $P$ does not contain a signed poset isomorphic to any of those shown
in Figure 45 as an induced subposet.

|  |
---|---|---
|  |
|  |

$a$$b$$c$$-c$$-b$$-a$ (a)

$b$$a$$c$$-c$$-a$$-b$ (b)

$a$$b$$c$$-c$$-b$$-a$ (c)

$a$$b$$-b$$c$$-a$$-c$ (d)

$a$$c$$b$$-c$$-b$$-a$ (e)

$a$$-c$$b$$-b$$c$$-a$ (f)

$b$$a$$-c$$c$$-a$$-b$ (g)

$a$$b$$c$$-a$$-b$$-c$ (h)

$a$$c$$b$$-a$$-b$$-c$ (i)

Figure 45: The excluded posets, part one

|  |
---|---|---

$a$$b$$c$$-c$$-b$$-a$ (a)

$a$$b$$-b$$-a$$-c$$c$ (b)

$a$$b$$-b$$c$$d$$-d$$-c$$-a$ (c)

|
---|---

$a$$b$$c$$d$$-d$$-c$$-b$$-a$ (d)

$a$$b$$c$$d$$e$$-e$$-d$$-c$$-b$$-a$ (e)

$-n$$2$$-(n-2)$$4$$-(n-4)$$1$$-(n-1)$$3$$-(n-3)$$5$$3$$n-1$$-1$$n-2$$-2$$n$
(f) for $n\geq 3$ odd

$1$$2$$3$$4$$n-1$$n$$-1$$-2$$-3$$-4$$-(n-2)$$-(n-1)$$-n$$-(n-3)$$n-2$$n-3$ (g)
for $n\geq 4$ even

Figure 46: The excluded posets, part two

Key to the proof will be the following lemma.

######  Lemma 18.11.

Suppose $P^{\prime}$ is a signed poset that contains an induced subposet $P$,
such that $P$ has a connected order ideal $J$ which intersects at least two
other connected ideals of $P$ nontrivially. Then $P^{\prime}$ also has such an
ideal.

###### Proof.

It suffices to examine the case where $P$ is a signed poset on $n$ and
$P^{\prime}$ is a signed poset on $n+1$. Recall that an ideal is determined by
the antichain of its maximal elements. Consequently, every connected ideal $J$
of $P$ corresponds to an ideal $J^{\prime}$ in $\widehat{G}(P^{\prime})$
having the same determining antichain.

Claim: $J^{\prime}$ is an isotropic order ideal of $\widehat{G}(P^{\prime})$
and thus an ideal of $P^{\prime}$.

Suppose not. There are two cases.

  * •

First suppose $j_{1}$ and $j_{2}$ are maximal elements of $J$ (and thus of
$J^{\prime}$) with $(n+1)<j_{1}$ and $-(n+1)<j_{2}$. Then one must have
$-j_{2}<j_{1}$ and $-j_{1}<j_{2}$. Since $P$ is an induced subposet of
$P^{\prime}$, the same relation holds in $P$, which means $J$ was not
isotropic, a contradiction.

  * •

Suppose $j\in J^{\prime}$ such that $(n+1)<j$ and $-(n+1)<j$. Then $-j<j$,
which contradicts that $J$ was isotropic.

Thus, $J^{\prime}$ must be an ideal of $P^{\prime}$.

If $J$ and $K$ intersect nontrivially in $P$ and $J^{\prime}$ and $K^{\prime}$
are the corresponding ideals of $P^{\prime}$. Certainly $J^{\prime}\cap
K^{\prime}\neq\emptyset$. Suppose $J^{\prime}\subset K^{\prime}$, i.e.
$J^{\prime}$ and $K^{\prime}$ do not intersect nontrivially. Since $P$ is an
induced subposet of $P^{\prime}$, that would force $J\subset K$, contradicting
that $J$ and $K$ intersect nontrivially.

Consequently, if $P$ has a connected ideal $J$ that intersects connected
ideals $K_{1}$ and $K_{2}$ nontrivially, $J^{\prime}$ intersects
$K_{1}^{\prime}$ and $K_{2}^{\prime}$ nontrivially, completing the proof. ∎

While the above lemma makes the sufficiency of Theorem 18.10 straightforward,
the necessity argument is made easier by two further lemmas.

######  Lemma 18.12.

Suppose $n\geq 3$ and $P\subset\Phi_{B_{n}}$ is a signed poset such that no
$i<-i$. Then $P$ is not an initial complete intersection.

###### Proof.

One may assume $P$ is connected. Begin by observing that a member of the
infinite families in Figures 46(f) and 46(g) cannot be an initial complete
intersection as a consequence of Proposition 18.6. Since $P$ has no $i<-i$,
for $i\in\pm[x]$, there must be an $i\in[n]$ such that there is an isotropic
path from $i$ to $-i$. Since $i$ and $-i$ are not comparable, this path must
have at least two intermediate vertices and have a maximum and minimum other
than $i$ and $-i$. Then the image of this path under the involution will form
a cycle $C$ which is an induced subposet of $P$. Taking the smallest induced
subposet of $C$ which remains a cycle gives a member of one of the infinite
families in Figures 46(f) and 46(g), so by Lemma 18.11, $P$ is not an initial
complete intersection. ∎

Féray and Reiner characterized posets in type A such that
$R_{P}^{\mathrm{wt}}$ is a complete intersection as being those posets
avoiding the intersection of the three type A posets Figures 45(l),(m),(n)
with $\Phi_{A_{n-1}}$. While
$S^{\mathrm{wt}}_{P}/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$ being
a complete intersection implies $S^{\mathrm{wt}}_{P}/I^{\mathrm{wt}}_{P}$ is a
complete intersection (see [36, Exercise 3.3]), the converse is not, in
general, true. For example, consider the signed poset in Figure 47. The
semigroup ring $R_{P}^{\mathrm{wt}}$ has a complete intersection presentation:

$0\to(U_{\bar{4}}U_{14}-U_{1},U_{\bar{3}}U_{134}-U_{14},U_{\bar{4}}U_{134}-U_{13})\to\\\
k[U_{1},U_{2},U_{\bar{3}},U_{\bar{4}},U_{13},U_{14},U_{134}]\to
R_{P}^{\mathrm{wt}}\to 0.$

However,
$\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}=(U_{\bar{4}}U_{14},U_{\bar{3}}U_{134},U_{\bar{4}}U_{134})$,
so $S^{\mathrm{wt}}_{P}/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$ is
not a complete intersection.

$1$$2$$-3$$-4$$4$$3$$-2$$-1$ Figure 47: A signed poset $P$ where
$R_{P}^{\mathrm{wt}}$ is a complete intersection, but $P$ is not an initial
complete intersection.

The following lemma is an immediate consequence of Proposition 10.2 and
Theorem 10.5 of [21].

######  Lemma 18.13.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset such that $i<-i$ for all $i$.
Then $P$ is an initial complete intersection if and only if $P$ does not
contain any of the signed posets in Figures 45(l),(m),(n) as an induced
subposet.

One is now ready to prove Theorem 18.10.

###### Proof of Theorem 18.10.

The necessity follows from checking that all the posets in the list have at
least one connected isotropic order ideal that intersects at least two other
connected ideals nontrivially and applying Lemma 18.11.

For the sufficiency, Lemmas 18.12 and 18.13 allow one to immediately reduce to
the case where there is $i\in\pm[n]$ such that $i<-i$ in $P$ and there is some
$j$ such that $j$ and $-j$ are incomparable. There are a number of cases.

  1. (a)

Suppose $i\in\widehat{G}(P)$ is such that $i$ and $-i$ are incomparable and
$i$ is not comparable to any $a<-a$. Since $\widehat{G}(P)$ is connected,
there must be some path from $i$ to an $a$ such that $a<-a$ such that no
intermediate vertex is less than its negative. Call the vertices on this path
$i,v_{1},v_{2},\ldots,v_{k},a$. Let $m$ be minimal such that $v_{m}>a$. Then
one must have $v_{m-1}<v_{m}$. Taking the induced subposet on
$v_{m-1},v_{m},a$ gives a signed poset isomorphic to Figure 45(g).

  2. (b)

Suppose every $i\in\widehat{G}(P)$ such that $i$ and $-i$ are incomparable is
comparable to some $a$ such that $a<-a$.

    1. (b.i)

Suppose $\widehat{G}(P)$ has a connected isotropic order ideal $J$ containing
two elements $i,j$ which are not comparable to their negatives and there is no
$a$ such that $a<-a$ and $i,j>a$. Since $J$ is connected, there must be a path
from $i$ to $j$ contained entirely in $J$. If $j<i$, then the induced subposet
on $i,j,a$ gives a signed poset isomorphic to Figure 45(g).

If $i$ and $j$ are not comparable, one may assume without loss of generality
that the path between them in $J$ passes through only elements which are less
than their negatives. Since $i$ and $j$ are not both comparable to any $a$
with $a<-a$, the path between them must have at least three intermediate
vertices, $a_{1},a_{2},a_{3}$, as shown in Figure 48.

$i$$a_{1}$$a_{2}$$a_{3}$$j$ Figure 48: The path described in the second half
of case (b)(i) in the proof of Theorem 18.10

Since $a_{1}<-a_{1}$ and $i$ and $-i$ are incomparable, one knows the first
vertex along the path must be below $i$. Then taking the induced subposet on
$i,a_{2},a_{3}$ gives a signed poset isomorphic to that in Figure 45(k).

    2. (b.ii)

Assume instead that $J$ contains $i,j$ not comparable to their negatives and
that there is an $a$ with $a<-a$ with $i,j>a$. Taking the induced subposet on
$i,j,a$ gives one of seven possibilities, which are all forbidden posets,
namely Figures 45(a),(b),(c),(d),(f),(h),(i).

    3. (b.iii)

Suppose $J$ is a connected ideal containing precisely one element not less
than its negative, call it $i$. One knows that $I_{\leq}(-i)$ is isotropic, so
$J$ and $I_{\leq}(-i)$ intersect nontrivially.

Claim: $i$ is a maximal element of $J$.

Suppose not and $j\in J$ with $j>i$. Since $J$ contains only one element not
less than its negative, i.e. $i$, $j<-j$, but the symmetry in $\widehat{G}(P)$
then forces $i<-i$, a contradiction.

Having proved the claim, there are two final cases to check.

      1. (b.iii.1)

Suppose $J\cap K=\emptyset$. Then one must have $-i\in K$. Since $K\neq
I_{\leq}(-i)$, there must be $a,b\in K$ with $b,-i>a$ and $a<-a,b<-b$.
However, the induced subposet on $a,b,-i$ is isomorphic to the signed poset of
Figure 45(k).

      2. (b.iii.2)

Suppose $J\cap K\neq\emptyset$. Therefore, there must be $j,\ell$ in $J+K$
with $j<i,\ell$. Since $j,\ell\in J+K$, one must have that $j<-j$ and
$\ell<-\ell$. One then has that the induced subposet on $j,k,\ell$ is
isomorphic to one of the signed posets shown in Figures 45(j) and (k).

∎

####  18.3 Constructing the complete intersection posets

In [21], Féray and Reiner also characterized the posets for which
$R_{P}^{\mathrm{wt}}$ is a complete intersection as being the _forests with
duplication_ , those posets which could be constructed via operations they
called _disjoint union_ , _duplication of a hanger_ and _hanging_. An
analogous construction exists for signed posets which are initial complete
intersections. Due to the reductions in Section 18.1, one can dispense with
the disjoint union operation in the construction.

Recall that if $P$ is a signed poset, $I_{<}(a)=\\{b\in P\colon b<a\\}$ and
$I_{\leq}(a)=\\{b\in P\colon b\leq a\\}$, with the comparisons being made in
$\widehat{G}(P)$. Let $P_{<a}$ denote the subposet of $P$ induced by
$\\{i\colon\pm i\in I_{<}(a)\\}$ and let $P_{\leq a}$ denote the subposet of
$P$ induced by $\\{i\colon\pm i\in I_{\leq}(a)\\}$.

######  Definition 18.14.

Suppose $P$ is a signed poset and $i\in\widehat{G}(P)$ is such that $i<-i$.
Then $i$ is said to be a _hanger_ if, for each $a\in I_{<}(i)$, the induced
subset on those vertices $\pm j$ such that $j<i$ (which may be empty) and for
each $b\in G(P)\smallsetminus G(P_{<i})$, any path from $a$ to $b$ must pass
through $i$.

######  Definition 18.15.

A signed poset $P$ is said to have been _obtained by hanging $P_{1}$ below $a$
in $P_{2}$_ if

  * •

$a$ is a hanger in $P$,

  * •

$P_{1}=P_{<}(a)$, and

  * •

$P_{2}$ is the induced subposet of $P$ on $i$ such that $\pm i\not<a$ in
$\widehat{G}(P)$.

See Figure 49 for an example of hanging.

$1$$2$$-3$$3$$-2$$-1$$1$$2$$-3$$3$$-2$$-1$$-4$$4$hanging Figure 49: Obtaining
a new poset by hanging a single element 4 below 2

When $P$ is a type A signed poset, this definition of hanging coincides with
the hanging of [21] in the sense that if $P^{\prime}$ is obtained from $P$ by
hanging, $P^{\prime}\cap\Phi_{A_{n-1}}$ gives the poset that is the result of
the same hanging in $P\cap\Phi_{A_{n-1}}$.

######  Lemma 18.16.

Suppose $P$ is obtained by hanging $P_{1}$ below $a$ in $P_{2}$ and both
$P_{1}$ and $P_{2}$ are initial complete intersections. Then $P$ is also an
initial complete intersection.

###### Proof.

Suppose $P$ is obtained by hanging $P_{1}$ below $a$ in $P_{2}$ and both
$P_{1}$ and $P_{2}$ are initial complete intersections. The connected
(isotropic) order ideals of $P$ are of three types:

  1. (I)

$J$ for $J\in J_{\mathrm{conn}}(P_{1})$

  2. (II)

$J\cup P_{1}$ for $J\in J_{\mathrm{conn}}(P_{2})$ with $a\in J$.

  3. (III)

$J$ for $J\in J_{\mathrm{conn}}(P_{2})$ with $a\notin J$.

Ideals of type I can only intersect other ideals of type I nontrivially and
can therefore only intersect at most one other ideal nontrivially. A
nontrivially intersecting pair involving ideals of type II or III corresponds
to a nontrivially intersecting pair in $P_{2}$, so these ideals can also only
be involved in at most nontrivially intersecting pair. Consequently, one has
that $P$ is an initial complete intersection. ∎

The next move also closely parallels [21].

######  Definition 18.17.

The signed poset $P^{\prime}$ is said to have been obtained from $P$ by
_duplicating the hanger_ $a$ if
$\widehat{G}(P^{\prime})=\widehat{G}(P)\cup\\{a^{\prime},-a^{\prime}\\}$ with
$i<a^{\prime}$ or $a^{\prime}<i$ whenever $i<a$ or $a<i$ in $\widehat{G}(P)$.

Figure 50 gives an example of duplicating a hanger.

$1$$2$$-2$$-1$$1$$2$$3$$-2$$-3$$-1$duplicating $2$ Figure 50: Obtaining a new
poset by duplicating the hanger 2

######  Lemma 18.18.

Suppose $P^{\prime}$ is obtained from $P$ by duplicating a hanger and $P$ is
an initial complete intersection. Then $P^{\prime}$ is also an initial
complete intersection.

###### Proof.

Begin by noting that if $a$ is a hanger in $P$, it must be that $I_{\leq a}$
intersects no other ideal nontrivially. Then the connected isotropic order
ideals of $P^{\prime}$ are

  * •

$J$ for $J\in J_{\mathrm{conn}}(P)$ with $a\notin J$

  * •

$J\cup\\{a^{\prime}\\}$ if $J\in J_{\mathrm{conn}}(P)$ with $a\in J$

  * •

$I_{\leq}(a^{\prime})$.

The only nontrivially intersecting pair in $P^{\prime}$ that does not
correspond to a nontrivially intersecting pair in $P$ (i.e. the only pair
created by the duplication) is $\\{I_{\leq}(a),I_{\leq}(a^{\prime})\\}$,
meaning $P^{\prime}$ must also have
$S^{\mathrm{wt}}_{P^{\prime}}/\mathrm{in}I_{P^{\prime}}^{\mathrm{wt}}$ a
complete intersection. ∎

While hanging and duplication are translations of the type A definitions, an
additional move is required to construct all the posets that are initial
complete intersections.

######  Definition 18.19.

The signed poset $P^{\prime}$ is said to have been obtained from $P$ by a
_type B hanging_ of $n+1$ from $i$ if
$P^{\prime}=\overline{P\cup\\{e_{i}+\epsilon e_{n+1}\\}}^{PLC}$, where $i<-i$
and $\epsilon\in\\{\pm 1\\}$, and $I_{\leq}(i)$ is a maximal ideal with
respect to inclusion.

Figure 51 gives an example of type B hanging. The important thing to notice is
that the requirement $I_{\leq}(i)$ be a maximal ideal severely limits the new
ideals in $P^{\prime}$ compared to $P$.

$1$$2$$-2$$-1$$1$$2$$-2$$-1$$3$$-3$type B hanging Figure 51: Obtaining a
signed poset by the type B hanging of 3 above 1

######  Lemma 18.20.

Suppose $P^{\prime}$ is a signed poset obtained from $P$ by a type B hanging.
If $P$ is an initial complete intersection, then $P^{\prime}$ is also.

###### Proof.

Suppose $P^{\prime}$ is obtained from $P$ by a type B hanging. Then the
connected isotropic order ideals are

  * •

$J$ for $J\in J_{\mathrm{conn}}(P)$

  * •

$I_{\leq}(\epsilon(n+1))$

  * •

$\\{-\epsilon(n+1)\\}$.

It is clear that $I_{\leq}(\epsilon(n+1))$ and $\\{-\epsilon(n+1)\\}$
intersect nontrivially and each intersect no other ideal nontrivially. The
other ideals cannot be involved in more than one nontrivial intersection
because they were not in $P$. Thus $P^{\prime}$ must be an initial complete
intersection. ∎

######  Theorem 18.21.

Up to isomorphism, the signed posets with $\widehat{G}(P)$ connected which are
initial complete intersections are precisely those which can be constructed by
hanging, duplication of a hanger and type B hanging from the two posets in
Figure 52.

$1$$-1$ (a)

$1$$2$$-2$$-1$ (b)

Figure 52: The posets from which the initial complete intersections are built

###### Proof.

The proof proceeds by induction, with base cases of $n=1$ and $n=2$. Up to
isomorphism, there is only one signed poset for $n=1$, namely
$P_{1}=\\{+e_{1}\\}$ (see Figure 52a) and its toric ideal is trivial, so it is
certainly an initial complete intersection. For $n=2$, there are four signed
posets which are initial complete intersections: those obtained from $P_{1}$
by hanging, duplication of a hanger and type B hanging, and the signed poset
in Figure 52b, whose toric ideal is $(U_{12}U_{1\bar{2}}-U_{1}^{2})$, which is
principal, so the signed poset is an initial complete intersection.

Suppose $P$ is a signed poset on $n\geq 3$ and $\widehat{G}(P)$ is connected
and $P$ is an initial complete intersection. Consider $a\in\widehat{G}(P)$ and
let $I(a)=\\{i\in\pm[n]:i<a\\}$. From Lemma 18.12, one knows that one may
assume $a<-a$. There are three cases.

  1. (a)

Suppose that $a$ is not minimal and for all $a^{\prime}\in\widehat{G}(P)$ not
comparable to $a$ with $I_{<}(a^{\prime})$ isotropic, $I_{<}(a)\cap
I_{<}(a^{\prime})=\emptyset$. Then define two induced subposets of $P$:
$P_{<a}$, the induced subposet on $i$ such that $\pm i<a$, and
$P\smallsetminus P_{<a}$, the induced subposet on $i$ such that $\pm i\not<a$.
Furthermore, $P$ is obtained by hanging $P_{<a}$ below $P\smallsetminus
P_{<a}$.

  2. (b)

Suppose there exists $a^{\prime}\in\widehat{G}(P)$ not comparable to $a$ with
$I_{<}(a^{\prime})$ isotropic and $I_{<}(a)\cap
I_{<}(a^{\prime})\neq\emptyset$.

Claim: $a^{\prime}<-a^{\prime}$.

Suppose not. Then both $I_{<}(a^{\prime})$ and $I_{<}(-a^{\prime})$ are
isotropic and intersect nontrivially. Then since $a<-a$, it must be that
$a\neq-a^{\prime}$, meaning $I_{<}(a^{\prime})$ intersects both
$I_{<}(-a^{\prime})$ and $I_{<}(a)$ nontrivially, contradicting that $P$ had
$S/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$ a complete intersection.

As in type A (see [21, Theorem 10.6 ]), one decomposes $P$ into four induced
subposets:

$P=\widehat{P}\sqcup P_{<a,a^{\prime}}\sqcup(P_{<a}\smallsetminus
P_{<a,a^{\prime}})\sqcup(P_{<a^{\prime}}\smallsetminus P_{<a,a^{\prime}}),$

where $P_{<a,a^{\prime}}$ is the induced subposet on $i$ such that $\pm i<a$
and $\pm i<a^{\prime}$ in $\widehat{G}(P)$, $(P_{<a}\smallsetminus
P_{<a,a^{\prime}})$ is the induced subposet on $i$ such that $\pm i<a$ and
$\pm i\not<a^{\prime}$, similarly for $(P_{<a^{\prime}}\smallsetminus
P_{<a,a^{\prime}})$ and $\widehat{P}=P\smallsetminus(P_{<a}\sqcup
P_{<a^{\prime}})$.

Observe that any element less than either $a$ or $a^{\prime}$ is less than its
negative. The signed posets $P_{<a}\smallsetminus P_{<a^{\prime}}$ and
$P_{<a^{\prime}}\smallsetminus P_{<a}$ may be empty, but $P_{<a,a^{\prime}}$
is not.

One constructs a signed poset $Q$ using hanging, duplication and type B
hanging and then shows that $Q=P$. There are two cases of the construction.

     * •

Suppose there is $b\in\widehat{P}$ with $a<b$, $a^{\prime}\not<b$ and
$b\not<-b$. Construct $Q$ as follows:

       1. (1)

Start with $\widehat{P}\smallsetminus\\{a^{\prime},b\\}$.

       2. (2)

Hang $P_{<a,a^{\prime}}$ below $a$ in
$\widehat{P}\smallsetminus\\{a^{\prime},b\\}$.

       3. (3)

Duplicate $a$.

       4. (4)

Hang $P_{<a}\smallsetminus P_{<a^{\prime}}$ and $P_{<a^{\prime}}\smallsetminus
P_{<a}$ below $a$ and $a^{\prime}$, respectively.

       5. (5)

Type B hang $b$ from $c$, where $c\lessdot b$ in $P$.

     * •

Suppose no such $b$ exists. Construct $Q$ as follows:

       1. (1)

Start with $\widehat{P}\smallsetminus\\{a^{\prime},b\\}$.

       2. (2)

Hang $P_{<a,a^{\prime}}$ below $a$ in
$\widehat{P}\smallsetminus\\{a^{\prime},b\\}$.

       3. (3)

Duplicate $a$.

       4. (4)

Hang $P_{<a}\smallsetminus P_{<a^{\prime}}$ and $P_{<a^{\prime}}\smallsetminus
P_{<a}$ below $a$ and $a^{\prime}$, respectively.

In each case, each of the induced subposets used
($P_{<a,a^{\prime}},\widehat{P}\smallsetminus\\{a^{\prime}\\},\widehat{P}\smallsetminus\\{a^{\prime},b\\},P_{<a}\smallsetminus
P_{<a^{\prime}},P_{<a^{\prime}}\smallsetminus P_{<a},$) is an initial complete
intersection courtesy of Lemma 18.11, since they are induced subposets of an
initial complete intersection. Consequently, $Q$ is an initial complete
intersection.

Next, one needs to show that $P$ and $Q$ are really the same poset. Certainly
the restrictions of $P$ and $Q$ to $P_{<a,a^{\prime}}$, $P_{<a}\smallsetminus
P_{<a,a^{\prime}}$ and $P_{<a^{\prime}}\smallsetminus P_{<a,a^{\prime}}$ are
the same. It remains to check that when restricted to the vertices of
$\widehat{P}$, $P$ and $Q$ are the same. Suppose $c>a$ and $c\not>a^{\prime}$
in $\widehat{P}$. Consider two cases.

    1. (b.i)

Suppose $c<-c$. Recall that there must be at least one element, call it
$\ell$, such that $\ell<a$ and $\ell<a^{\prime}$. Moreover, $\ell<-\ell$. This
means that the induced subposet on $a,b,\ell$ is isomorphic to Figure 45(j).

    2. (b.ii)

Suppose $c\not<-c$. The existence of such a $c$ means $Q$ was constructed
using the first construction. If $c$ is not the $b$ from the construction, one
has that $a\in I_{\leq}(c)\cap I_{\leq}(b)$. However, $I_{\leq}(c)$ and
$I_{\leq}(-c)$ also intersect nontrivially, a contradiction, unless $b=-c$. If
$b=-c$, recall that there must be some $\ell\in P_{<a,a^{\prime}}$. Taking the
induced subposet of $a,a^{\prime},b,\ell$ gives the signed poset shown in
Figure 53.

$a$$b$$-b$$-a$$\ell$$a^{\prime}$$-a^{\prime}$ Figure 53: An induced subposet
when $c>a,c\not>a^{\prime},c<-c$ in case (b.ii) of the proof of Theorem 18.21

However, this poset is not an initial complete intersection, since
$\\{b,a,\ell\\}$ intersects both $\\{-b,a,\ell\\}$ and $\\{\ell,a^{\prime}\\}$
nontrivially, a contradiction, courtesy of Lemma 18.11.

It remains to check that given $x,y$ in two different pieces of the
decomposition, they share the same relation in $P$ and in $Q$. The cases break
down as they did in type A.

    1. (b.I)

Suppose $x\in P_{<a}\smallsetminus P_{<a^{\prime}}$ and $y\in
P_{a^{\prime}}\smallsetminus P_{<a}$. Then $x$ and $y$ must be incomparable in
both $P$ and $Q$.

    2. (b.II)

Suppose $x\in P_{<a}\smallsetminus P_{<a^{\prime}}$ (or
$P_{<a^{\prime}}\smallsetminus P_{<a}$) and $y\in P_{<a,a^{\prime}}$. In this
case, $x$ and $y$ are incomparable in $Q$ by construction. If $x<_{P}y$, then
$x\in P_{<a,a^{\prime}}$, a contradiction. If $y<_{P}x$, then the induced
subposet of $a,x,y$ is isomorphic to the signed poset in Figure 45(j),
contradiction. Thus, one has that $x$ and $y$ must also be incomparable in
$P$.

    3. (b.III)

Suppose $x\in P_{<a}\smallsetminus P_{<a^{\prime}}$ and $y\in\widehat{P}$.
Neither $y\leq_{Q}x$ nor $y\leq_{P}x$ is possible. Observe that $x\leq_{Q}y$
if and only if $a\leq_{P}y$ and $a\leq_{P}y$ implies $x\leq_{P}y$. Therefore,
one must show that if $a\not\leq_{P}y$ then $x\not\leq_{P}y$. Suppose not, and
$a\not\leq_{P}y$, but $x\leq_{P}y$. One knows that there exists $\ell\in
P_{<a,a^{\prime}}$. Suppose $\ell\not\leq_{P}y$. If $\ell<\ell^{\prime}$, then
the subposet of $P$ induced by $a,a^{\prime},x,y,\ell$ is isomorphic to the
signed poset in Figure 45(n) a contradiction. If $\ell$ and $-\ell$ are not
comparable, the subposet of $P$ induced by $a,y,x$ is isomorphic to the signed
poset in Figure 45(j) or (k), a contradiction in both cases.

Thus, one must have that $\ell\leq_{P}y$. There are two cases:

       * •

$y\not\geq_{P}a^{\prime}$. Then the subposet of $P$ induced by
$a,a^{\prime},y,\ell$ is isomorphic to the signed poset in Figure 45(m), a
contradiction.

       * •

$y\geq_{P}a^{\prime}$. Then the subposet of $P$ induced by $a,y,\ell$ is
isomorphic to the signed poset in Figure 45(j), a contradiction.

    4. (b.IV)

Suppose $x\in P_{<a^{\prime}}\smallsetminus P_{<a}$ and $y\in\widehat{P}$.
This case is the same as the previous case with the roles of $a$ and
$a^{\prime}$ exchanged.

    5. (b.V)

Suppose $x\in P_{<a,a^{\prime}}$ and $y\in\widehat{P}$. Begin by observing
that $y\leq_{Q}x$ and $y\leq_{P}x$ are impossible. One has that $x\leq_{Q}y$
if and only if $a\leq_{P}y$ or $a^{\prime}\leq_{P}y$. Both $a\leq_{P}y$ and
$a^{\prime}\leq_{P}y$ imply $x\leq_{P}y$, so one needs to show that if neither
$a\not\leq_{P}y$ nor $a^{\prime}\not\leq_{P}y$, then $x\not\leq_{P}y$. Suppose
not. If $y<-y$, the subposet induced by $x,y,a,a^{\prime}$ is isomorphic to
the forbidden signed poset in Figure 45(m).

If $y$ and $-y$ are not comparable, then the subposet induced by $x,y,a$ is
isomorphic to the signed poset in either Figure 45(j) or (k), a contradiction.

  3. (c)

Suppose that $a$ _is_ minimal and for all $a^{\prime}\in\widehat{G}(P)$ not
comparable to $a$ with $I_{<}(a^{\prime})$ isotropic, $I_{<}(a)\cap
I_{<}(a^{\prime})=\emptyset$. One may assume that no $a$ exists that falls
into either of the earlier two cases. Then every $a\in\widehat{G}(P)$ with
$a<-a$ is a minimal element of $\widehat{G}(P)$. One can then divide $[n]$
into $a_{1},\ldots,a_{k}$ and $b_{1},\ldots,b_{j}$, where $a_{i}<-a_{i}$ and
$b_{\ell}$ and $-b_{\ell}$ are not comparable. Since $P$ is an initial
complete intersection, no $a_{i}$ lies below more than one $b_{\ell}$. There
are two cases.

     * •

Suppose $a_{1}$ and $a_{2}$ are covered by all the same elements (i.e.
$-a_{1},\ldots,-a_{k}$ and possibly $b_{\ell}$ for some $\ell$). Let
$P^{\prime}=P\smallsetminus\\{a_{1}\\}$. Then $a_{2}$ is a hanger in
$P^{\prime}$ and $P$ is obtained from $P^{\prime}$ by duplicating $a_{2}$.

     * •

Suppose there is no such $a_{1},a_{2}$. Then, there must be an $a_{i}$ such
that $b_{\ell}>a_{i}$ and $b_{\ell}$ covers no other $a_{s}$. Let
$P^{\prime}=P\smallsetminus\\{b_{\ell}\\}$. Then $P$ is obtained from
$P^{\prime}$ by a type B hanging of $b_{\ell}$ from $a_{i}$.

∎

####  18.4 Computing Rational Functions

Having characterized the initial complete intersections, attention now turns
to calculating various rational function identities. One knows from
Proposition 18.2 that the $U_{J}U_{K}$ as $\\{J,K\\}$ runs over $\Pi(P)$ form
a minimal generating set for
$\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$. When $P$ is an initial
complete intersection, one then has

$\mathrm{Hilb}(R_{P}^{\mathrm{wt}},\bm{x})=\mathrm{Hilb}(S^{\mathrm{wt}}_{P}/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P},\bm{x})=\dfrac{\prod_{\\{J,K\\}\in\Pi(P)}(1-\bm{x}^{J}\bm{x}^{K})}{\prod_{J\in
J_{\mathrm{conn}}(P)}(1-\bm{x}^{J})}.$ (11)

One can then apply Proposition 6.14 to obtain $\Phi_{P}$.

######  Corollary 18.22.

Suppose $P$ is an initial complete intersection. Then

$\Phi_{P}(\bm{x})=\dfrac{\prod_{\\{J,K\\}\in\Pi(P)}\langle\bm{x},\chi_{J}+\chi_{K}\rangle}{\prod_{J\in
J_{\mathrm{conn}}(P)}\langle\bm{x},\chi_{J}\rangle}.$

$1$$3$$2$$-2$$-1$$-3$ Figure 54: A signed poset

For example, consider the signed poset in Figure 54. Then
$S^{\mathrm{wt}}_{P}=k[U_{1},U_{3},U_{12},U_{1\bar{2}}]$ and
$\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}=(U_{12}U_{1\bar{2}})$. Then

$\mathrm{Hilb}(S^{\mathrm{wt}}_{P}/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P},\bm{x})=\dfrac{1-x_{1}^{2}}{(1-x_{1})(1-x_{3})(1-x_{1}x_{2})(1-x_{1}x_{2}^{-1})},$

and applying Proposition 6.14 gives

$\Phi_{P}(\bm{x})=\dfrac{2x_{1}}{x_{1}x_{3}(x_{1}+x_{2})(x_{1}-x_{2})}.$

On the other hand, Table 5 gives the linear extensions of $P$, their descent
sets and their major index.

$w$ | $\mathrm{Des}(w)$ | $\mathrm{maj}(w)$
---|---|---
$\begin{pmatrix}1&2&3\\\ 1&2&3\end{pmatrix}$ | $\emptyset$ | $0$
$\begin{pmatrix}[r]1&2&3\\\ 1&-2&3\end{pmatrix}$ | $\\{2\\}$ | $2$
$\begin{pmatrix}[r]1&2&3\\\ 1&3&2\end{pmatrix}$ | $\\{2\\}$ | $2$
$\begin{pmatrix}[r]1&2&3\\\ 1&3&-2\end{pmatrix}$ | $\\{3\\}$ | $3$
$\begin{pmatrix}1&2&3\\\ 3&1&2\end{pmatrix}$ | $\\{1\\}$ | $1$
$\begin{pmatrix}[r]1&2&3\\\ 3&1&-2\end{pmatrix}$ | $\\{1,3\\}$ | $4$
Table 5: Linear extensions for the signed poset of Figure 54, their descents
and major index.

Then

$\Phi_{P}(\bm{x})=\sum_{w\in\mathcal{L}(P)}w\left(\dfrac{1}{x_{1}(x_{1}+x_{2})\cdots(x_{1}+\cdots+x_{n})}\right)\\\
=\dfrac{1}{x_{1}(x_{1}+x_{2})(x_{1}+x_{2}+x_{3})}+\dfrac{1}{x_{1}(x_{1}-x_{2})(x_{1}-x_{2}+x_{3})}+\dfrac{1}{x_{1}(x_{1}+x_{3})(x_{1}+x_{2}+x_{3})}\\\
+\dfrac{1}{x_{1}(x_{1}+x_{3})(x_{1}+x_{3}-x_{2})}+\dfrac{1}{x_{3}(x_{1}+x_{3})(x_{1}+x_{2}+x_{3})}+\dfrac{1}{x_{3}(x_{1}+x_{3})(x_{1}-x_{2}+x_{3})}\\\
=\dfrac{2}{(x_{1}-x_{2})(x_{1}+x_{2})x_{3}}=\dfrac{2x_{1}}{x_{1}x_{3}(x_{1}-x_{2})(x_{1}+x_{2})}.$

Altering the grading so $\deg U_{J}=|J|$ gives the following identity (c.f.
Proposition 18.1).

######  Corollary 18.23.

Suppose the signed poset $P$ is an initial complete intersection. Then

$\sum_{w\in\mathcal{L}(P)}q^{\mathrm{maj}(w)}=[n]!_{q}\dfrac{\prod_{\\{J,K\\}\in\Pi(P)}[|J|+|K|]_{q}}{\prod_{J\in
J_{\mathrm{conn}}(P)}[|J|]_{q}}.$

###### Proof.

Collapsing the ${\mathbb{Z}}^{n}$-grading to the ${{\mathbb{N}}}$-grading
where $\deg U_{J}=|J|$, (11) transforms into

$\mathrm{Hilb}(S^{\mathrm{wt}}_{P}/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P},q)=\sum_{f\in\mathcal{A}(P)}q^{|f|}=\dfrac{\prod_{\\{J,K\\}\in\Pi(P)}1-q^{|J|+|K|}}{\prod_{J\in
J_{\mathrm{conn}}(P)}1-q^{|J|}}.$

Using Proposition 18.1 one then has

$\sum_{w\in\mathcal{L}(P)}q^{\mathrm{maj}(w)}=[n]!_{q}\dfrac{\prod_{\\{J,K\\}\in\Pi(P)}1-q^{|J|+|K|}}{\prod_{J\in
J_{\mathrm{conn}}(P)}1-q^{|J|}}=[n]!_{q}\dfrac{\prod_{\\{J,K\\}\in\Pi(P)}[|J|+|K|]_{q}}{\prod_{J\in
J_{\mathrm{conn}}(P)}[|J|]_{q}},$

as claimed. ∎

Looking at the signed poset from Figure 54 once again, one has

$\sum_{w\in\mathcal{L}(P)}q^{\mathrm{maj}(w)}=[3]!_{q}\dfrac{[2+2]_{q}}{[1]_{q}[1]_{q}[2]_{q}[2]_{q}}=\dfrac{[4]_{q}[3]_{q}}{[2]_{q}}=1+q+2q^{2}+q^{3}+q^{4},$

matching the previous tabulation (see Table 5).

Lastly, one can alter the grading of
$S^{\mathrm{wt}}_{P}/\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$ a
third time, taking $\deg U_{J}=1$ to obtain the following.

######  Corollary 18.24.

Suppose $P$ is a signed poset which is an initial complete intersection. Then

$\sum_{f\in\mathcal{A}(P)}t^{\nu(f)}=\dfrac{(1-t^{2})^{|\Pi(P)|}}{(1-t)^{|J_{\mathrm{conn}}(P)|}},$

where $\nu(f)$ is the number of ideals in the unique expression of $f$ as a
sum of nontrivially intersecting connected ideals.

##  Chapter Unfinished Business

###  19 Two triangulations of the weight cone

As was the case for Féray and Reiner in type A, the ideal
$\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$ suggests a triangulation
of the weight cone $K_{P}^{\mathrm{wt}}$. They explained (see [21, §11])

  * •

that $K_{P}^{\mathrm{wt}}$ is a maximal cone in the normal fan of the graphic
zonotope associated to the graph underlying the Hasse diagram,

  * •

that the normal fan of the graphic zonotope is refined by the normal fan of
the graph associahedron of the same graph and,

  * •

lastly, that $\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$ is the
Stanley-Reisner ideal for the simplicial complex describing the triangulation
of the weight cone by the the normal fan of the graph associahedron.

This section explains these two triangulations of the weight cone, the first
indexed by linear extensions, the second by sets of pairwise trivially
intersecting connected ideals, explains Zaslavsky’s signed graph analogue of
the graphic zonotope and how it gives a triangulation of
$K_{P}^{\mathrm{wt}}$, analogous to type A, but leaves open the problem of
finding the correct definition of signed graph associahedron.

######  Definition 19.1.

A _triangulation_ of the cone $K$ is a collection
$T=\\{\sigma_{1},\ldots,\sigma_{k}\\}$ of simplicial cones such that

  * •

$\bigcup\sigma_{i}=K$;

  * •

if $\sigma\in T$, then every face of $\sigma$ is in $T$;

  * •

for any $\sigma_{i},\sigma_{j}\in T$,$\sigma_{i}\cap\sigma_{j}$ is a common
face of $\sigma_{i}$ and $\sigma_{j}$

The first triangulation of $K_{P}^{\mathrm{wt}}$ to be discussed is one
suggested by Proposition 10.3, that where the maximal cones are unions of the
$w\Phi^{+}$-partitions for $w\in\mathcal{L}(P)$, called the _$P$ -partition
triangulation_. We next describe a second triangulation called the _trivially
intersecting ideals triangulation_.

######  Proposition 19.2.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset. The weight cone
$K_{P}^{\mathrm{wt}}$ is triangulated by cones
$\\{\sigma_{A}\colon\mathop{\mathrm{span}}A\\}$, where $A$ runs over all sets
of connected ideals of $P$, the elements of which pairwise intersect
trivially.

Proposition 19.2 is an immediate consequence of [58, Theorem 8.3 ]. Moreover,
as mentioned in Section 18.2, since
$\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$ is square-free, it is the
Stanley-Reisner ideal of the complex on $J_{\mathrm{conn}}(P)$ whose faces are
given by sets of ideals which pairwise intersect trivially.

Figure 55 shows the two different triangulations: the $P$-partition
triangulation in Figure 55(d), where the maximal cones are indexed by the
linear extensions of $P$, and the non-intersecting ideals triangulation of
Proposition 19.2 in Figure 55(c), where the maximal cones are indexed by
signed posets (more on these signed posets later).

$1$$2$$3$$+$$-$$+$$+$$+$ (a) Hasse diagram

$\bar{2}01$$\bar{2}0\bar{1}$$0\bar{2}\bar{1}$$0\bar{2}1$$02\bar{1}$$021$$20\bar{1}$$201$
(b) The acyclotope $\mathcal{Z}[\Sigma_{P}]$

(c) $K_{P}^{\mathrm{wt}}$ triangulated by sets of pairwise trivially
intersecting ideals

(d) $K_{P}^{\mathrm{wt}}$ triangulated by $\mathcal{N}(\mathcal{Z}[\pm
K_{n}^{B}])$

Figure 55: The two different triangulations of $K_{P}^{\mathrm{wt}}$

In type A, the analogous triangulation is explained by Féray and Reiner in
terms of the graphic zonotope.

######  Definition 19.3.

A _zonotope_ is a polytope that is the Minkowski sum of line segments. In
particular, if $G$ is a graph, then

$\mathcal{Z}[G]=\sum_{e=(i,j)\in G}[-(e_{i}-e_{j}),e_{i}-e_{j}],$

is called the _graphic zonotope_.

Féray and Reiner note that the maximal cones of the normal fan
$\mathcal{N}(\mathcal{Z}[G])$ are indexed by acyclic orientations of $G$. In
particular, if $P$ is a poset and $G$ is the graph underlying its Hasse
diagram, there is an orientation, call it $\omega$, of $G$ such that the
corresponding maximal cone of the normal fan, $\mathcal{N}_{\omega}$, is, in
fact, $K_{P}^{\mathrm{wt}}$.

######  Definition 19.4.

Let $G$ be a graph. Its _graphical building set_ $\mathcal{B}(G)$ is the
collection of sets of vertices $J$ where the vertex-induced subgraph $G|_{J}$
is connected.

The graphical building set is used to define the graph associahedron of [12].

######  Definition 19.5.

Suppose $G$ is a graph. Its _graph associahedron_ is

$\mathcal{P}_{G}=\sum_{J\in\mathcal{B}(G)}\mathop{\mathrm{conv}}\\{e_{j}\colon
j\in J\\}.$

Féray and Reiner show the following.

######  Proposition 19.6 ([21, Proposition 11.7]).

Suppose $P$ is a poset on $[n]$, $G$ is the graph underlying its Hasse diagram
and $w$ is the orientation of $G$ giving the Hasse diagram of $P$. Then the
simplicial complex $\Delta_{P}$ having
$\mathop{\mathrm{in}_{\preceq}}I^{\mathrm{wt}}_{P}$ as its Stanley-Reisner
ideal describes the triangulation of the cone $\mathcal{N}_{w}$ in the fan
$\mathcal{N}(\mathcal{Z}[G])$ by cones of the normal fan
$\mathcal{N}(\mathcal{P}_{G})$.

The maximal cones of this triangulation are indexed by
$\mathcal{B}(G)$-forests, forests $F$ in which every principal ideal $F_{\leq
i}$ is a connected ideal of $P$ and, whenever $i$ and $j$ are incomparable in
$F$, then the ideal $F_{\leq i}$ and $F_{\leq j}$ of $P$ is disconnected.

One would hope for a similar understanding of the triangulation of Proposition
19.2. In the signed poset case, one can avail oneself of the analogue of the
graphic zonotope—the acyclotope of Zaslavsky.

######  Definition 19.7.

Suppose $\Sigma$ is a signed graph and $\tau$ is a bidirection orienting
$\Sigma$. Then _acyclotope_ of $\Sigma$, written $\mathcal{Z}[\Sigma]$ is the
polytope

$\mathcal{Z}[\Sigma]=\sum_{e\in E}[-x_{\tau}(e),x_{\tau}(e)],$

where $x_{\tau}(e)$ is the column vector associated to $e$ in the incidence
matrix of $\Sigma$.

Figure 55(b) gives an example of the acyclotope of the signed poset in Figure
55(a).

######  Proposition 19.8.

The hyperplane arrangement $H(\Sigma)$ whose fan corresponds to the normal fan
of the graphic zonotope is:

$\begin{array}[]{ll}x_{i}=\sigma(e)x_{j}&\text{for an edge $e=(i,j)$}\\\
x_{i}=0&\text{for a loop or half edge $e=(i,i)$, $e=(i,-)$}\end{array}$

The regions of $H(\Sigma)$ correspond to various orientations of $\Sigma$.

######  Definition 19.9.

A _cycle_ of an oriented signed graph is a matroid circuit such that there is
no vertex $v$ such that all $\tau(v,e)$ coincide as $e$ runs over the edges of
the circuit incident to $v$. An orientation that contains no cycles will be
said to be _acyclic_.

Zaslavsky generalized a result of Greene to the signed graph/acyclotope case
in the following.

######  Theorem 19.10 ([63, Theorem 4.4 ]).

Suppose $\Sigma$ is a signed graph. There is a one-to-one correspondence
between the regions of $H[\Sigma]$ and acyclic orientations of $\Sigma$.

As an example, consider the signed poset, $P$, shown in Figure 56. Figure
56(c) shows $\Sigma_{P}$, the signed graph underlying the Hasse diagram of $P$
(Figure 56(a)). As $\Sigma_{P}$ has a single unbalanced cycle, so every
orientation will be acyclic.

$1$$2$$+$$-$$+$$+$ (a) Hasse diagram

$1$$2$$-2$$-1$ (b) $\widehat{G}(P)$

$1$$2$$+$$-$ (c) $\Sigma_{P}$

Figure 56: $P=\\{+e_{1}-e_{2},+e_{1}+e_{2},+e_{1}\\}$

${(0,0)}$${(1,-1})$${(2,0)}$${(1,1)}$ (a) the zonotope $\mathcal{Z}[\Sigma]$

$x_{1}=x_{2}$$x_{1}=-x_{2}$$1$$2$$-$$+$$+$$+$$1$$2$$+$$+$$-$$-$$1$$2$$+$$+$$-$$-$$1$$2$$+$$-$$+$$+$
(b) $\mathcal{N}(\mathcal{Z}[\Sigma])$ with maximal cones labeled by acyclic
orientations of $\Sigma$

Figure 57: The zonotop $\mathcal{Z}[\Sigma]$ and its Newton polytope for
$\Sigma$ from Figure 56

Figure 57 shows the acyclotope $\mathcal{Z}[\Sigma_{P}]$ and its normal fan
$\mathcal{N}(\mathcal{Z}[\Sigma_{P}])$. Recall from Proposition 10.3 that the
$P$-partitions of a signed poset are the disjoint union of the
$w\Phi^{+}$-partitions for each $w\in\mathcal{L}(P)$. This corresponds to the
triangulation of $\mathcal{N}(\mathcal{Z}[\Sigma_{P}])$ by the normal fan of
the acyclotope of the complete graph, as in type A. There are a number of
possible choices for a complete signed graph, but taking the lead from the
type A braid arrangement, there is a clear choice.

######  Definition 19.11.

Let $\pm K_{n}^{B}$ be the signed graph whose vertices are $[n]$ and whose
edges are:

  * •

an edge $\\{i,j\\}$ signed $+$ for all pairs $i,j\in[n]$,

  * •

an edge $\\{i,j\\}$ signed $-$ for all pairs $i,j\in[n]$

  * •

a half edge at $i$ signed $+$ for each $i\in[n]$.

Call $\pm K_{n}^{B}$ a _complete signed graph_.

One could opt to replace the half edges with loops, but in that case an
orientation of a complete signed graph would, strictly speaking, correspond to
$\Phi_{C_{n}}$ roots rather than $\Phi_{B_{n}}$, though it would not alter the
normal fan of the zonotope. Figure 58 shows $\pm K_{2}^{B}$, $\mathcal{Z}(\pm
K_{2}^{B})$ and $\mathcal{N}(\mathcal{Z}(\pm(K_{2}^{B}))$.

$1$$2$$+$$-$$+$$+$ (a) $\pm K_{2}^{B}$

${(0,0)}$${(0,1)}$${(1,2)}$${(2,2)}$${(3,1)}$${(3,0)}$${(2,-1)}$${(1,-1)}$ (b)
$\mathcal{Z}[\pm K_{2}^{B}]$

$x_{1}=0$$x_{2}=0$$x_{1}=x_{2}$$x_{1}=-x_{2}$$(12)$$(21)$$(2\bar{1})$$(\bar{1}2)$$(\bar{1}\bar{2})$$(\bar{2}\bar{1})$$(\bar{2}1)$$(1\bar{2})$
(c) $\mathcal{N}(\mathcal{Z}[\pm K_{2}^{B}])$

Figure 58: The complete signed graph $\pm K_{2}^{B}$, its acyclotope and the
normal fan of the acyclotope

One sees that the example $K_{P}^{\mathrm{wt}}$ is triangulated by the cones
spanned by $(12)\Phi_{2}^{+}$ and $(21)\Phi_{2}^{+}$, which are, in fact, the
linear extensions of $P$.

######  Proposition 19.12.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset with $\Sigma$ the signed
graph underlying its Hasse diagram. Then $\mathcal{N}(\mathcal{Z}[\Sigma])$ is
refined by $\mathcal{N}(\mathcal{Z}[\pm K_{n}^{B}])$ and, if $\tau$ is the
orientation of $\Sigma$ corresponding to $P$, then $\mathcal{N}_{\tau}$ is
triangulated by the cones of $\mathcal{N}(\mathcal{Z}[\pm K_{n}^{B}])$
corresponding to the linear extensions of $P$.

###### Proof.

First, it is straightforward to see that $\mathcal{N}(\mathcal{Z}[\Sigma])$ is
refined by $\mathcal{N}(\mathcal{Z}[\pm K_{n}^{B}])$. The hyperplanes defining
$\mathcal{N}(\mathcal{Z}[\Sigma])$ are a subset of the hyperplanes defining
$\mathcal{N}(\mathcal{Z}[\pm K_{n}^{B}])$.

Now, suppose $\tau$ is the orientation of $\Sigma$ corresponding to $P$. By
construction, the maximal cones of $\mathcal{N}(\mathcal{Z}[\pm K_{n}^{B}])$
are the $w\Phi^{+}$-partition cones for the elements of the Weyl group, i.e.
the signed permutations. Thus, if the maximal cone corresponding to $w$ lies
in $\mathcal{N}_{\tau}$, by definition, $w$ will be a linear extension of $P$.
∎

As was noted above, one can index the maximal cones of the triangulation from
Proposition 19.2 by certain signed posets, namely those whose weight cones are
the maximal cones of the triangulation. By construction, these signed posets
have simplicial and unimodular weight cones. They should be the signed
analogue of type A’s $\mathcal{B}(G)$-forests.

######  Question 19.13.

What is the appropriated analogue of Carr and Devadoss’s graph associahedron
for signed graphs? Does its normal fan give the triangulation of
$K_{P}^{\mathrm{wt}}$ given in Proposition 19.2?

At the moment, it appears an appropriate signed graph associahedron can be
obtained by shaving the $n$-cube at faces corresponding to “signed tubes” of
the signed graph, with a proof proceeding as that of Carr and Devadoss.
However, all the details have yet to be written down.

###  20 The type C weight cone

Thus far, consideration of the weight cone, ideals and $P$-partitions has been
restricted to those signed posets $P\subset\Phi_{B_{n}}$. However, though one
can read the ideals from either $\widehat{G}_{B}(P)$ or
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$, understanding one does not
immediately lead to understanding the other, as the next two examples
illustrate.

$1$$2$$-2$$-1$ Figure 59: $P=\\{e_{1}-e_{2},e_{1}+e_{2},e_{1}\\}$

Consider the signed posets $P=\\{e_{1}-e_{2},e_{1}+e_{2},e_{1}\\}$ and
${P^{\scriptscriptstyle\vee}}=\\{e_{1}-e_{2},e_{1}+e_{2},2e_{2}\\}$. Figure 59
illustrates $\widehat{G}(P)$. The connected ideals of $P$ and
${P^{\scriptscriptstyle\vee}}$ are shown in Table 6. Recall that ideals live
in the coweight lattice, so the maximal ideals of $P$ and
${P^{\scriptscriptstyle\vee}}$ do not coincide.

ideal in $\widehat{G}(P)$ | ideal in $P$ | ideal in ${P^{\scriptscriptstyle\vee}}$
---|---|---
$\\{1\\}$ | $(1,0)$ | $(1,0)$
$\\{1,2\\}$ | $(1,1)$ | $(\frac{1}{2},\frac{1}{2})$
$\\{1,-2\\}$ | $(1,-1)$ | $(\frac{1}{2},-\frac{1}{2})$
Table 6: Type $B$ and $C$ ideals for $\widehat{G}(P)$ from Figure 59

Inspecting the ideals reveals that all three connected ideals are required to
generate the semigroup $K_{P}^{\mathrm{wt}}\cap L^{\mathrm{cowt}}_{B}$.
However, only $(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},-\frac{1}{2})$ are
required to generate $K_{P}^{\mathrm{wt}}\cap L^{\mathrm{cowt}}_{C}$.
Consequently, one sees $K_{P}^{\mathrm{wt}}$ is unimodular with respect to
$L^{\mathrm{cowt}}_{C}$ but _not_ with respect to $L^{\mathrm{cowt}}_{B}$. In
both cases, one has the triangulation into cones defined by collections of
pairwise trivially intersecting ideals, even though $K_{P}^{\mathrm{wt}}$ is
already simplicial and unimodular when viewed in type C.

Alone this is not enough to conclude that the arguments from Chapter Root and
weight semigroup rings for signed posets do not go through almost immediately
in type C. Consider the signed poset in Figure 60.

$1$$3$$2$$-2$$-1$$-3$ Figure 60: $\widehat{G}({P^{\scriptscriptstyle\vee}})$
for
${P^{\scriptscriptstyle\vee}}=\\{e_{1}-e_{2},e_{1}+e_{2},e_{1}+e_{3},e_{3}-e_{2},2e_{1}\\}$

The connected ideals are
$\\{1\\},\\{3\\},\\{1,-2\\},\\{1,2,3\\},\\{1,-2,-3\\}$. The toric ideal is the
kernel of $\varphi\colon S^{\mathrm{wt}}_{{P^{\scriptscriptstyle\vee}}}\to
k[x_{1}^{\pm 1},x_{2}^{\pm 1},x_{3}^{\pm 1}]$ defined by

$\displaystyle\varphi(U_{1})$ $\displaystyle=x_{1}^{2}$
$\displaystyle\varphi(U_{3})$ $\displaystyle=x_{3}^{2}$
$\displaystyle\varphi(U_{1\bar{2}})$ $\displaystyle=x_{1}^{2}x_{2}^{-2}$
$\displaystyle\varphi(U_{123})$ $\displaystyle=x_{1}x_{2}x_{3}$
$\displaystyle\varphi(U_{1\bar{2}\bar{3}})$
$\displaystyle=x_{1}x_{2}^{-1}x_{3}^{-1}.$

Using Macaulay2, one sees that

$\ker\varphi=(U_{1\bar{2}}U_{123}^{2}-U_{1}^{2}U_{3},U_{123}U_{1\bar{2}\bar{3}}-U_{1}),$

meaning that while the generators of the toric ideal are still indexed by
pairs of nontrivially intersecting connected ideals, one needs to tweak the
definition of $\mathrm{syz}(U_{J},U_{K})$ somewhat. Proposition 18.2 used that
the leading term of $\mathrm{syz}(U_{J},U_{K})$ in type B is quadratic, which
is clearly not the case here, though it is likely that a similar result holds
and many of the results of Chapter Root and weight semigroup rings for signed
posets could be pushed through into type C.

###  21 On characterizing the $R_{P}^{\mathrm{rt}}$ complete intersections

In this section, attention returns to the root cone and its semigroup (refer
to Chapter Root and weight semigroup rings for signed posets for previous
discussion). [8, Theorem 7.7 ] had shown that $\Psi_{P}$ factored for posets
they called “gluings of diamonds along chains”, of which strongly planar
posets were a subset. A poset was said to be a gluing of diamonds along chains
if it could be decomposed into a collection of diamonds by means of
disconnecting chains. A diamond was a cycle with a unique maximum and minimum.
A disconnecting chain is a chain in the poset that partitions the vertices
into three groups: the chain itself, and two other sets such that the paths
between the two sets much pass through a vertex of the chain. For example,
consider the poset in Figure 61.

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$ Figure 61: A poset which has
$R_{P}^{\mathrm{rt}}$ a complete intersection but is not gluing of diamonds
along chains

Certainly, it can be broken apart via disconnecting chains in the posets shown
in Figure 62.

|
---|---
|

$2$$3$$4$$5$$10$ (a)

$3$$4$$5$$6$$7$ (b)

$1$$2$$3$$4$$9$ (c)

$1$$2$$7$$8$ (d)

Figure 62: The poset of Figure 61 broken into unicycle components

The poset in Figure 62(d) is not a diamond, so is not covered by Boussicault
and Féray’s result. However, the fact the poset of Figure 61 is a complete
intersection is explained by [9, Theorem 8.6 ], who give an algebraic
explanation for the factorization via opening/closing notches—locating
Boussicault and Féray’s disconnecting chains can serve as a guide for which
notches to open.

An immediate consequence of [9, Theorem 8.6] is that a poset which can be
broken apart into unicyclic components (those having one or no cycles) by
opening notches must have $R_{P}^{\mathrm{rt}}$ a complete intersection. It
turns out that this suffices to characterize the posets for which
$R_{P}^{\mathrm{rt}}$ is a complete intersection, which was conjectured by the
author of this thesis and V. Reiner and proved by [43].

######  Theorem 21.1 ([43, Theorem 3]).

Suppose $P$ is a poset. Its root cone semigroup ring $R_{P}^{\mathrm{rt}}$ is
a complete intersection if and only if the Hasse diagram of $P$ can be
obtained from unicyclic posets by repeated gluings along chains.

One would hope for a similar result in the signed poset case. Like in the
poset case, it is clear that strong planarity is not a necessary condition for
$R_{P}^{\mathrm{rt}}$ to be a complete intersection. After all, consider the
signed poset in Figure 63. Its toric ideal is principal and there is no notch
that can be opened, so $R_{P}^{\mathrm{rt}}$ must be a complete intersection,
though the poset is clearly not strongly planar.

$1$$2$$-1$$-2$ Figure 63: A signed poset which is not strongly planar, but has
$R_{P}^{\mathrm{rt}}$ a complete intersection

Unicyclic posets, of course, have principal root cone toric ideals.
Furthermore, cycles in the Hasse diagram of the poset correspond to circuits
in its matroid. This turns out to be the key fact to identifying the correct
signed analogue of “unicyclic”.

######  Conjecture 21.2.

Suppose $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) is a signed poset.
$R_{P}^{\mathrm{rt}}$ (resp. $R_{{P^{\scriptscriptstyle\vee}}}^{\mathrm{rt}}$)
is a complete intersection if and only if $\widehat{G}(P)$ can be broken into
biconnected components $P_{1},\ldots,P_{k}$, each of which has a Hasse diagram
with at most one circuit, by opening a series of signed notches.

## References

  * [1] Martin Aigner  “Combinatorial theory” 234, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]  Springer-Verlag, Berlin-New York, 1979, pp. viii+483
  * [2] Kazutoshi Ando, Satoru Fujishige and Toshio Nemoto  “The minimum-weight ideal problem for signed posets”  In _J. Oper. Res. Soc. Japan_ 39.4, 1996, pp. 558–565
  * [3] Alexander Barvinok and James E. Pommersheim  “An Algorithmic Theory of Lattice Points in Polyhedra”  In _New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-97)_ 38, Math. Sci. Res. Inst. Publ.  Cambridge: Cambridge University Press, 1999, pp. 91–147
  * [4] Alexander I. Barvinok  “Computing the Volume, Counting Integral Points, and Exponential Sums”  In _Discrete & Computational Geometry_, 1993  URL: http://www.springerlink.com/index/UH13667243Q06665.pdf
  * [5] Anders Björner and Michelle L. Wachs  “$q$-hook length formulas for forests”  In _Journal of Combinatorial Theory - Series A_ 52.2, 1989, pp. 165–187
  * [6] Tristram Bogart, Anders N. Jensen and Rekha R. Thomas  “The circuit ideal of a vector configuration”  In _J. Algebra_ 309.2, 2007, pp. 518–542  DOI: 10.1016/j.jalgebra.2006.07.025
  * [7] Adrien Boussicault  “Action du groupe symétrique sur certaines fractions rationnelles suivi Puissances paires du Vandermonde”, 2009
  * [8] Adrien Boussicault and Valentin Féray  “Application of graph combinatorics to rational identities of type A”  In _Electronic Journal of Combinatorics_ 16, 2009, pp. 1–39
  * [9] Adrien Boussicault, Valentin Féray, Alain Lascoux and Victor Reiner  “Linear extension sums as valuations of cones”  In _Journal of Algebraic Combinatorics_ 35.4, 2012, pp. 33  arXiv: http://arxiv.org/abs/1008.3278
  * [10] Graham Brightwell and Peter Winkler  “Counting linear extensions”  In _Order_ , 1991, pp. 225–242  URL: http://www.springerlink.com/index/P395864591L07770.pdf
  * [11] Michel Brion and Michèle Vergne  “Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue”  In _Ann. Sci. École Norm. Sup. (4)_ 32.5, 1999, pp. 715–741  DOI: 10.1016/S0012-9593(01)80005-7
  * [12] Michael P. Carr and Satyan L. Devadoss  “Coxeter complexes and graph-associahedra”  In _Topology Appl._ 153.12, 2006, pp. 2155–2168  DOI: 10.1016/j.topol.2005.08.010
  * [13] Eduardo Cattani, Raymond Curran and Alicia Dickenstein  “Complete intersections in toric ideals”  In _Proceedings of the American Mathematical Society_ 135.2, 2007, pp. 329–335
  * [14] Frédéric Chapoton, Florent Hivert, Jean-Christophe Novelli and Jean-Yves Thibon  “An operational calculus for the mould operad”  In _Int. Math. Res. Not. IMRN_ , 2008, pp. Art. ID rnn01822  DOI: 10.1093/imrn/rnn018
  * [15] Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein  “Introduction to Algorithms, Third Edition”  The MIT Press, 2009
  * [16] Sebastian A. Csar  “Root and weight semigroup rings for signed posets”, 2014
  * [17] Arthur M. Duval  “Simplicial posets:f-vectors and free resolutions”, 1991
  * [18] David Eisenbud  “Commutative Algebra with a View Toward Algebraic Geometry” 150, Graduate Texts in Mathematics  New York: Springer-Verlag, 1995, pp. xvi+785  DOI: 10.1007/978-1-4612-5350-1
  * [19] David Eisenbud and Bernd Sturmfels  “Binomial Ideals”  In _Duke Mathematical Journal_ 84.1 Durham, NC: Duke University Press, 1935-, 1994, pp. 1–45  URL: http://arxiv.org/abs/alg-geom/9401001
  * [20] Viviana Ene and Jürgen Herzog  “Gröbner bases in commutative algebra” 130, Graduate Studies in Mathematics  Providence, RI: American Mathematical Society, 2012, pp. xii+164
  * [21] Valentin Féray and Victor Reiner  “$P$-partitions revisited”  In _J. Commut. Algebra_ 4.1, 2012, pp. 101–152  DOI: 10.1216/JCA-2012-4-1-101
  * [22] Klaus G. Fischer, Walter Morris and Jay Shapiro  “Affine semigroup rings that are complete intersections”  In _Proceedings of the American Mathematical Society_ 125.11, 1997, pp. 3137–3145  URL: http://www.ams.org/proc/1997-125-11/S0002-9939-97-03920-8/
  * [23] Klaus G. Fischer and Jay Shapiro  “Mixed matrices and binomial ideals”  In _Journal of Pure and Applied Algebra_ 113.1, 1996, pp. 39–54  URL: http://www.sciencedirect.com/science/article/pii/0022404995001441
  * [24] Steven D. Fischer  “Signed poset homology and q-analog Möbius functions.”, 1993, pp. 100
  * [25] 4ti2 team  “4ti2—A software package for algebraic, geometric and combinatorial problems on linear spaces”, Available at www.4ti2.de
  * [26] William Fulton  “Introduction to toric varieties” The William H. Roever Lectures in Geometry 131, Annals of Mathematics Studies  Princeton, NJ: Princeton University Press, 1993, pp. xii+157
  * [27] Isidoro Gitler, Enrique Reyes and Rafael H. Villarreal  “Ring graphs and complete intersection toric ideals”  In _Discrete Mathematics_ 310, 2010, pp. 430–441
  * [28] Isidoro Gitler, Enrique Reyes and Rafael H. Villarreal  “Ring graphs and toric ideals”  In _Electronic Notes in Discrete Mathematics_ 28, 2007, pp. 393–400
  * [29] Paul Gordan  “Les invariants des Formes Binaires”  In _Journal de Mathématiques Pures et Appliqués_ 5.6, 1900, pp. 141–156
  * [30] Daniel R. Grayson and Michael E. Stillman  “Macaulay2, a software system for research in algebraic geometry”  http://www.math.uiuc.edu/Macaulay2/  URL: http://www.math.uiuc.edu/Macaulay2/
  * [31] Curtis Greene  “A rational-function identity related to the Murnaghan-Nakayama formula for the characters of $S_{n}$”  In _J. Algebraic Combin._ 1.3, 1992, pp. 235–255  DOI: 10.1023/A:1022435901373
  * [32] Gert-Martin Greuel and Gerhard Pfister  “A Singular introduction to commutative algebra”  Berlin: Springer, 2008, pp. xx+689
  * [33] Frank Harary  “On the notion of balance of a signed graph.”  In _The Michigan Mathematical Journal_ 2.2, 1953, pp. 143–146  DOI: 10.1307/mmj/1028989917
  * [34] Jürgen Herzog  “Generators and relations of abelian semigroups and semigroup rings.”  In _Manuscripta Math._ 3, 1970, pp. 175–193
  * [35] Jürgen Herzog  “Hilbert Functions”, pp. 331–354
  * [36] Jürgen Herzog and Takayuki Hibi  “Monomial ideals” 260, Graduate Texts in Mathematics  London: Springer-Verlag London Ltd., 2011, pp. xvi+305  DOI: 10.1007/978-0-85729-106-6
  * [37] Anargyros Katsabekis, Marcel Morales and Apostolos Thoma  “Binomial generation of the radical of a lattice ideal”  In _Journal of Algebra_ 324.6 Elsevier Inc., 2010, pp. 1334–1346  DOI: 10.1016/j.jalgebra.2010.07.005
  * [38] F.S. MacAulay  “Some Properties of Enumeration in the Theory of Modular Systems”  In _Proceedings of the London Mathematical Society_ s2-26.1, 1927, pp. 531–555  DOI: 10.1112/plms/s2-26.1.531
  * [39] José Martínez-Bernal and Rafael H. Villarreal  “Toric ideals generated by circuits”  In _Algebra Colloq._ 19.4, 2012, pp. 665–672  DOI: 10.1142/S1005386712000533
  * [40] Nickolas J. Michelacakis and Apostolos Thoma  “On the geometry of complete intersection toric varieties”  In _Archiv der Mathematik_ 87.2, 2006, pp. 113–123  DOI: 10.1007/s00013-005-1652-z
  * [41] Ezra Miller and Bernd Sturmfels  “Combinatorial commutative algebra” 227, Graduate Texts in Mathematics  New York: Springer-Verlag, 2005, pp. xiv+417
  * [42] Marcel Morales and Apostolos Thoma  “Complete intersection lattice ideals”  In _Journal of Algebra_ , 2005  URL: http://www.sciencedirect.com/science/article/pii/S0021869304005654
  * [43] Walter D. Morris  “Complete Intersection Affine Semigroup Rings Arising from Posets”, http://math.gmu.edu/$\sim$wmorris/compintgraph.pdf, 2013
  * [44] Hidefumi Ohsugi and Takayuki Hibi  “Indispensable binomials of finite graphs”  In _J. Algebra Appl._ 4.4, 2005, pp. 421–434  DOI: 10.1142/S0219498805001265
  * [45] Hidefumi Ohsugi and Takayuki Hibi  “Toric ideals and their circuits”  In _J. Commut. Algebra_ 5.2, 2013, pp. 309–322
  * [46] Hidefumi Ohsugi and Takayuki Hibi  “Toric ideals generated by quadratic binomials”  In _J. Algebra_ 218.2, 1999, pp. 509–527  DOI: 10.1006/jabr.1999.7918
  * [47] Victor Reiner  “Quotients of Coxeter complexes and $P$-partitions”, 1990
  * [48] Victor Reiner  “Signed posets”  In _Journal of Combinatorial Theory - Series A_ 62.2, 1993, pp. 324–360  DOI: 10.1016/0097-3165(93)90052-A
  * [49] Enrique Reyes, Christos Tatakis and Apostolos Thoma  “Minimal generators of toric ideals of graphs”  In _Adv. in Appl. Math._ 48.1, 2012, pp. 64–78  DOI: 10.1016/j.aam.2011.06.003
  * [50] J.. Rosales and Pedro A. García-Sánchez  “On complete intersection affine semigroups”  In _Communications in Algebra_ 23.14, 1995, pp. 5395–5412
  * [51] Richard P. Stanley  “Combinatorics and commutative algebra” 41, Progress in Mathematics  Boston, MA: Birkhäuser Boston Inc., 1996, pp. x+164
  * [52] Richard P. Stanley  “Enumerative combinatorics. Vol. 1” With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original 49, Cambridge Studies in Advanced Mathematics  Cambridge: Cambridge University Press, 1997, pp. xii+325
  * [53] Richard P. Stanley  “Hilbert functions of graded algebras”  In _Advances in Math._ 28.1, 1978, pp. 57–83
  * [54] Richard P. Stanley  “Ordered Structures and Partitions”, Memoirs of the American Mathematical Society 119  Providence, RI: American Mathematical Society, 1972
  * [55] W.. Stein  “Sage Mathematics Software (Version 6.1.1)” http://www.sagemath.org  The Sage Development Team, 2014
  * [56] John R. Stembridge  “Coxeter cones and their $h$-vectors”  In _Advances in Mathematics_ 217.5, 2008, pp. 1935–1961  DOI: 10.1016/j.aim.2007.09.002
  * [57] Bernd Sturmfels  “Asymptotic Analysis of Toric Ideals”  In _Mem. Fac. Sci. Kyushu Univ. Ser. A_ 46.2, 1992, pp. 217–228  URL: https://www.jstage.jst.go.jp/article/kyushumfs/46/2/46%5C_2%5C_217/%5C_pdf
  * [58] Bernd Sturmfels  “Gröbner Bases and Convex Polytopes”  Providence, RI: American Mathematical Society, 1996
  * [59] Wolmer V. Vasconcelos  “Computational methods in commutative algebra and algebraic geometry” 2, Algorithms and Computation in Mathematics  Berlin: Springer-Verlag, 1998, pp. xii+394  DOI: 10.1007/978-3-642-58951-5
  * [60] Wolmer V. Vasconcelos  “Computational methods in commutative algebra and algebraic geometry” 2, Algorithms and Computation in Mathematics  Berlin: Springer-Verlag, 1998, pp. xii+394  DOI: 10.1007/978-3-642-58951-5
  * [61] Rafael H. Villarreal  “Rees algebras of edge ideals”  In _Communications in Algebra_ 23.9, 1995, pp. 3513–3524
  * [62] Hassler Whitney  “On the abstract properties of linear dependence”  In _American Journal of Mathematics_ 57.3 JSTOR, 1935, pp. 509–533  URL: http://www.jstor.org/stable/2371182
  * [63] Thomas Zaslavsky  “Orientation of Signed Graphs”  In _European Journal of Combinatorics_ 12.4, 1991, pp. 361–375  URL: http://www.math.binghamton.edu/zaslav/Tpapers/osg.eujc.pdf
  * [64] Thomas Zaslavsky  “Signed graphs”  In _Discrete Applied Mathematics_ 4.1, 1982, pp. 47–74  DOI: 10.1016/0166-218X(82)90033-6
  * [65] Günter M. Ziegler  “Lectures on polytopes” 152, Graduate Texts in Mathematics  Springer-Verlag, New York, 1995, pp. x+370  DOI: 10.1007/978-1-4613-8431-1