# Root and weight semigroup rings for signed posets

Sebastian Alexander Csar

###### Abstract

We consider a pair of semigroups associated to a signed poset, called the root
semigroup and the weight semigroup, and their semigroup rings,
$R_{P}^{\mathrm{rt}}$ and $R_{P}^{\mathrm{wt}}$, respectively.

Theorem 14.5 gives generators for the toric ideal of affine semigroup rings
associated to signed posets and, more generally, oriented signed graphs. These
are the subrings of Laurent polynomials generated by monomials of the form
$t_{i}^{\pm 1},t_{i}^{\pm 2},t_{i}^{\pm 1}t_{j}^{\pm 1}$. This result appears
to be new and generalizes work of [9], [27] and [61]. Theorem 15.12 shows that
strongly planar signed posets $P$ have rings $R_{P}^{\mathrm{rt}}$,
$R_{{P^{\scriptscriptstyle\vee}}}^{\mathrm{rt}}$ which are complete
intersections, with Corollary 15.20 showing how to compute $\Psi_{P}$ in this
case. Theorem 17.3 gives a Gröbner basis for the toric ideal of
$R_{P}^{\mathrm{wt}}$ in type B, generalizing [21, Proposition 6.4 ]. Theorems
18.10 and 18.21 giving two characterizations (via forbidden subposets versus
via inductive constructions) of the situation where this Gröbner basis gives a
complete intersection presentation for its initial ideal, generalizing [21,
Theorems 10.5, 10.6 ].

University of Minnesota Mathematics Victor Reiner, advisor 2014 August

It is perhaps obligatory to start the acknowledgements by thanking your
advisor. To that end, I thank Vic Reiner for his guidance, patience and
willingness to learn to parse the varied meanings of “all right”. I am
similarly indebted to Dennis Stanton and Gregg Musiker.

I thank Nathan Williams, Kevin Dilks and Thomas McConville for fielding my
random questions, Maggie Ewing and David Morawski for the standing work
appointments, Alanna Hoyer-Leitzl for her thoughtful support, Heidi Goodson,
Jeff Cruse, Patrick Campbell and Baiying Liu for their conversations in the
office and Julie Leifeld for deciding we should be friends.

I am grateful to the FLBs first for letting me join the group and second for
being such wonderful friends. I thank Jenna Ingalls for being my best friend
with all that entails, and my parents and brother for their endless love and
support. For my grandad.

##  Chapter Introduction

A partially ordered set, or _poset_ , is a set $P$ together with a binary
relation $<$ such that

  * •

$<$ is antisymmetric: if $x<y$ then $y\not<x$ for all $x,y\in P$, and

  * •

$<$ is transitive: if $x<y$ and $y<z$ then $x<z$ for all $x,y,z\in P$.

Consider the poset in Figure 1. The underlying set is $P=\\{1,2,3,4,5\\}$ and
$x<y$ if there is a path from $x$ to $y$ travelling upwards along each edge in
the path.

$1$$2$$3$$4$$5$ Figure 1: A poset

For example, $1<3$, but $1\not<4$.

An element of a poset $y\in P$ is said to _cover_ $x\in P$ if $x<y$ and there
is no $z\in P$ such that $x<z<y$. One writes $x\lessdot y$ to emphasize that
the relation between $x$ and $y$ is a _covering relation_.

A _linear extension_ of a poset is an extension $\prec$ of $<$ to a total
order, i.e. a linear order of the elements of $P$ by $\prec$ so that if $x<y$
then $x\prec y$. The linear extensions of our example are

$\begin{split}1\prec 2\prec 3\prec 4\prec 5\\\ 1\prec 2\prec 4\prec 3\prec
5\\\ 2\prec 1\prec 3\prec 4\prec 5\\\ 2\prec 1\prec 4\prec 3\prec 5\\\ 2\prec
4\prec 1\prec 3\prec 5\end{split}$

The set of linear extensions of $P$ is denoted $\mathcal{L}(P)$. When $P$ is a
poset on $[n]\coloneqq\\{1,2,\ldots,n\\}$, it is natural to regard the linear
extensions as permutations of $[n]$. When $P$ is the poset in Figure 1 one
then has $\mathcal{L}(P)=\\{12345,12435,21345,21435,24135\\}$.

While finding a linear extension of a finite poset is straightforward—it is
what is known as _topological sorting_ in computer science and can be done in
linear time (see, for instance [15, §22.4])—counting the number of linear
extensions proves rather more difficult. Brightwell and Winkler showed in [10]
that the problem is $\\#P$-complete. As a consequence, computing rational
functions which are sums over linear extensions proves difficult. This leads
to two basic questions:

  * •

When and how can the linear extensions of a poset be counted without listing
them?

  * •

When and how can such a rational function be computed without listing all the
linear extensions?

###  1 A root system perspective on posets

In [9], Boussicault, Féray, Lascoux and Reiner used a view of posets as sets
of type A roots to explain how a pair of rational functions which are sums
over linear extensions can be evaluated, at least in certain cases. They
transform a poset $P$ into the collection of type A roots
$\\{e_{i}-e_{j}\colon i<_{P}j\\}$. Under this scheme, $P$ from Figure 1
corresponds to
$\\{e_{1}-e_{3},e_{1}-e_{5},e_{2}-e_{3},e_{2}-e_{4},e_{2}-e_{5},e_{3}-e_{5},e_{4}-e_{5}\\}$.
The two rational functions they considered are, for a poset $P$ on
$[n]\coloneqq\\{1,2,\ldots,n\\}$,

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

and

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

where the linear extensions are viewed as permutations acting on the indices
of $x_{i}$.

The function $\Psi_{P}$ had previously been considered by [31], where he gave
an evaluation for _strongly planar_ posets, i.e. those posets whose Hasse
diagrams remain planar after the addition of minimum and maximum elements
$\hat{0}$ and $\hat{1}$.

######  Theorem 1.1 (Greene, [31]).

Suppose $P$ is a strongly planar poset. Then

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

where $\rho$ runs over all the bounded regions enclosed by the Hasse diagram
of $P$ and $i\lessdot j$ runs over all covering relations of the poset.

In our example, computing $\Psi_{P}$ as a sum over the linear extensions gives

$\displaystyle\Psi_{P}(\bm{x})=$
$\displaystyle\sum_{w\in\mathcal{L}(P)}w\left(\dfrac{1}{(x_{1}-x_{2})(x_{2}-x_{3})(x_{3}-x_{4})(x_{4}-x_{5})}\right)$
$\displaystyle=$
$\displaystyle\dfrac{1}{(x_{1}-x_{2})(x_{2}-x_{3})(x_{3}-x_{4})(x_{4}-x_{5})}+\dfrac{1}{(x_{1}-x_{2})(x_{2}-x_{4})(x_{4}-x_{3})(x_{3}-x_{5})}$
$\displaystyle+\dfrac{1}{(x_{2}-x_{1})(x_{1}-x_{3})(x_{3}-x_{4})(x_{4}-x_{5})}+\dfrac{1}{(x_{2}-x_{1})(x_{1}-x_{4})(x_{4}-x_{3})(x_{3}-x_{5})}$
$\displaystyle+\dfrac{1}{(x_{2}-x_{4})(x_{4}-x_{1})(x_{1}-x_{3})(x_{3}-x_{5})}$
$\displaystyle=$
$\displaystyle\dfrac{x_{2}-x_{5}}{(x_{1}-x_{3})(x_{2}-x_{3})(x_{2}-x_{4})(x_{3}-x_{5})(x_{4}-x_{5})}.$

On the other hand,

$\Psi_{P}(\bm{x})=\dfrac{\prod_{\rho}(x_{\min(\rho)}-x_{\max(\rho)})}{\prod_{i\lessdot
j}(x_{i}-x_{j})}=\dfrac{x_{2}-x_{5}}{(x_{1}-x_{3})(x_{2}-x_{3})(x_{2}-x-4)(x_{3}-x_{5})(x_{4}-x_{5})}.$

The function $\Phi_{P}$ was considered in the case of forests by [14], who
proved the following.

######  Theorem 1.2 (Chapoton, Hivert, Novelli, Thibon, [14]).

Suppose $P$ is a forest (i.e. every element is covered by at most one other
element). Then

$\Phi_{P}(\bm{x})=\prod_{i=1}^{n}\dfrac{1}{\sum_{j\leq_{P}i}x_{j}}.$

Boussicault, Féray, Lascoux and Reiner then defined a pair of dual cones, the
root cone $K_{P}^{\mathrm{rt}}={\mathbb{R}}_{+}P$ and the weight cone
$K_{P}^{\mathrm{wt}}={\mathbb{R}}_{+}\\{\chi_{J}\colon J\in J(P)\\}$, where
$J(P)$ is the set of _order ideals_ $J$, subsets $J\subset P$ with the
condition that if $y\in J$ and $x<_{P}y$, then $x\in J$, and $\chi_{J}$ is the
characteristic vector of $J$. They made two important realizations:

  * •

$\Psi_{P}$ and $\Phi_{P}$ are the Laplace transform valuations of
$K_{P}^{\mathrm{rt}}$ and $K_{P}^{\mathrm{wt}}$, and

  * •

$\Psi_{P}$ and $\Phi_{P}$ can be recovered from the Hilbert series of the
semigroup rings

$R_{P}^{\mathrm{rt}}=k[K_{P}^{\mathrm{rt}}\cap{\mathbb{Z}}^{n}]\quad\text{and}\quad
R_{P}^{\mathrm{wt}}=k[K_{P}^{\mathrm{wt}}\cap{\mathbb{Z}}^{n}],$

respectively.

These two observations enabled Boussicault, Féray, Lascoux and Reiner in [9]
and Féray and Reiner in [21] to use Proposition 6.14 to obtain the following
two results.

######  Theorem 1.3 (Boussicault, Féray, Lascoux, Reiner).

Suppose $P$ is a strongly planar poset. Then $R_{P}^{\mathrm{rt}}$ is a
complete intersection,

$\mathrm{Hilb}(R_{P}^{\mathrm{rt}},\bm{x})=\dfrac{\prod_{\rho}(1-x_{\min(\rho)}x_{\max(\rho)}^{-1})}{\prod_{i\lessdot_{P}j}(1-x_{i}x_{j}^{-1})}$

and

$\Psi_{P}(\bm{x})=\dfrac{\prod_{\rho}(x_{\min(\rho)}-x_{\max(\rho)})}{\prod_{i\lessdot_{P}j}(x_{i}-x_{j})}.$

In [21], Féray and Reiner described a class of posets generalizing forests
called _forests with duplication_ , which are precisely those posets such that
$R_{P}^{\mathrm{wt}}$ is a complete intersection, and compute $\Phi_{P}$ in
this case.

######  Theorem 1.4 (Féray and Reiner).

A poset $P$ is a forest with duplication if and only if $R_{P}^{\mathrm{wt}}$
is a complete intersection, in which case

$\mathrm{Hilb}(R_{P}^{\mathrm{wt}},\bm{x})=\dfrac{\prod_{\\{J,K\\}\in\Pi(P)}(1-\bm{x}^{J}\bm{x}^{k})}{\prod_{J\in
J_{\mathrm{conn}}(P)}(1-\bm{x}^{J})}$

and

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

where $J_{\mathrm{conn}}(P)$ is the set of connected order ideals of $P$,
$\Pi(P)$ is the set of pairs of connected order ideals which intersect
nontrivially ($J\cap K\neq\emptyset$ and neither $J\subset K$ nor $K\subset
J$) and $\langle\bm{x},\chi_{J}\rangle=\sum_{i\in J}x_{i}$.

Return again to the poset in Figure 1. One has that the root cone semigroup
ring is presented as

$k[U_{13},U_{23},U_{24},U_{35},U_{45}]/(U_{23}U_{35}-U_{24}U_{45})$

with the map $U_{ij}\mapsto x_{i}x_{j}^{-1}\in k[x_{1}^{\pm
1},\ldots,x_{n}^{\pm 1}]$, and the weight cone semigroup ring is presented as

$k[U_{1},U_{2},U_{24},U_{123},U_{1234},U_{12345}]/(U_{123}U_{24}-U_{2}U_{1234})$

with the map $U_{J}\mapsto\prod_{i\in J}x_{i}\in k[x_{1},\ldots,x_{n}]$. Since
both of the presentation ideals, $(U_{23}U_{35}-U_{24}U_{45})$ and
$(U_{123}U_{24}-U_{2}U_{1234})$, are principal, both $R_{P}^{\mathrm{rt}}$ and
$R_{P}^{\mathrm{wt}}$ are complete intersections. Both rings are naturally
${\mathbb{Z}}^{5}$-graded, and this ${\mathbb{Z}}^{5}$-grading coincides with
an ${{\mathbb{N}}}^{5}$-grading of $R_{P}^{\mathrm{wt}}$. One then has

$\mathrm{Hilb}(R_{P}^{\mathrm{rt}},\bm{x})=\dfrac{(1-x_{2}x_{5}^{-1})}{(1-x_{1}x_{3}^{-1})(1-x_{2}x_{3}^{-1})(1-x_{2}x_{4}^{-1})(1-x_{3}x_{5}^{-1})(1-x_{4}x_{5}^{-1})},$

so

$\Psi_{P}(\bm{x})=\dfrac{(x_{2}-x_{5})}{(x_{1}-x_{3})(x_{2}-x_{3})(x_{2}-x_{4})(x_{3}-x_{5})(x_{4}-x_{5})},.$

as seen above. On the other hand,

$\mathrm{Hilb}(R_{P}^{\mathrm{wt}},\bm{x})=\dfrac{(1-x_{1}x_{2}^{2}x_{3}x_{4})}{(1-x_{1})(1-x_{2})(1-x_{2}x_{4})(1-x_{1}x_{2}x_{3})(1-x_{1}x_{2}x_{3}x_{4})(1-x_{1}x_{2}x_{3}x_{4}x_{5})},$

so

$\Phi_{P}(\bm{x})=\dfrac{x_{1}+2x_{2}+x_{3}+x_{4}}{x_{1}x_{2}(x_{2}+x_{4})(x_{1}+x_{2}+x_{3})(x_{1}+x_{2}+x_{3}+x_{4})(x_{1}+x_{2}+x_{3}+x_{4}+x_{5})}.$

Computing $\Phi_{P}(\bm{x})$ as a sum over linear extensions, one has

$\Phi_{P}(\bm{x})=\\\
\sum_{w\in\mathcal{L}(P)}w\left(\dfrac{1}{x_{1}(x_{1}+x_{2})(x_{1}+x_{2}+x_{3})(x_{1}+x_{2}+x_{3}+x_{4})(x_{1}+x_{2}+x_{3}+x_{4}+x_{5})}\right)\\\
=\dfrac{1}{x_{1}(x_{1}+x_{2})(x_{1}+x_{2}+x_{3})(x_{1}+x_{2}+x_{3}+x_{4})(x_{1}+x_{2}+x_{3}+x_{4}+x_{5})}\\\
+\dfrac{1}{x_{1}(x_{1}+x_{2})(x_{1}+x_{2}+x_{4})(x_{1}+x_{2}+x_{4}+x_{3})(x_{1}+x_{2}+x_{4}+x_{3}+x_{5})}\\\
+\dfrac{1}{x_{2}(x_{2}+x_{1})(x_{2}+x_{1}+x_{3})(x_{2}+x_{1}+x_{3}+x_{4})(x_{2}+x_{1}+x_{4}+x_{3}+x_{5})}\\\
+\dfrac{1}{x_{2}(x_{2}+x_{1})(x_{2}+x_{4}+x_{1})(x_{2}+x_{1}+x_{4}+x_{3})(x_{2}+x_{1}+x_{4}+x_{3}+x_{5})}\\\
+\dfrac{1}{x_{2}(x_{2}+x_{4})(x_{2}+x_{4}+x_{1})(x_{2}+x_{1}+x_{4}+x_{3})(x_{2}+x_{4}+x_{1}+x_{3}+x_{5})}\\\
=\dfrac{x_{1}+2x_{2}+x_{3}+x_{4}}{x_{1}x_{2}(x_{1}+x_{2}+x_{3})(x_{2}+x_{4})(x_{1}+x_{2}+x_{3}+x_{4})(x_{1}+x_{2}+x_{3}+x_{4}+x_{5})},$

matching the above computation.

Féray and Reiner also observed that when the ${{\mathbb{N}}}^{n}$-grading on
$R_{P}^{\mathrm{wt}}$ is collapsed to an ${{\mathbb{N}}}$-grading, taking
$\deg x_{i}=1$ for all $i$, one has

$\mathrm{Hilb}(R_{P}^{\mathrm{wt}},q)=\dfrac{\sum_{w\in\mathcal{L}(P)}q^{\mathrm{maj}(w)}}{(1-q)(1-q^{2})\cdots(1-q^{n})}.$

In particular, when $P$ is a forest, this recovers the $q$-hook formula of
Björner and Wachs [5]. For the poset in Figure 1, one has

$\mathrm{Hilb}(R_{P}^{\mathrm{wt}},q)=\dfrac{1-q^{5}}{(1-q)^{2}(1-q^{2})(1-q^{3})(1-q^{4})(1-q^{5})}\\\
=\dfrac{1+q^{3}+q+q^{4}+q^{2}}{(1-q)(1-q^{2})(1-q^{3})(1-q^{4})(1-q^{5})}=\dfrac{\sum_{w\in\mathcal{L}(P)}q^{\mathrm{maj}(w)}}{(1-q)(1-q^{2})(1-q^{3})(1-q^{4})(1-q^{5})}.$

###  2 The signed poset story

The story for signed posets is really quite similar to that for posets and we
will generalize all of the above results to this context. Signed posets will
be defined formally in Definition 7.2. They come in pairs,
$P\subset\Phi_{B_{n}}$ and ${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$,
since $\Phi_{B_{n}}$ and $\Phi_{C_{n}}$ are dual root systems. Inspecting the
definitions of $\Psi_{P}$ and $\Phi_{P}$ in type A, one sees that the
denominators of the fraction on which $w$ acts correspond to a choice of
simple roots and the corresponding fundamental dominant coweights,
respectively. This observation leads one to define

$\displaystyle\Psi_{P}(\bm{x})$
$\displaystyle=\sum_{w\in\mathcal{L}(P)}w\left(\dfrac{1}{(x_{1}-x_{2})(x_{2}-x_{3})\cdots(x_{n-1}-x_{n})x_{n}}\right)\text{and}$
$\displaystyle\Psi^{\scriptscriptstyle\vee}_{{P^{\scriptscriptstyle\vee}}}(\bm{x})$
$\displaystyle=\sum_{w\in\mathcal{L}({P^{\scriptscriptstyle\vee}})}w\left(\dfrac{1}{(x_{1}-x_{2})(x_{2}-x_{3})\cdots(x_{n-1}-x_{n})2x_{n}}\right)$

to parallel $\Psi$ in type A and

$\displaystyle\Phi_{P}(\bm{x})$
$\displaystyle=\sum_{w\in\mathcal{L}(P)}w\left(\dfrac{1}{x_{1}(x_{1}+x_{2})\cdots(x_{1}+\cdots+x_{n})}\right)\text{and}$
$\displaystyle\Phi^{\scriptscriptstyle\vee}_{{P^{\scriptscriptstyle\vee}}}(\bm{x})$
$\displaystyle=\sum_{w\in\mathcal{L}({P^{\scriptscriptstyle\vee}})}w\left(\dfrac{1}{x_{1}(x_{1}+x_{2})+\cdots+(x_{1}+\cdots+x_{n-1})(\frac{1}{2}x_{1}+\cdots+\frac{1}{2}x_{n})}\right)$

to parallel $\Phi$ in type $A$.

One can likewise define root and weight cones for signed posets (see Section
12) and use the corresponding semigroup rings to evaluate $\Psi$ and $\Phi$ in
some cases where the semigroup ring is a complete intersection.

$4$$3$$1$$2$$-2$$-1$$-3$$-4$$\rho$$\sigma,\iota(\sigma)$$\iota(\rho)$ Figure
2: A signed poset

By way of example, consider Figure 2. As will be explained in Section 8, it is
a representation of a signed poset
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$, with
${P^{\scriptscriptstyle\vee}}=\\{+e_{1}-e_{2},+e_{1}+e_{2},+e_{3}-e_{2},+e_{4}-e_{3},+e_{4}-e_{1},+e_{4}-e_{2},+e_{2}+e_{4},+e_{3}+e_{4},+e_{1}+e_{3},+e_{1}+e_{4},+2e_{1},+2e_{4}\\}$.
Note that there is an involutive poset anti-automorphism, $\iota$, exchanging
$i$ and $-i$. The poset is strongly planar (this will mean
${P^{\scriptscriptstyle\vee}}$ is strongly planar) and the three regions it
encloses fall into two orbits under the involution: $\\{\rho,\iota(\rho)\\}$
and $\\{\sigma=\iota(\sigma)\\}$. Theorem 15.12 will show that this implies
the following complete intersection presentation for
$R_{{P^{\scriptscriptstyle\vee}}}^{\mathrm{rt}}$:

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

with the map $U_{ij}\mapsto
x_{|i|}^{\mathrm{sgn}(i)}x_{|j|}^{\mathrm{sgn}(j)}\in k[x_{1}^{\pm
1},\ldots,x_{n}^{\pm 1}]$, where $\mathrm{sgn}(i)=1$ and
$\mathrm{sgn}(\bar{i})=-1$, i.e.

$\displaystyle U_{12}\mapsto x_{1}x_{2}$ $\displaystyle U_{1\bar{2}}\mapsto
x_{1}x_{2}^{-1}$ $\displaystyle U_{\bar{1}4}\mapsto x_{1}^{-1}x_{4}$
$\displaystyle U_{\bar{2}3}\mapsto x_{2}^{-1}x_{3}$ $\displaystyle
U_{\bar{3}4}\mapsto x_{3}^{-1}x_{4}.$

Suppose the polynomial ring is ${\mathbb{Z}}^{4}$-graded with

$\displaystyle\deg U_{12}$ $\displaystyle=(1,1,0,0)$ $\displaystyle\deg
U_{1\bar{2}}$ $\displaystyle=(1,-1,0,0)$ $\displaystyle\deg U_{\bar{1}4}$
$\displaystyle=(-1,0,0,1)$ $\displaystyle\deg U_{\bar{2}3}$
$\displaystyle=(0,-1,1,0)$ $\displaystyle\deg U_{\bar{3}4}$
$\displaystyle=(0,0,-1,1).$

Then

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

Looking at the numerator, one sees that it corresponds to
$1-x_{|\max(\rho)|}^{-\mathrm{sgn}(\max(\rho))}x_{|\min(\rho)|}^{\mathrm{sgn}(\min(\rho))}$,
and the terms of the denominator correspond to (pairs of) covering relations
in the poset of Figure 2, paralleling Theorem 1.3. This is Theorem 15.12, and
Section 15.4 will explain the computation of $\Psi$ for certain signed posets
in types B and C.

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

As in type A, the weight cone semigroup will allow one to calculate $\Phi$,
but there is a wrinkle. Consider the poset shown in Figure 3. The weight cone
semigroup ring $R_{P}^{\mathrm{wt}}$ is presented as

$R_{P}^{\mathrm{wt}}\cong
k[U_{1},U_{12}U_{123},U_{\bar{3}}]/(U_{123}U_{\bar{3}}-U_{12}),$

via the map $U_{J}\mapsto\bm{x}^{J}=\prod_{j\in J}x_{|j|}^{\mathrm{sgn}(j)}\in
k[x_{1}^{\pm 1},\ldots,x_{n}^{\pm n}]$, i.e.

$\displaystyle U_{1}$ $\displaystyle\mapsto x_{1}$ $\displaystyle U_{12}$
$\displaystyle\mapsto x_{1}x_{2}$ $\displaystyle U_{123}$
$\displaystyle\mapsto x_{1}x_{2}x_{3}$ $\displaystyle U_{\bar{3}}$
$\displaystyle\mapsto x_{3}^{-1}.$

When the polynomial ring is graded by ${\mathbb{Z}}^{3}$ with

$\displaystyle\deg U_{1}$ $\displaystyle=(1,0,0)$ $\displaystyle\deg U_{12}$
$\displaystyle=(1,1,0)$ $\displaystyle\deg U_{123}$ $\displaystyle=(1,1,1)$
$\displaystyle\deg U_{\bar{3}}$ $\displaystyle=(0,0,-1),$

the Hilbert series is

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

The ideals of the signed poset are $\\{1\\},\\{1,2\\},\\{1,2,3\\},\\{-3\\}$
(see Section 9), corresponding to each term of the denominator of the Hilbert
series. The numerator corresponds to the one pair of ideals which intersect
nontrivially (to be defined in Section 17), $\\{1,2,3\\}$ and $\\{-3\\}$,
which combine to form $\\{1,2\\}$. While this parallels Theorem 1.4, it turns
out that the “correct” thing to do is to consider an initial ideal, as doing
so allows one to migrate through various specializations of the grading—the
toric ideal from Theorem 17.3 is not necessarily homogeneous in gradings other
than the one by ${\mathbb{Z}}^{n}$.

Chapter Root and weight semigroup rings for signed posets summarizes necessary
background on cones, semigroups and root systems. Further background
information will be introduced as needed. Chapter Root and weight semigroup
rings for signed posets reviews the definition of signed posets, as well as
ideals, $P$-partitions and linear extensions of signed posets. Section 8
explains how signed posets can be represented as posets and oriented signed
graphs. Fischer’s representation of a signed poset $P\subset\Phi_{B_{n}}$ from
[24] is modified for purposes of Chapter Root and weight semigroup rings for
signed posets. Section 12 defines the root and weight cones of a signed poset
and gives dual characterizations for when they are each pointed, full-
dimensional and simplicial.

Chapters Root and weight semigroup rings for signed posets and Root and weight
semigroup rings for signed posets proceed independently of one another.
Chapter Root and weight semigroup rings for signed posets discusses the root
cone semigroup. Section 14 discusses the toric ideal of the semigroup
associated to a signed graph (Theorem 14.5) and uses that result to describe
generating sets for the toric ideals of the root cone semigroup,
posets/digraphs and graphs. Section 15 describes a situation when
$R_{P}^{\mathrm{rt}}$ is a complete intersection and Section 15.4 computes
$\Psi_{P}$.

Chapter Root and weight semigroup rings for signed posets discusses the weight
cone semigroup in type B. Section 17.3 gives a presentation for the weight
cone semigroup ring. Section 18 turns its attention to computing $\Phi_{P}$
and

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

This is again a question of complete intersections, but not of the weight cone
semigroup ring. Instead, modding out by an initial ideal of the toric ideal
preserves the Hilbert series, but allows a presentation involving an
${{\mathbb{N}}}$-graded homogeneous ideal. Signed posets with the property
that modding out by the initial ideal of the toric ideal gives a complete
intersection will be called _initial complete intersections_. Section 18.2
characterizes these signed posets as those avoiding certain induced subposets,
while Section 18.3 explains how these posets can be constructed via any
sequence of three moves.

Chapter Root and weight semigroup rings for signed posets discusses a few
loose ends: understanding two triangulations of the weight cone, the type C
weight cone and characterizing when $R_{P}^{\mathrm{rt}}$ is a complete
intersection.

###  3 Summary of the Main Results

  * •

Theorem 14.5 giving generators for the toric ideal of affine semigroup rings
associated to signed posets and, more generally, oriented signed graphs. These
are the subrings of Laurent polynomials generated by monomials of the form
$t_{i}^{\pm 1},t_{i}^{\pm 2},t_{i}^{\pm 1}t_{j}^{\pm 1}$. This result appears
to be new and generalizes work of [9], [27] and [61].

  * •

Theorem 15.12 showing that strongly planar signed posets $P$ have rings
$R_{P}^{\mathrm{rt}}$, $R_{{P^{\scriptscriptstyle\vee}}}^{\mathrm{rt}}$ which
are complete intersections, with Corollary 15.20 showing how to compute
$\Psi_{P}$ in this case.

  * •

Theorem 17.3 giving a Gröbner basis for the toric ideal of
$R_{P}^{\mathrm{wt}}$ in type B, generalizing [21, Proposition 6.4 ].

  * •

Theorems 18.10 and 18.21 giving two characterizations (via forbidden subposets
versus via inductive constructions) of the situation where this Gröbner basis
gives a complete intersection presentation for its initial ideal, generalizing
[21, Theorems 10.5, 10.6 ].

###  4 A remark on notation

  * •

To save space and improve readability, negative numbers will sometimes be
written as $\bar{1}$ rather than $-1$.

  * •

The parentheses and brackets will sometimes be dropped from vectors and sets
in figures, e.g. $100$ instead of $(1,0,0)$.

  * •

There will be many vertices of graphs that come in pairs $i,-i$ for $i\in[n]$
and polynomial rings where the variables are indexed by $[n]$. If $v$ is a
variable, understand $x_{v}$ to mean the variable $x_{|v|}$.

  * •

The characteristic vector of a set $J\subset\\{\pm 1,\ldots,\pm n\\}$ such
that $J$ does not contain both $i$ and $-i$ for any $i$ is the vector
$\chi_{J}$ whose coordinates are defined by

$(\chi_{J})_{i}=\begin{cases}1&\text{if }i\in J\\\ -1&\text{if }-i\in J\\\
0&\text{else}\end{cases}$

  * •

If $J\subset\\{\pm 1,\ldots,\pm n\\}$ is such that there is no $i$ such that
both $i,-i\in J$, then

$\langle\bm{x},J\rangle=\langle\bm{x},\chi_{J}\rangle=\sum_{i}(\chi_{J})_{i}x_{i}.$

##  Chapter Some Background

This chapter reviews requisite material on polyhedral cones, semigroups and
root systems. Chapter Root and weight semigroup rings for signed posets will
fit these ideas together in the discussion of signed posets. Further
background material will be introduced as needed.

###  5 Polyhedral Cones

Associated to a signed poset will be two polyhedral cones, the root cone and
the weight cone. Consequently, this section reviews some basic facts about
polyhedral cones. One can also refer to [26] for further information on
polyhedral cones.

######  Definition 5.1.

A _polyhedral cone_ $K\subset{\mathbb{R}}^{n}$ is the intersection of finitely
many half-spaces determined by hyperplanes $H_{\alpha}=\\{x\colon\langle
x,\alpha\rangle=0\\}$. The $H_{\alpha}$ are the _supporting hyperplanes_ of
$K$. Alternatively, a cone may be characterized as the positive span of some
finite collection of vectors, $W$, with the positive span being denoted
${\mathbb{R}}_{+}W$. A set of _extreme rays_ of a cone $K$ are vectors
comprising a set $W$, minimal with respect to inclusion, such that
$K={\mathbb{R}}_{+}W$.

There are a number of properties that can be used to describe a cone.

######  Definition 5.2.

The _dimension_ of a cone $K$, denoted $\dim K$ is the dimension of the vector
space spanned by its extreme rays, and $K$ is said to be _full-dimensional_ if
$K\subset{\mathbb{R}}^{n}$ and $\dim K=n$. A cone is said to be _pointed_ when
it does not contain a line. It is said to be _rational_ with respect to a
full-rank lattice $L\subset{\mathbb{R}}^{n}$ when the $\alpha$ determining the
supporting hyperplanes lie in $L$. It is _simplicial_ if the extreme rays are
linearly independent. A (simplicial) cone is _unimodular_ with respect to a
lattice if the primitive vectors along the extreme rays form a basis of the
lattice.

For example, consider the cone in Figure 4.

$x$$y$$z$$(1,-1,1)$$(1,1,1)$$(-1,1,1)$$(-1,-1,1)$ Figure 4: Cone spanned by
$(1,1,1)$, $(-1,1,1)$, $(-1,-1,1)$ and $(1,-1,1)$

The extreme rays are $(1,1,1)$, $(-1,1,1)$, $(-1,-1,1)$ and $(1,-1,1)$ and the
cone is pointed and rational. Since it is a cone in ${\mathbb{R}}^{3}$ with
four extreme rays, it cannot be simplicial. Likewise, it cannot be unimodular
with respect to the lattice ${\mathbb{Z}}^{3}$.

######  Definition 5.3.

If $K$ is a cone, its _dual_ or _polar cone_ is

$K^{*}=\\{x\in{\mathbb{R}}^{n}\colon\langle x,a\rangle\geq 0\ \forall a\in
K\\}.$

The dual is sometimes defined with a $\leq$ rather than a $\geq$, but the
appeal of this choice of $\geq$ will become clear in Section 12 and one can
insert a minus sign where appropriate when the $\leq$ definition is more apt.
The following facts about cones and their duals are well known.

  * •

A cone is the dual of its dual, i.e. $K^{**}=K$.

  * •

A cone $K$ is full-dimensional if and only if its dual $K^{*}$ is pointed.

  * •

A cone $K$ is simplicial if and only if its dual $K^{*}$ is also simplicial.

####  5.1 Rational Functions and Cones

Recall that the goal is to prove some rational function identities for signed
posets. A key step will be to understand the functions $\Psi$ and $\Phi$ as
valuations of cones.

In [4], Barvinok considers an exponential integral and exponential sum over a
polyhedral cone, $K$:

$\int_{K}e^{-\langle
x,u\rangle}\,du\quad\text{and}\quad\sum_{K\cap{\mathbb{Z}}^{n}}e^{-\langle
x,u\rangle},$

where $x\in{\mathbb{R}}^{n}$ and $du$ is Lebesgue measure on
${\mathbb{R}}^{n}$. Each gives a rational function, in $x_{i}$ and
$X_{i}=e^{x_{i}}$, respectively (see Propositions 5.4 and 6.8).

######  Proposition 5.4 ([4, Proposition 2.4]).

Let $K$ be a pointed, full-dimensional polyhedral cone in ${\mathbb{R}}^{n}$.
Then, for all $x\in\operatorname{Int}K^{*}$, the integral

$\int_{K}e^{-\langle x,u\rangle}\,du$

exists and determines a function $s(K;x)$, which is rational in
$x\in{\mathbb{C}}^{n}$.

Furthermore, if $K$ is not pointed or if $K$ is not full-dimensional,
$s(K;x)=0$.

For example, consider the cone in Figure 4 once again. To compute the
integral, it is easiest to split $K$ into two simplicial cones $K_{1}$ and
$K_{2}$:

$\displaystyle K_{1}$
$\displaystyle=\text{span}_{{\mathbb{R}}_{+}}\\{(1,-1,1),(-1,-1,1),(-1,1,1)\\}$
$\displaystyle K_{2}$
$\displaystyle=\text{span}_{{\mathbb{R}}_{+}}\\{(-1,1,1),(1,-1,1),(1,1,1)\\}.$

Since $K_{1}$ and $K_{2}$ are both simplicial, every element of each cone can
be written uniquely as a linear combination of the extreme rays. Then, if one
denotes the extreme rays of $K_{1}$ by $u_{1},u_{2},u_{3}$ and the extreme
rays of $K_{2}$ by $v_{1},v_{2},v_{3}$, for a fixed
$x\in\operatorname{Int}K^{*}$, one can compute the integral as follows.

$\int_{K}e^{-\langle x,u\rangle}\,du=\int_{K_{1}}e^{-\langle
x,u\rangle}\,du+\int_{K_{2}}e^{-\langle x,u\rangle}\,du-\int_{K_{1}\cap
K_{2}}e^{-\langle x,u\rangle}\,du.$

The last integral is $0$ since $K_{1}\cap K_{2}$ is not full-dimensional. Then

$\displaystyle\int_{K_{1}}e^{-\langle x,u\rangle}\,du$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}e^{-\langle
x,a_{1}u_{1}+a_{2}u_{2}+a_{3}u_{3}\rangle}\,da_{3}da_{2}da_{1}$
$\displaystyle=\lim_{b_{1},b_{2},b_{3}\to\infty}\int_{0}^{b_{1}}\int_{0}^{b_{2}}\int_{0}^{b_{3}}e^{-\langle
x,a_{1}u_{1}\rangle}e^{-\langle x,a_{2}u_{2}\rangle}e^{-\langle
x,a_{3}u_{3}\rangle}\,da_{3}da_{2}da_{1}$
$\displaystyle=\lim_{b_{1},b_{2},b_{3}\to\infty}\int_{0}^{b_{1}}e^{-\langle
x,a_{1}u_{1}\rangle}\int_{0}^{b_{2}}e^{-\langle
x,a_{2}u_{2}\rangle}\int_{0}^{b_{3}}e^{-a_{3}\langle
x,u_{3}\rangle}\,da_{3}da_{2}da_{1}$
$\displaystyle=\lim_{b_{1},b_{2},b_{3}\to\infty}\int_{0}^{b_{1}}e^{-\langle
x,a_{1}u_{1}\rangle}\int_{0}^{b_{2}}e^{-\langle
x,a_{2}u_{2}\rangle}\left[\dfrac{-1}{\langle x,u_{3}\rangle}e^{-a_{3}\langle
x,u_{3}\rangle}\right]_{0}^{b_{3}}\,da_{2}da_{1}$
$\displaystyle=\lim_{b_{1},b_{2},b_{3}\to\infty}\left[\dfrac{-1}{\langle
x,u_{1}\rangle}e^{-a_{1}\langle
x,u_{1}\rangle}\right]_{0}^{b_{1}}\left[\dfrac{-1}{\langle
x,u_{2}\rangle}e^{-a_{2}\langle
x,u_{2}\rangle}\right]_{0}^{b_{2}}\left[\dfrac{-1}{\langle
x,u_{3}\rangle}e^{-a_{3}\langle x,u_{3}\rangle}\right]_{0}^{b_{3}}$

Since $x\in\operatorname{Int}K^{*}$, one knows that $\langle
x,u_{1}\rangle,\langle x,u_{2}\rangle,\langle x,u_{3}\rangle\geq 0$.
Therefore,

$\lim_{b_{i}\to\infty}e^{-b_{i}\langle x,u_{i}\rangle}=0$

for $i=1,2,3$. Thus, one has

$\int_{K_{1}}e^{-\langle x,u\rangle}\,du=\dfrac{1}{\langle
x,u_{1}\rangle}\dfrac{1}{\langle x,u_{2}\rangle}\dfrac{1}{\langle
x,u_{3}\rangle}=\dfrac{1}{(-x_{1}+x_{2}-x_{3})}\dfrac{1}{(x_{1}+x_{2}-x_{3})}\dfrac{1}{(x_{1}-x_{2}-x_{3})}.$

A similar calculation with $K_{2}$ gives

$\int_{K_{2}}e^{-\langle
x,u\rangle}\,du=\dfrac{1}{(x_{1}-x_{2}-x_{3})}\dfrac{1}{(-x_{1}+x_{2}-x_{3})}\dfrac{1}{(-x_{1}-x_{2}-x_{3})}.$

One then has

$\int_{K}e^{-\langle
x,u\rangle}\,du=\dfrac{1}{(-x_{1}+x_{2}-x_{3})}\dfrac{1}{(x_{1}+x_{2}-x_{3})}\dfrac{1}{(x_{1}-x_{2}-x_{3})}\\\
+\dfrac{1}{(x_{1}-x_{2}-x_{3})}\dfrac{1}{(-x_{1}+x_{2}-x_{3})}\dfrac{1}{(-x_{1}-x_{2}-x_{3})}\\\
=\dfrac{1}{(x_{1}-x_{2}-x_{3})(-x_{1}+x_{2}-x_{3})}\left(\dfrac{1}{-x_{1}-x_{2}-x_{3}}+\dfrac{1}{x_{1}+x_{2}-x_{3}}\right)$

######  Proposition 5.5 ([4, (2.1) ]).

Suppose $K$ is a simplicial cone whose extreme rays are
$\\{u_{1},\ldots,u_{n}\\}$. Then

$s(K;\bm{x})=|u_{1}\wedge\cdots\wedge
u_{n}|\prod_{i=1}^{n}\langle\bm{x},u_{i}\rangle^{-1},$

where $|u_{1}\wedge\cdots\wedge u_{n}|$ is the volume of the parallelopiped
formed by the $u_{i}$.

Discussion of the exponential sum will be postponed until the Section 6.

###  6 Semigroups

The next important concept is that of the semigroup.

######  Definition 6.1.

A _semigroup_ is a set together with an associative binary operation. An
_affine semigroup_ is a semigroup which is isomorphic to a finitely-generated
subsemigroup of ${\mathbb{Z}}^{n}$ under addition.

In general, unlike monoids, semigroups need not have an identity element.
However, the semigroups of interest here will have an identity and the binary
operation, $+$, will be commutative. The semigroups considered in Chapters
Root and weight semigroup rings for signed posets and Root and weight
semigroup rings for signed posets are affine semigroups as a consequence of
Gordan’s Lemma (see [20, Proposition 5.14]).

######  Theorem 6.2 (Gordan’s Lemma).

Suppose $K$ is a rational polyhedral cone in ${\mathbb{R}}^{n}$ and $A$ is a
subgroup of ${\mathbb{Z}}^{n}$. Then $K\cap A$ is an affine semigroup.

As an example, suppose $K$ is the cone in Figure 5 whose extreme rays are
$(1,0)$ and $(0,1)$ and suppose $A$ is
${\mathbb{Z}}(1,1)\subset{\mathbb{Z}}^{2}$.

Figure 5: The cone spanned by $(1,0)$ and $(0,1)$ intersected with
${\mathbb{Z}}^{2}$

Then $K\cap A=\\{(i,i)\colon i\in{\mathbb{Z}}_{\geq 0}\\}$.

####  6.1 Semigroup Rings and Toric Ideals

One can move from the semigroup world to the somewhat more familiar world of
rings by considering semigroup rings.

######  Definition 6.3.

Suppose $A\subset{\mathbb{Z}}^{n}$ is an affine semigroup generated by
$\\{a_{1},\ldots,a_{m}\\}$. Let $L=k[t_{1}^{\pm 1},\ldots,t_{n}^{\pm 1}]$ be
the Laurent polynomial ring. The _semigroup ring_ of $A$ is the subring of $L$
spanned by $\bm{t}^{a_{i}}$, $i=1,\ldots,m$, where
$\bm{t}^{a_{i}}=t_{1}^{a_{i1}}t_{2}^{a_{i2}}\cdots t_{n}^{a_{in}}$ when
$a_{i}=(a_{i1},a_{i2},\ldots,a_{in})$. Denote the semigroup ring of $A$ by
$k[A]$.

It is often more convenient to work with a presentation of the semigroup ring
as a quotient by the toric ideal than to view the semigroup ring as a subring
of the Laurent polynomial ring.

######  Definition 6.4.

Let $S=k[x_{1},\ldots,x_{m}]$ be a polynomial ring in $m$ variables and $A$ a
semigroup generated by $\\{a_{1},\ldots,a_{m}\\}$. Define a map $\varphi\colon
S\to L$ by

$\varphi(x_{i})=\bm{t}^{a_{i}}.$

One then has that $k[A]\cong S/\ker\varphi$. The kernel of $\varphi$ is the
ideal known as the _toric ideal_ of $A$, denoted $I_{A}$.

######  Definition 6.5.

A _binomial_ is a polynomial which is a difference of two monomials and a
_binomial ideal_ is an ideal that is generated by binomials.

Comprehensive discussion of binomial ideals can be found in [19].

The following definition and notation is useful for understanding the
generators of toric ideals.

######  Definition 6.6.

If $u\in{\mathbb{Z}}^{n}$, define two vectors, the positive and negative
supports of $u$ as follows. The _positive support_ $u^{+}$ is given by

$u^{+}_{i}=\begin{cases}u_{i}&\text{if $u_{i}>0$}\\\ 0&\text{else}\end{cases}$

Likewise, the _negative support_ $u^{-}$ is

$u^{-}_{i}=\begin{cases}u_{i}&\text{if $u_{i}<0$}\\\ 0&\text{else}\end{cases}$

Then to each $u\in{\mathbb{Z}}^{n}$, one can associate a binomial
$\bm{x}^{u^{+}}-\bm{x}^{u^{-}}$.

The following proposition is well known and the proof may be found in [58,
Lemma 4.1 ] or [20, Lemma 5.2 ].

######  Proposition 6.7.

Suppose $A$ is an affine semigroup and $I_{A}$ its toric ideal. Then $I_{A}$
is a binomial ideal generated by $\bm{x}^{u^{+}}-\bm{x}^{u^{-}}$ for $u\in\ker
M$ where $M$ is the matrix whose columns are the generators of $A$.

As an example, consider the semigroup whose generators are $(1,-1,1)$,
$(-1,-1,1)$, $(-1,1,1)$, $(1,1,1)$. (This is the semigroup obtained by
intersecting the cone of Figure 4 with ${\mathbb{Z}}^{3}$.) The matrix $M$ is
then

$\begin{pmatrix}[r]1&-1&-1&1\\\ -1&-1&1&1\\\ 1&1&1&1\\\ \end{pmatrix}$

The kernel of $M$ is one-dimensional and spanned by $(1,-1,1,-1)^{\top}$. Let
$S=k[x_{1},x_{2},x_{3},x_{4}]$ and $\varphi\colon S\to k[t_{1}^{\pm
1},t_{2}^{\pm 1},t_{3}^{\pm 1}]$ be defined by

$\displaystyle\varphi(x_{1})$ $\displaystyle=t_{1}t_{2}^{-1}t_{3}$
$\displaystyle\varphi(x_{2})$ $\displaystyle=t_{1}^{-1}t_{2}^{-1}t_{3}$
$\displaystyle\varphi(x_{3})$ $\displaystyle=t_{1}^{-1}t_{2}t_{3}$
$\displaystyle\varphi(x_{4})$ $\displaystyle=t_{1}t_{2}t_{3}$

The kernel of $\varphi$ is then the principal ideal $(x_{1}x_{3}-x_{2}x_{4})$.
Certainly this ideal is contained in the kernel. One knows from Proposition
6.7 that $\ker\varphi=(\bm{x}^{u^{+}}-\bm{x}^{u^{-}})$ for
$u=m(1,-1,1,-1)^{\top}\in\ker M\cap{{\mathbb{N}}}^{n}$, with
$m\in{\mathbb{Z}}_{>0}$, so one shows that
$\bm{x}^{u^{+}}-\bm{x}^{u^{-}}\in(x_{1}x_{3}-x_{2}x_{4})$. This is
straightforward:

$x_{1}^{m}x_{3}^{m}-x_{2}^{m}x_{4}^{m}=\\\
(x_{1}x_{3}-x_{2}x_{4})((x_{1}x_{3})^{m-1}+(x_{1}x_{3})^{m-2}x_{2}x_{4}+\cdots+x_{1}x_{3}(x_{2}x_{4})^{m-2}+(x_{2}x_{4})^{m-1})\\\
\in(x_{1}x_{3}-x_{2}x_{4}),$

so $(x_{1}x_{3}-x_{2}x_{4})=\ker\varphi$.

####  6.2 A Rational Function

The exponential sum

$\sum_{K\cap{\mathbb{Z}}^{n}}e^{-\langle x,u\rangle}$

discussed in [4] and mentioned in Section 5.1 is then a sum over the elements
of an affine semigroup. The following proposition is the analogue of
Proposition 5.4.

######  Proposition 6.8 ([4, Proposition 4.4 ]).

Suppose $K$ is a pointed rational polyhedral cone in ${\mathbb{R}}^{n}$. Then
for $x\in\operatorname{Int}K^{*}$, the series

$\sum_{K\cap{\mathbb{Z}}^{n}}e^{-\langle x,u\rangle}$

converges and determines a function $\sigma(K;x)$ which is rational in
$X_{i}=e^{x_{i}}$, $i=1,\ldots,n$. Furthermore, there exists a representation

$\sigma(K;x)=\dfrac{P(x)}{\prod_{i=1}^{m}(1-e^{-\langle x,u_{i}\rangle})},$

where $P(x)$ is a Laurent polynomial in $X_{i}$ and the $u_{i}$ are the
extreme rays of $K$. If $K$ is not pointed, $\sigma(K;x)=0$.

Moreover, we can understand this sum as the Hilbert series of the semigroup
ring.

######  Definition 6.9.

Suppose $k$ is a field and $R$ is a finitely-generated $k$-algebra. Suppose
further that $R$ is graded by some index set $I$ equipped with an addition,
i.e. $R=\bigoplus_{\alpha\in I}R_{\alpha}$ with $R_{\alpha}R_{\beta}\subset
R_{\alpha+\beta}$. Its _Hilbert series_ is

$\mathrm{Hilb}(R,x)=\sum_{\alpha\in I}\dim(R_{\alpha})x^{\alpha}=\sum_{r\in
R}x^{\deg r}.$

A natural grading of the semigroup ring of an affine semigroup $A$ is the
${\mathbb{Z}}^{n}$-grading where $\deg(\bm{x}^{a})=(a_{1},\ldots,a_{n})$ for
$a\in A$. Then one has that

$\mathrm{Hilb}(k[A],\bm{x})=\sum_{a\in A}x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots
x_{n}^{a_{n}}.$

Note that we are abusing notation here and using $x_{i}$ in both the
presentation of the semigroup ring and the Hilbert series, though these are
actually different $x_{i}$.

Moreover, if one supposes that $A$ arose from Gordan’s Lemma as the
intersection of a pointed rational cone $K$ with ${\mathbb{Z}}^{n}$, one has
that

$\mathrm{Hilb}(k[A],\bm{x})=\sum_{a\in A}x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots
x_{n}^{a_{n}}=\sum_{a\in
K\cap{\mathbb{Z}}^{n}}x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}},$

and this last sum is precisely the sum
$\sum_{K\cap{\mathbb{Z}}^{n}}e^{-\langle X,u\rangle}$ after the change of
coordinates $x_{i}=e^{-X_{i}}$.

One can compute the Hilbert series of a graded ring from a minimal finite free
resolution courtesy of the following well-known fact (see, for example, [51,
Theorem I.11.3 ]).

######  Proposition 6.10.

Suppose $M$ is a graded $A$-module and

$0\to F_{t}\to F_{t-1}\to\cdots\to F_{1}\to F_{0}\to M\to 0$

is a finite free resolution of $M$ by graded $A$-modules. Then

$\mathrm{Hilb}(M,x)=\sum_{i=0}^{t}(-1)^{i}\mathrm{Hilb}(F_{i},x).$

Consider the cone $K$ having extreme rays $(1,0,0),(1,1,0),(1,0,1),(1,1,1)$.
In fact, $K$ is the weight cone of a poset on three elements, so [9,
Proposition 7.1] gives that these vectors also (minimally) generate the
semigroup $K\cap{\mathbb{Z}}^{n}$, which will be denoted by $A$. Let
$S=k[a,b,c,d]$ with $\deg(a)=(1,0,0)$, $\deg(b)=(1,1,0)$, $\deg(c)=(1,0,1)$,
$\deg(d)=(1,1,1)$. The semigroup ring $k[A]$ is isomorphic to $S/(bc-ad)$ (see
[21, Proposition 6.4]). One then has the following (minimal) finite free
resolution for $S/I$:

$0\to S(-(2,1,1))\stackrel{{\scriptstyle\varphi}}{{\to}}S\to S/I\to 0$

where $\varphi$ is the map sending $1$ to $bc-ad$, and $S(-(2,1,1))$ is $S$
with the grading shifted so that $1\in S$ has degree $(2,1,1)$. One can then
compute the Hilbert series of $S/I$ (i.e. of $k[A]$) as

$\mathrm{Hilb}(S/I,\bm{x})=\mathrm{Hilb}(S,\bm{x})-\mathrm{Hilb}(S(-(2,1,1)),\bm{x})\\\
=\dfrac{1}{(1-x_{1})(1-x_{1}x_{3})(1-x_{1}x_{2})(1-x_{1}x_{2}x_{3})}\\\
-\dfrac{x_{1}^{2}x_{2}x_{3}}{(1-x_{1})(1-x_{1}x_{3})(1-x_{1}x_{2})(1-x_{1}x_{2}x_{3})}\\\
=\dfrac{1-x_{1}^{2}x_{2}x_{3}}{(1-x_{1})(1-x_{1}x_{3})(1-x_{1}x_{2})(1-x_{1}x_{2}x_{3})}$

On the other hand, one can compute the exponential sum directly. Let $K_{1}$
be the cone whose extreme rays are $(1,0,0),(1,1,0),(1,0,1)$, $K_{2}$ the cone
whose extreme rays are $(1,1,0),(1,0,1),(1,1,1)$. Then $K_{1}\cap K_{2}$ is
the cone whose extreme rays are $(1,1,0)$ and $(1,0,1)$. (See Figure 6.)

$x$$y$$z$$(1,0,0)$$(1,1,0)$$(1,0,1)$$(1,1,1)$$K_{2}$$K_{1}$ Figure 6: Cone
spanned by $(1,0,0)$, $(1,1,0)$, $(1,0,1)$ and $(1,1,1)$

One then has

$\sum_{K\cap{\mathbb{Z}}^{n}}e^{-\langle
x,u\rangle}=\sum_{K_{1}\cap{\mathbb{Z}}^{n}}e^{-\langle
x,u\rangle}+\sum_{K_{2}\cap{\mathbb{Z}}^{n}}e^{-\langle
x,u\rangle}-\sum_{K_{1}\cap K_{2}\cap{\mathbb{Z}}^{n}}e^{-\langle
x,u\rangle}.$

Making the change of variables so that $x_{i}=e^{-x_{i}}$, one has that

$\displaystyle\sum_{K_{1}\cap{\mathbb{Z}}^{3}}e^{-\langle x,u\rangle}$
$\displaystyle=\sum_{a_{1}=0}^{\infty}\sum_{a_{2}=0}^{\infty}\sum_{a_{3}=0}^{\infty}e^{-\langle
x,a_{1}(1,0,0)+a_{2}(1,1,0)+a_{3}(1,0,1)\rangle}$
$\displaystyle=\sum_{a_{1}=0}^{\infty}\sum_{a_{2}=0}^{\infty}\sum_{a_{3}=0}^{\infty}x_{1}^{a_{1}}(x_{1}x_{2})^{a_{2}}(x_{1}x_{3})^{a_{3}}$
$\displaystyle=\dfrac{1}{(1-x_{1}x_{3})(1-x_{1}x_{2})(1-x_{1})}.$

Similarly,

$\sum_{K_{2}\cap{\mathbb{Z}}^{3}}e^{-\langle
x,u\rangle}=\dfrac{1}{(1-x_{1}x_{2})(1-x_{1}x_{3})(1-x_{1}x_{2}x_{3})}$

and

$\sum_{K_{1}\cap K_{2}\cap{\mathbb{Z}}^{3}}e^{-\langle
c,x\rangle}=\dfrac{1}{(1-x_{1}x_{2})(1-x_{1}x_{3})}.$

One then has that

$\sum_{K\cap{\mathbb{Z}}^{n}}e^{-\langle
x,u\rangle}=\dfrac{1}{(1-x_{1}x_{3})(1-x_{1}x_{2})(1-x_{1})}+\dfrac{1}{(1-x_{1}x_{2})(1-x_{1}x_{3})(1-x_{1}x_{2}x_{3})}\\\
-\dfrac{1}{(1-x_{1}x_{2})(1-x_{1}x_{3})}\\\
=\dfrac{1-x_{1}^{2}x_{2}x_{3}}{(1-x_{1})(1-x_{1}x_{2})(1-x_{1}x_{3})(1-x_{1}x_{2}x_{3})},$

matching the Hilbert series computation.

####  6.3 Complete Intersections

Part of the goal of Chapters Root and weight semigroup rings for signed posets
and Root and weight semigroup rings for signed posets will be to identify
signed posets for which the computation of the Hilbert series of (one or the
other of) the relevant rings is particularly straightforward, namely when the
rings are complete intersections.

######  Definition 6.11.

Suppose $R$ is a ring. A sequence of elements $\theta_{1},\ldots,\theta_{k}\in
R$ is said to be a _regular sequence_ if $\theta_{i+1}$ is a non-zero divisor
in $R/(\theta_{1},\ldots,\theta_{i})$ for $i=0,\ldots,k-1$. A quotient ring
$R/I$ is said to be a _complete intersection_ if $I$ is generated by a regular
sequence.

The following proposition simplifies the computation of the Hilbert series of
a complete intersection.

######  Proposition 6.12.

Suppose $R$ is a graded ring and $\theta$ is a non-zero divisor in $R$ and
$\deg\theta\neq 0$. Then

$\mathrm{Hilb}(R/(\theta),\bm{x})=(1-\bm{x}^{\deg\theta})\mathrm{Hilb}(R,\bm{x}).$

Iterating this relation gives the following.

######  Corollary 6.13.

Suppose $R$ is a ring and $(\theta_{1},\ldots,\theta_{k})$ is a regular
sequence. Then

$\mathrm{Hilb}(R/(\theta_{1},\ldots,\theta_{k}),\bm{x})=\mathrm{Hilb}(R,\bm{x})\prod_{i=1}^{k}(1-\bm{x}^{\deg\theta_{i}}).$

Boussicault, Féray, Lascoux and Reiner explain in Section 2.4 of [9] how the
valuations $s(K;c)$ from Section 5.1 and $\sigma(K;c)$ from Section 6.2 are
connected via a residue operation. This will be particularly relevant in the
case where the semigroup ring is a complete intersection as a result of the
following proposition.

######  Proposition 6.14.

Suppose $L$ is a lattice and let $K$ be a pointed $L$-rational cone for which
$R=k[K\cap L]$ is a complete intersection with

$R\cong S/I=k[U_{1},\ldots,U_{k}]/(\theta_{1},\ldots,\theta_{d}),$

where $\theta_{1},\ldots,\theta_{d}$ are $L$-homogeneous elements of degrees
$\delta_{1},\ldots,\delta_{d}$ forming a regular sequence. Then

$\mathrm{Hilb}(R;\bm{X})=\sigma(K;\bm{X})=\dfrac{\prod_{i=1}^{d}(1-\bm{X}^{\delta_{i}})}{\prod_{j=1}^{d}(1-\bm{X}^{u_{j}})},$

and if $K$ is full-dimensional,

$s(K;x)=\dfrac{\prod\langle x,\delta_{j}\rangle}{\prod\langle
x,u_{j}\rangle},$

where $\deg U_{j}=u_{j}$.

Section 13 will show that the two rational functions $\Psi$ and $\Phi$ on a
signed poset can be understood as the valuation $s(-;x)$ on certain cones.
Proposition 6.14 enables one to compute these rational functions in some cases
by establishing that a certain ring is a complete intersection and finding a
regular sequence generating the toric ideal.

As an example, consider the semigroup generated by

$(1,0,0),(0,0,-1),(1,1,0),(1,-1,-1),(1,1,1).$

This will be one of the semigroups considered in Chapter Root and weight
semigroup rings for signed posets. Let $K$ be the cone spanned by the
semigroup generators and $S=k[x_{1},x_{2},x_{3},x_{4},x_{5}]$ with $\deg
x_{1}=(1,0,0)$, $\deg x_{2}=(0,0,-1)$, $\deg x_{3}=(1,1,0)$, $\deg
x_{4}=(1,-1,-1)$ and $\deg x_{5}=(1,1,1)$. The toric ideal of the semigroup
is, courtesy of Macaulay2, $I=(x_{2}x_{5}-x_{3},x_{4}x_{5}-x_{1}^{2})$, and
one can check that the generators of the toric ideal form a regular sequence,
so the semigroup ring $R\cong S/I$ is a complete intersection.

Then, from Corollary 6.13 (or the first half of Proposition 6.14, which is
obtained by repeatedly applying the corollary), one has

$\mathrm{Hilb}(R,\bm{x})=\dfrac{(1-x_{1}x_{2})(1-x_{1}^{2})}{(1-x_{1})(1-x_{3}^{-1})(1-x_{1}x_{2})(1-x_{1}x_{2}^{-1}x_{3}^{-1})(1-x_{1}x_{2}x_{3})}$

and

$s(K,x)=\dfrac{(x_{1}+x_{2})(2x_{1})}{x_{1}(-x_{3})(x_{1}+x_{2})(x_{1}-x_{2}-x_{3})(x_{1}+x_{2}+x_{3})}.$

###  7 Root Systems

To extend the notion of posets, one must first recall some definitions
regarding root systems.

######  Definition 7.1.

A _(crystallographic) root system_ is a set $\Phi\subset{\mathbb{R}}^{n}$ such
that

  1. (a)

$\Phi$ spans ${\mathbb{R}}^{n}$;

  2. (b)

if $\alpha\in\Phi$, then $-\alpha\in\Phi$ and $\pm\alpha$ are the only
multiples of $\alpha$ in $\Phi$;

  3. (c)

$\Phi$ is closed under the reflection $\sigma_{\alpha}$ across the hyperplane
perpendicular to $\alpha$ for each $\alpha\in\Phi$;

  4. (d)

for all $\alpha,\beta\in\Phi$, one has

$2\dfrac{\langle\alpha,\beta\rangle}{\langle\alpha,\alpha\rangle}\in{\mathbb{Z}},$

where $\langle-,-\rangle$ is the standard inner product on ${\mathbb{R}}^{n}$.

Condition (d) is known as the _crystallographic condition_. This condition
gives the existence of root and weight lattices, which will be of use later.

######  Definition 7.2.

A subset $\Delta$ of a root system $\Phi$ is a choice of _simple roots_ if
$\Delta$ spans ${\mathbb{R}}^{n}$ and partitions the root system into those
roots lying in their positive integer span—the _positive roots_ , denoted
$\Phi^{+}$—and those lying in their negative integer span—the _negative roots_
, $\Phi^{-}$.

The discussion of signed posets will focus on two root systems, $\Phi_{B_{n}}$
and $\Phi_{C_{n}}$. The roots are

  * •

$\\{\pm e_{i}\pm e_{j}\colon i,j\in[n]\\}\cup\\{\pm e_{i}\colon i\in[n]\\}$
for $\Phi_{B_{n}}$, and

  * •

$\\{\pm e_{i}\pm e_{j}\colon i,j\in[n]\\}\cup\\{\pm 2e_{i}\colon i\in[n]\\}$
for $\Phi_{C_{n}}$.

The simple roots will be taken to be

  * •

$\\{+e_{i}-e_{i+1}\colon i=1,\ldots,n-1\\}\cup\\{+e_{n}\\}$ for
$\Phi_{B_{n}}$, and

  * •

$\\{+e_{i}-e_{i+1}\colon i=1,\ldots,n-1\\}\cup\\{+2e_{n}\\}$ for
$\Phi_{C_{n}}$.

This choice of simple roots gives the following positive roots.

  * •

$\\{+e_{i}+e_{j}\colon i,j\in[n]\\}\cup\\{+e_{i}-e_{j}\colon
i<j\\}\cup\\{+e_{i}\colon i\in[n]\\}$ for $\Phi_{B_{n}}$, and

  * •

$\\{+e_{i}+e_{j}\colon i,j\in[n]\\}\cup\\{+e_{i}-e_{j}\colon
i<j\\}\cup\\{+2e_{i}\colon i\in[n]\\}$ for $\Phi_{C_{n}}$.

Subsequent chapters will parallel work of [9] and [21] addressing the type A
case. The $\Phi_{A_{n-1}}$ roots are

$\\{e_{i}-e_{j}\colon i,j\in[n],i\neq j\\},$

and the choice of simple roots used is $\\{e_{i}-e_{i+1}\colon
i=1,\ldots,n-1\\}$, giving $\\{e_{i}-e_{j}\colon i,j\in[n],i<j\\}$ as the
positive roots.

######  Definition 7.3.

Given a root system $\Phi$, the _(integral) weights_ are the
$\mu\in{\mathbb{R}}^{n}$ such that, for each $\alpha\in\Phi$,

$2\dfrac{\langle\mu,\alpha\rangle}{\langle\alpha,\alpha\rangle}\in{\mathbb{Z}}.$

If one has made a choice of simple roots, say
$\\{\alpha_{1},\ldots,\alpha_{n}\\}$, there is then a distinguished set of
weights, the _fundamental dominant weights_ $\mu_{1},\ldots,\mu_{n}$ uniquely
defined by the conditions

$2\dfrac{\langle\mu_{i},\alpha_{j}\rangle}{\langle\alpha_{j},\alpha_{j}\rangle}=\delta_{ij},$

where $\delta_{ij}$ is the Kronecker delta.

For $\Phi_{B_{n}}$, the weights are the elements of ${\mathbb{Z}}^{n}$ plus
$\left(\pm\frac{m}{2},\pm\frac{m}{2},\ldots,\pm\frac{m}{2}\right)$ for
$m\in{\mathbb{Z}}$. For the choice of simple roots made above, the fundamental
dominant weights are

$(1,0,\ldots,0),(1,1,0,\ldots,0),\ldots,(1,\ldots,1,0),\left(\frac{1}{2},\ldots,\frac{1}{2}\right).$

For $\Phi_{C_{n}}$, the weights are the elements of ${\mathbb{Z}}^{n}$ and,
with the choice of simple roots made above, the fundamental dominant weights
are $(1,0,\ldots,0)$,$(1,1,0,\ldots,0),\ldots,(1,\ldots,1,0)$,$(1,\ldots,1)$.

######  Definition 7.4.

Suppose $\Phi$ is a root system. Its _dual root system_ ,
$\Phi^{\scriptscriptstyle\vee}$, is the root system whose roots are

$\alpha^{\scriptscriptstyle\vee}=\dfrac{2\alpha}{\langle\alpha,\alpha\rangle}$

for $\alpha\in\Phi$. The roots of $\Phi^{\scriptscriptstyle\vee}$ are the
_coroots_ of $\Phi$ and the weights of $\Phi^{\scriptscriptstyle\vee}$ are the
_coweights_ of $\Phi$.

One always has that
$\Phi^{{\scriptscriptstyle\vee}{\scriptscriptstyle\vee}}=\Phi$. Note that
$\Phi_{B_{n}}$ and $\Phi_{C_{n}}$ are dual root systems. The roots, weights
and coweights form lattices, called the _root_ , _weight_ and _coweight_
lattices. As an example, consider the root and weight/coweight lattices for
$\Phi_{B_{2}}$ and $\Phi_{C_{2}}$ in Figure 7.

|
---|---
|

(a) The $\Phi_{B_{2}}$ root lattice

(b) The $\Phi_{C_{2}}$ root lattice

(c) The $\Phi_{B_{2}}$ coweight/$\Phi_{C_{2}}$ weight lattice

(d) The $\Phi_{B_{2}}$ weight/$\Phi_{C_{2}}$ coweight lattice

Figure 7: Root and Weight Lattices

####  7.1 The Weyl Group and Signed Permutations

######  Definition 7.5.

The reflections $\sigma_{\alpha}$ across the hyperplanes $H_{\alpha}$
perpendicular to $\alpha\in\Phi$ form a group called the _Weyl group_ and
denoted by $W$.

If one removes the hyperplanes $H_{\alpha}$, one divides ${\mathbb{R}}^{n}$
into connected components which are open simplicial cones known as _Weyl
chambers_. The Weyl group $W$ acts simply transitively on the Weyl chambers.

The Weyl group of $\Phi_{A_{n-1}}$ is the familiar symmetric group $S_{n}$.
Since $\Phi_{B_{n}}$ and $\Phi_{C_{n}}$ are dual root systems they share the
same reflections and thus the same Weyl group. This is the group of _signed
permutations_ , known as the _hyperoctahedral group_.

######  Definition 7.6.

A signed permutation of $[n]$ is a permutation $w$ of $\\{\pm 1,\ldots,\pm
n\\}$ such that $w(-i)=-w(i)$.

Consequently, signed permutations may be written in a two-line notation
specifying the images of $\\{1,\ldots,n\\}$. For example,

$w=\begin{pmatrix}1&2&3&4\\\ 2&-3&4&-1\end{pmatrix}$ (1)

is the signed permutation where $w(1)=2$, $w(2)=-3$, $w(3)=4$, $w(4)=-1$ and
$w(-i)=-w(i)$ for $i=1,\ldots,4$.

The signed permutation analogues of a well-known permutation statistic will
prove useful in Chapter Root and weight semigroup rings for signed posets.
Name our choice of simple roots as $\alpha_{i}=e_{i}-e_{i+1}$ for
$i=1,\ldots,n-1$ and $\alpha_{n}=e_{n}$.

######  Definition 7.7.

A signed permutation $w$ is said to have a _descent_ at $i$ if
$w(\alpha_{i})\in\Phi^{-}$. The set of descents of $w$ is denoted
$\mathrm{Des}(w)$. The _major index_ of $w$ is

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

For example, the signed permutation $w$ shown in (1) has descents at $2$ and
$4$, so $\mathrm{maj}(w)=6$.

##  Chapter Signed Posets

This chapter will discuss background on signed posets and introduce the root
cone and weight cone.

Signed posets were introduced by Reiner in [47] and [48]. Before defining
signed posets, it is useful to define one piece of notation which will
reappear later.

######  Definition 7.1.

Suppose $\Phi$ is a root system and $S\subset\Phi$. Then the _positive linear
closure_ , $\overline{S}^{PLC}$, is the set of $\alpha\in\Phi$ which are non-
negative linear combinations of elements of $S$, i.e.
$\overline{S}^{PLC}={\mathbb{R}}_{+}S\cap\Phi$.

######  Definition 7.2.

If $\Phi$ is a root system, a _$\Phi$ -poset_ is a subset $P\subset\Phi$ such
that

  1. (a)

if $\alpha\in P$, then $-\alpha\notin P$

  2. (b)

$P=\overline{P}^{PLC}$.

A poset $P$ can be understood as the $\Phi_{A_{n-1}}$-poset
$\\{e_{i}-e_{j}\colon i<_{P}j\\}$. This view of posets reveals condition (a)
in the definition as the analogue of antisymmetry and condition (b) as the
analogue of transitivity.

The $\Phi$-posets are the _parsets_ of [47]. Since $\Phi_{B_{n}}$ and
$\Phi_{C_{n}}$ are dual root systems, $P\subset\Phi_{B_{n}}$ is a
$\Phi_{B_{n}}$-poset if and only if
${P^{\scriptscriptstyle\vee}}=\\{\alpha^{\vee}\colon\alpha\in
P\\}\subset\Phi_{C_{n}}$ is a $\Phi_{C_{n}}$-poset. Signed posets were defined
to be the $\Phi_{B_{n}}$-posets, but in light of this duality, both
$\Phi_{B_{n}}$\- and $\Phi_{C_{n}}$-posets will be called _signed posets_ ,
with $\Phi_{B_{n}}$ or $\Phi_{C_{n}}$ being specified when necessary.

One can check that
$P=\\{+e_{1}-e_{2},+e_{1}-e_{3},+e_{2}+e_{3},+e_{1}+e_{3},+e_{1}\\}\subset\Phi_{B_{3}}$
is a signed poset. In this case,
${P^{\scriptscriptstyle\vee}}=\\{+e_{1}-e_{2},+e_{1}-e_{3},+e_{2}+e_{3},+e_{1}+e_{3},+2e_{1}\\}\subset\Phi_{C_{3}}$.
On the other hand, $P\smallsetminus\\{+e_{1}\\}$ is _not_ a signed poset as it
is not closed under positive linear combinations remaining in $\Phi_{B_{3}}$
since $\frac{1}{2}((e_{1}-e_{3})+(e_{1}+e_{3}))=e_{1}$.

######  Definition 7.3.

Two signed posets $P$ and $P^{\prime}$ are _isomorphic_ if there is a signed
permutation $w$ such that $wP=P^{\prime}$.

Note that, strictly speaking, the definition of $\Phi$-poset allows a poset
$P\subset\Phi\subset{\mathbb{R}}^{m}$ where $P$ is supported on
$\\{e_{i}\colon i\in A\\}$ for some $A\subsetneq[m]$. In this case, $P$ is
isomorphic to some $P^{\prime}\subset\Phi^{\prime}\subset{\mathbb{R}}^{n}$ for
$|A|=n$.

###  8 Representing Signed Posets

There are two representations of signed posets, one as an oriented signed
graph and the other as a poset on $\pm[n]$, that prove useful in different
contexts. First, though, one needs to define oriented signed graphs.

######  Definition 8.1.

A _signed graph_ $\Sigma$ is a pair $(\Gamma,\sigma)$ where $\Gamma$ is a
graph with vertex set $V$ and edge set $E$ and $\sigma$ is a map $\sigma\colon
E\to\\{\pm\\}$, assigning a sign to each edge.

######  Definition 8.2.

An _oriented signed graph_ is a signed graph $\Sigma$ together with a
_bidirection_ , $\tau$, assigning signs to the incidences of $\Gamma$ in such
a way as to be compatible with $\sigma$, i.e. $\tau\colon
I(\Gamma)\to\\{\pm\\}$ such that

$\sigma(e)=-\tau(v,e)\tau(w,e)$

when $e$ is an edge between vertices $v$ and $w$ ($v$ and $w$ need not be
distinct).

As an example, consider the signed graph $\Sigma$ and an orientation of
$\Sigma$ by the bidirection $\tau$ in Figure 8. Notice that the bidirected
edges of $\Sigma$ correspond to elements of $\Phi_{C_{n}}$: the edge $(u,v)$
corresponds to $\tau(u)e_{u}+\tau(v)e_{v}$.

${1}$${2}$${3}$$+$$-$$+$$-$$+$ (a) The signed graph $\Sigma$

${1}$${2}$${3}$$+$$-$$+$$+$$+$$-$$+$$+$$+$$+$ (b) $\Sigma$ oriented by $\tau$

Figure 8: A signed graph and a bidirection

To construct the Hasse diagram of a signed poset, one first constructs an
oriented signed graph as follows. If $P\subset\Phi_{B_{n}}$ (resp.
$P\subset\Phi_{C_{n}}$), the vertices of the graph are $[n]$. The bidirected
edges are as in Table 1.

edge | poset element
---|---
$i$$j$$+$$-$ |  $e_{i}-e_{j}$$\;\in\;$ | $P$
$i$$j$$+$$+$ |  $e_{i}+e_{j}$$\;\in\;$ | $P$
$i$$j$$-$$-$ |  $-e_{i}-e_{j}$$\;\in\;$ | $P$
$i$$+$$+$ |  $e_{i}\in P$, $2e_{i}$$\;\in\;$ | ${P^{\scriptscriptstyle\vee}}$
$i$$-$$-$ |  $-e_{i}\in P$, $-2e_{i}$$\;\in\;$ | ${P^{\scriptscriptstyle\vee}}$
Table 1: Association between bidirected edges and elements of a signed poset

However, because signed posets are closed under positive linear combinations,
the presence of some elements is implied by others.

######  Definition 8.3.

Suppose $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) is a signed poset. Let
$\Gamma$ be the oriented signed graph whose edges are obtained from Table 1.
Let $\Sigma_{P}$ be the oriented signed graph obtained by removing the implied
edges from $\Gamma$. Then $\Sigma_{P}$ is the _Hasse diagram_ of $P$.

It is not immediately obvious that the Hasse diagram should be well-defined.
That it is well-defined is explained by [48, p. 329 ].

Consider the oriented signed graph in Figure 8(b). It corresponds to the
signed poset
$\\{+e_{1}-e_{2},+e_{1}-e_{3},+e_{2}+e_{3},+e_{1}+e_{3},+e_{1}\\}$. The Hasse
diagram is shown in Figure 9.

${1}$${2}$${3}$$+$$-$$+$$-$$+$$+$ Figure 9: The Hasse diagram of
$\\{+e_{1}-e_{2},+e_{1}-e_{3},+e_{2}+e_{3},+e_{1}+e_{3},+e_{1}\\}$

The other representation of a signed poset was introduced by [24] and then
again independently by [2].

######  Definition 8.4.

Suppose $P\subset\Phi_{B_{n}}$ and
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$ are a pair of dual signed
posets. Define a poset $\widehat{G}_{B}(P)$ on
$[-n,n]=\\{-n,\ldots,1,0,1,\ldots,n\\}$ by taking the transitive closure of
the relations determined by Table 2. Similarly, define a poset
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$ on $\pm[n]=\\{\pm 1,\ldots,\pm
n\\}$ by taking the transitive closure of the relations determined by Table 3.

$\begin{array}[]{r@{\;<\;}lcr@{\;<\;}lcl}i&j&\text{and}&-j&-i&\text{for}&+e_{i}-e_{j}\in
P\\\ i&-j&\text{and}&j&-i&\text{for}&+e_{i}+e_{j}\in P\\\
-i&j&\text{and}&-j&i&\text{for}&-e_{i}-e_{j}\in P\\\
i&0&\text{and}&0&-i&\text{for}&+e_{i}\in P\\\
-i&0&\text{and}&0&i&\text{for}&-e_{i}\in P\\\ \end{array}$ Table 2: Relations
defining $\widehat{G}_{B}(P)$
$\begin{array}[]{r@{\;<\;}lcr@{\;<\;}lcl}i&j&\text{and}&-j&-i&\text{for}&+e_{i}-e_{j}\in
P\\\ i&-j&\text{and}&j&-i&\text{for}&+e_{i}+e_{j}\in P\\\
-i&j&\text{and}&-j&i&\text{for}&-e_{i}-e_{j}\in P\\\ \pm i&\mp
i&\lx@intercol\hfil\text{for}\lx@intercol&\pm 2e_{i}\in P\\\ \end{array}$
Table 3: Relations defining $\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$

Understand $\widehat{G}(P)$ to mean either $\widehat{G}_{B}(P)$ or
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$ as appropriate. In Chapter
Root and weight semigroup rings for signed posets it will be convenient to
revert to Fischer’s original definition and use
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$ to represent a
$P\subset\Phi_{B_{n}}$. Proposition 9.3 shows that this abuse is acceptable.
We will call $\widehat{G}(P)$ the _Fischer poset_ of the signed poset. The
Fischer posets for
$P^{\star}=\\{+e_{1}-e_{2},+e_{1}-e_{3},+e_{2}+e_{3},+e_{1}+e_{3},+e_{1}\\}$
are shown in Figure 10. (This poset is being named $P^{\star}$ since it will
feature as an example in this section and the next.) Both Fischer and Ando,
Fujishige and Nemoto defined $\widehat{G}_{C}(P)$ as being associated to
$P\subset\Phi_{B_{n}}$. However, adding the $0$ vertex in type B and defining
$\widehat{G}_{B}(P)$ and $\widehat{G}_{C}(P)$ separately simplifies Chapter
Root and weight semigroup rings for signed posets.

$1$$2$$3$$-3$$-2$$0$$-1$ (a) $\widehat{G}_{B}(P^{\star})$

$1$$2$$3$$-3$$-2$$-1$ (b) $\widehat{G}_{C}(P^{\star{\scriptscriptstyle\vee}})$

Figure 10: $\widehat{G}_{B}(P)$ and $\widehat{G}_{C}(P)$ for
$P^{\star}=\\{+e_{1}-e_{2},+e_{1}-e_{3},+e_{2}+e_{3},+e_{1}+e_{3},+e_{1}\\}$

Both $\widehat{G}_{B}(P)$ and $\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$
are equipped with involutions, denoted $\iota$, sending $i$ to $-i$ and the
edge $i\to j$ to $-j\to-i$, making $\iota$ a poset anti-automorphism. In
$\widehat{G}_{B}(P)$, this involution fixes $0$, and in
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$ it fixes edges of the form
$i\to-i$. That $\widehat{G}_{B}(P)$ and
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$ are equipped with this
involution means that they connect to the theory of coverings of signed
graphs. This will be discussed in Section 14. Somewhat confusingly, the
oriented signed graphs which they cover are _not_ necessarily the Hasse
diagram just described in Definition 8.3. The Hasse diagram will be useful in
Section 12 and then dispensed with until Chapter Root and weight semigroup
rings for signed posets, as the Fischer poset will prove to be the more
convenient notion, lending itself to analogies to the type A case of [9] and
[21].

Note that when $P$ does not contain an element of the form $\pm e_{i}$, the
element $0$ is not comparable to any other elements of $\widehat{G}(P)$.

The following two propositions characterizing $\widehat{G}_{B}(P)$ and
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$ are immediate consequences of
the definitions.

######  Proposition 8.5.

A poset on $[-n,n]$ is $\widehat{G}_{B}(P)$ of a $\Phi_{B_{n}}$-poset $P$ if
and only if the following two conditions hold:

  * •

for all $i,j\in\\{-n,\ldots,n\\}$, if $i<j$ then $-j<-i$.

  * •

for all $i\in\\{-n,\ldots,n\\}$, if $i<-i$ then $i<0<-i$.

######  Proposition 8.6.

A poset on $\pm[n]$ is $\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$ for a
$\Phi_{C_{n}}$-poset ${P^{\scriptscriptstyle\vee}}$ if and only if the
following two conditions hold:

  * •

if $i<j$ then $-j<-i$ for all $i,j\in\pm[n]$ and

  * •

if $i<-i$ and $j<-j$ then $i<-j$ and $j<-i$ for all $i,j\in\pm[n]$.

######  Definition 8.7.

A signed poset $P^{\prime}$ is an _induced subposet_ of another signed poset
$P$ if $P^{\prime}\subset P$ and there is an $A\subsetneq[n]$ such that
$P^{\prime}\subset P\cap\text{span}_{\mathbb{R}}\\{e_{i}\colon i\in A\\}$.

Equally, $P^{\prime}$ is an induced subposet of $P$ if
$\widehat{G}(P^{\prime})$ is an induced subposet of $\widehat{G}(P)$.

###  9 Ideals and $P$-partitions in Signed Posets

Recall that an order ideal of a poset $P$ is a subset $I\subset P$ such that
if $x\in I$ and $y<x$, then $y\in I$. On the other hand, when one views a
poset $P$ on $\\{1,\ldots,n\\}$ as a collection of roots in $\Phi_{A_{n-1}}$,
the characteristic vectors of the ideals are precisely those
$J\in\\{1,0\\}^{n}$ such that $\langle J,\alpha\rangle\geq 0$ for all
$\alpha\in P$. This view motivates Boussicault, Féray, Lascoux and Reiner’s
definition of the weight cone in [9].

In [48], Reiner defined an _ideal_ of a signed poset $P\subset\Phi_{B_{n}}$ to
be a $J\in\\{1,-1,0\\}^{n}$ such that $\langle J,\alpha\rangle\geq 0$ for all
$\alpha\in P$. The two key observations to generalize these definitions to
$\Phi$-posets are

  * •

the projections of $\\{1,0\\}^{n}$ to the hyperplane $\sum_{i=1}^{n}x_{i}=0$
(i.e. the hyperplane of ${\mathbb{R}}^{n}$ in which $\Phi_{A_{n-1}}$ lives)
form the orbit of the $\Phi_{A_{n-1}}$-fundamental coweights under the action
of the Weyl group (recall that $\Phi_{A_{n-1}}$ is dual to itself, so its
weights and coweights are the same), and

  * •

$\\{1,-1,0\\}^{n}$ is the orbit of the fundamental coweights of $\Phi_{B_{n}}$
under the action of the Weyl group.

######  Definition 9.1.

Suppose $P\subset\Phi$ is a $\Phi$-poset. An _ideal_ of $P$ is an $f$ in the
orbit of the fundamental weights under the action of the Weyl group such that
$\langle f,\alpha\rangle\geq 0$ for all $\alpha\in P$. The set of ideals of
$P$ is denoted $J(P)$, as for posets.

As an example, consider once again the poset $P^{\star}$ shown in Figure 10.
The ideals of $P^{\star}$ are the elements of the poset shown in Figure 11.
One will note that these are the characteristic vectors of some of the order
ideals of $\widehat{G}(P)$.

######  Definition 9.2.

Suppose $P\subset\Phi_{B_{n}}$ (resp. $P\subset\Phi_{C_{n}}$) is a signed
poset. An order ideal, $I$, of $\widehat{G}(P)$ is said to be _isotropic_ when
the following two conditions hold:

  * •

$0\notin I$ (when $P\subset\Phi_{B_{n}})$

  * •

if $i\in I$, then $-i\notin I$ for $i\in\pm[n]$

Fischer and later Ando, Fujishige and Nemoto showed that the isotropic order
ideals correspond to the ideals of a signed poset. It turns out that it is
more convenient to look only at
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$ when talking about the ideals.
The correspondence between ideals of a signed poset and isotropic order ideals
and the fact that examining $\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$
suffices is encapsulated in the following proposition.

######  Proposition 9.3.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset and
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$ is the poset consisting of
the corresponding dual roots. Then

  1. (a)

the ideals of $P$ are the characteristic vectors of the isotropic order ideals
of $\widehat{G}_{B}(P)$,

  2. (b)

the ideals of ${P^{\scriptscriptstyle\vee}}$ are the characteristic vectors of
the isotropic order ideals of $\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$
with the exception that the order ideals of size $n$ correspond to one half
times their characteristic vectors and

  3. (c)

the set of isotropic order ideals of $\widehat{G}_{B}(P)$ is the same as the
set of isotropic order ideals of
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$, and the set of connected
isotropic order ideals of $\widehat{G}_{B}(P)$ is the same as the set of
connected isotropic order ideals of
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$.

In light of this view of the ideals of a signed poset, the vector of an ideal
and the corresponding (isotropic) set of elements of $\pm[n]$ will be used
interchangeably.

###### Proof.

  1. (a)

Suppose $J$ is an isotropic order ideal of $\widehat{G}_{B}(P)$. Suppose
$\chi_{J}$ is not an ideal of $f$. Then there is an $\alpha\in P$ such that
$\langle\chi_{J},\alpha\rangle<0$. For ease of notation, let
$\epsilon=\chi_{J}$. Note that for all $i$, $\epsilon_{i}\in\\{0,1,-1\\}$.
There are five cases.

     * •

Suppose $\alpha=e_{i}-e_{j}$. Then $\epsilon_{i}-\epsilon_{j}<0$. If
$\epsilon_{i}=0$, then $\epsilon_{j}=1$ so $j\in J$. However, one knows (from
$\alpha$) that $i<j$ in $\widehat{G}_{B}(P)$. If $\epsilon_{i}=-1$, then
$\epsilon_{j}\geq 0$. Therefore, $-i\in J$. However, $-j<-i$ in
$\widehat{G}_{B}(P)$, so $-j\in J$, contradicting that $J$ was isotropic.

     * •

Suppose $\alpha=e_{i}+e_{j}$. Then $\epsilon_{i}+\epsilon_{j}<0$. Without loss
of generality suppose $\epsilon_{i}=-1$, so $-i\in J$. Therefore, since $j<-i$
in $\widehat{G}_{B}(P)$, so $j\in J$, meaning $\epsilon_{j}=1$, a
contradiction since then $\epsilon_{i}+\epsilon_{j}=0$.

     * •

Suppose $\alpha=+e_{i}$. Then $\epsilon_{i}<0$, so $-i\in J$. Since $i<-i$ in
$\widehat{G}_{B}(P)$, one has $i\in J$, contradicting that $J$ was isotropic.

     * •

Suppose $\alpha=-e_{i}-e_{j}$. The symmetric argument to the case of
$\alpha=e_{i}+e_{j}$ shows this is impossible.

     * •

Suppose $\alpha=-e_{i}$. The symmetric argument to the case of $\alpha=e_{i}$
shows this is impossible.

Now suppose $f\in L^{\mathrm{cowt}}_{B}$ is an ideal. By construction, the set
in $\widehat{G}_{B}(P)$ corresponding to $f$, call it $J$, is isotropic.
Suppose $J$ is not an order ideal. Then there is $i,j\in[n]$ and
$\delta,\epsilon\in\\{\pm\\}$ such that $\delta i\lessdot\epsilon j$ and
$\epsilon j\in J$ and $\delta i\notin J$. In particular, $j\neq 0$. Then
$\delta e_{i}-\epsilon j\in P$ and $\langle\delta e_{i}-\epsilon
e_{j},f\rangle<0$, contradicting that $f$ was an ideal.

  2. (b)

The argument for type B also works for type C.

  3. (c)

Suppose $J$ is an isotropic order ideal in $\widehat{G}_{B}(P)$. Since $J$ is
isotropic, $0\notin J$. Consider $J$ as a subset of
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$. Suppose it is not an order
ideal. Then there is an $i\in\pm[n]$ and $j\in J$ such that $i<j$ and $i\notin
J$. Without loss of generality, one may assume $i,j>0$. Then, since $i<j$, one
has that $e_{i}-e_{j}\in{P^{\scriptscriptstyle\vee}}$, meaning $e_{i}-e_{j}\in
P$. However, since $i\notin J$, one must have $\langle
e_{i}-e_{j},\chi_{J}\rangle=-1$, contradicting that $J$ is an ideal of $P$.

The argument when $J$ is an isotropic order ideal in
$\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$ is exactly the same, except
one must also account for the possibility that $i=0$. However, if $i=0$, one
must have $-j<j$ by Proposition 8.5.

∎

######  Definition 9.4.

An ideal of a signed poset $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) will be called _connected_
if it corresponds to a connected isotropic order ideal in $\widehat{G}(P)$.
Denote the set of connected order ideals by $J_{\mathrm{conn}}(P)$.

See Figure 13 for an example of a signed poset where not every ideal is
connected.

In [48], an order was defined on the ideals of a signed poset by extending
componentwise the order $0<1,-1$. This order on the ideals corresponds to
ordering the corresponding isotropic order ideals by inclusion. As in the case
of posets, this order gives a meet-semilattice of ideals, denoted, like the
set of ideals, $J(P)$. Figure 11 shows $J(P)$ for
$P=\\{+e_{1}-e_{2},+e_{1}-e_{3},+e_{2}+e_{3},+e_{1}+e_{3},+e_{1}\\}$.

$(0,0,0)$$(1,0,0)$$(1,1,0)$$(1,0,1)$$(1,1,-1)$$(1,1,1)$$(1,-1,1)$$+1$$+2$$+3$$-3$$+3$$+2$$-2$
Figure 11: $J(P^{\star})$ for
$P^{\star}=\\{+e_{1}-e_{2},+e_{1}-e_{3},+e_{2}+e_{3},+e_{1}+e_{3},+e_{1}\\}$
with the edges annotated with the difference between the ideals

The definition of $P$-partition for a signed poset is the analogue of that for
a poset when a poset is viewed as an $\Phi_{A_{n-1}}$-poset. Recall that, from
this perspective, a $P$-partition is an $f\in{{\mathbb{N}}}^{n}$ such that
$\langle f,\alpha\rangle\geq 0$ for all $\alpha\in P$. In [48], Reiner defined
a $P$-partition for a signed poset $P\subset\Phi_{B_{n}}$ to be an
$f\in{\mathbb{Z}}^{n}$ such that $\langle f,\alpha\rangle\geq 0$ for all
$\alpha\in P$.

One makes a similar pair of observations regarding the $P$-partitions as one
did regarding ideals:

  * •

the projections of ${{\mathbb{N}}}^{n}$ to the hyperplane
$\sum_{i=0}^{n}x_{i}=0$ in ${\mathbb{R}}^{n}$ are the $\Phi_{A_{n-1}}$
coweights, and

  * •

the ${\mathbb{Z}}^{n}$ are the $\Phi_{B_{n}}$ coweights.

######  Definition 9.5.

Suppose $P$ is a $\Phi$-poset. A $P$-partition is an $f$ in the coweight
lattice of $\Phi$ such that $\langle f,\alpha\rangle\geq 0$ for all $\alpha\in
P$. The set of $P$-partitions is denoted $\mathcal{A}(P)$.

###  10 Linear Extensions

######  Definition 10.1.

A _linear extension_ of a $\Phi$-poset $P$ is an element $w$ of the Weyl group
of $\Phi$ such that $P\subset w\Phi^{+}$. The set of linear extensions is
denoted $\mathcal{L}(P)$.

In the case of signed posets, the Weyl group consists of the signed
permutations, as explained in Section 7.

######  Definition 10.2.

A signed poset is said to be _naturally labelled_ if the identity signed
permutation is a linear extension.

Note that since every signed poset has a linear extension (see the proof of
[48, Theorem 3.3]), every signed poset is isomorphic to a naturally labeled
signed poset. Thus, it is not deceptive that examples will usually be
naturally labelled.

The linear extensions can be read off $J(P)$ by recording the difference
between successive ideals in the maximal chains of $J(P)$. In the running
example of $P^{\star}$, these differences were noted in Figure 11. One sees
that $P^{\star}$ is naturally labelled and the linear extensions are

$\begin{pmatrix}1&2&3\\\ 1&2&3\\\ \end{pmatrix}\quad\begin{pmatrix}1&2&3\\\
1&3&2\\\ \end{pmatrix}\quad\begin{pmatrix}1&2&3\\\ 1&2&-3\\\
\end{pmatrix}\quad\begin{pmatrix}1&2&3\\\ 1&-2&3\\\ \end{pmatrix}$

In [48], Reiner proved an analogue of Stanley’s Fundamental Theorem of
$P$-partitions for $\Phi$-posets.

######  Proposition 10.3 ([48, Theorem 3.3]).

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

$\mathcal{A}(P)=\bigsqcup_{w\in\mathcal{L}(P)}\mathcal{A}(w\Phi^{+}).$

###  11 Embedding Posets as Signed Posets

In [48], Reiner defined the embedding of a poset as a signed poset. Recall
that a poset $P$ on $[n]$ can be viewed as a $\Phi_{A_{n-1}}$-poset by taking
$P=\\{e_{i}-e_{j}\colon i<_{P}j\\}$. The poset $P$ is then embedded in
$\Phi_{B_{n}}$ as the signed poset $P_{B}=P\cup\\{+e_{i}\colon
i\in[n]\\}\cup\\{+e_{i}+e_{j}\colon i,j\in[n]\\}$. Then
$P_{C}=(P\cup\\{+e_{i}\colon i\in[n]\\}\cup\\{+e_{i}+e_{j}\colon
i,j\in[n]\\})^{\scriptscriptstyle\vee}=P\cup\\{+2e_{i}\colon
i\in[n]\\}\cup\\{+e_{i}+e_{j}\colon i,j\in[n]\\}$. Call signed posets arising
in this manner _type A signed posets_. The ideals of $P_{B}$ are precisely the
ideals of $P$. Likewise, $P_{B}$ and $P$ share the same linear extensions (in
the sense that any linear extension of $P_{B}$ and of $P$ have the same action
on $[n]$).

######  Proposition 11.1.

Suppose $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) is such that $i<-i$ in
$\widehat{G}(P)$ for all $i\in[n]$. Then $P$ is a type A signed poset.

$1$$2$$3$ (a) $P$

$1$$2$$3$$-3$$-2$$3$$0$ (b)
$P\cup\\{+e_{1}+e_{2},+e_{1}+e_{3},+e_{2}+e_{3},+e1,+e_{2},+e_{3}\\}$

Figure 12: The poset $P=\\{e_{1}-e_{2},e_{1}-e_{3}\\}$ and its embedding as a
signed poset

As an example, consider the posets shown in Figure 12. $P$ is the intersection
of $P\cup\\{+e_{1}+e_{2},+e_{1}+e_{3},+e_{2}+e_{3},+e1,+e_{2},+e_{3}\\}$ with
the hyperplane $\\{x\in{\mathbb{R}}^{n}\colon x_{1}+\cdots+x_{n}=0\\}$. The
ideals of $P$ as a signed poset are precisely the ideals of $P$ as a poset.

###  12 The Root and Weight Cones and Their Extreme Rays

Associated to a signed poset $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) are two polyhedral cones:
the _root cone_ , denoted $K_{P}^{\mathrm{rt}}$, and the _weight cone_ ,
denoted $K_{P}^{\mathrm{wt}}$.

######  Definition 12.1.

Suppose $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) is a signed poset. Its
_root cone_ is the positive linear span of its elements:
$K_{P}^{\mathrm{rt}}={\mathbb{R}}_{+}P$.

Its _weight cone_ is the dual to the root cone:
$K_{P}^{\mathrm{wt}}=\\{f\in{\mathbb{R}}^{n}\colon\langle f,\alpha\rangle\geq
0\ \forall\alpha\in P\\}$.

Observe that if $P\subset\Phi_{B_{n}}$ and
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$ are dual posets, $P$ and
${P^{\scriptscriptstyle\vee}}$ share the same root and weight cones. The
definition of Hasse diagram gives a characterization of the extreme rays of
the root cone.

######  Proposition 12.2.

Suppose $P$ is a signed poset. Its root cone is an affine polyhedral cone
whose extreme rays are given by the edges of the Hasse diagram.

Proposition 12.2 generalizes the type A result of [9, Proposition 5.1(i) ].

######  Proposition 12.3.

Suppose $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) is a type A signed poset.
Let $P^{\prime}=P\cap\Phi_{A_{n-1}}$ be the poset on $[n]$ which $P$ embeds.
Then

$K^{\mathrm{rt}}_{P^{\prime}}=K_{P}^{\mathrm{rt}}\cap\\{x\in{\mathbb{R}}^{n}\colon
x_{1}+\cdots+x_{n}=0\\},$

and furthermore, extreme rays of $K^{\mathrm{rt}}_{P^{\prime}}$ are extreme
rays of $K_{P}^{\mathrm{rt}}$.

###### Proof.

Inspecting the definitions, one sees immediately that
$K^{\mathrm{rt}}_{P^{\prime}}\subset
K_{P}^{\mathrm{rt}}\cap\\{x\in{\mathbb{R}}^{n}\colon x_{1}+\cdots+x_{n}=0\\}$.

The extreme rays of $K^{\mathrm{rt}}_{P^{\prime}}$ correspond to the covering
relations of the Hasse diagram of $P^{\prime}$. To show that each covering
relation $\alpha\in P^{\prime}$ is an extreme ray of $K_{P}^{\mathrm{rt}}$, it
suffices to check that there cannot exist $\beta,\gamma\in P$ such that
$\gamma\in\Phi_{B_{n}}^{+}$ (resp. $\gamma\in\Phi_{C_{n}}^{+}$) and
$\alpha=\beta+\gamma$. However, since $\alpha=e_{i}-e_{j}$, this is
impossible. ∎

Characterizing the extreme rays of $K_{P}^{\mathrm{wt}}$ requires an
additional definition and a lemma regarding ideals.

######  Definition 12.4.

An ideal $J$ of $P$ is said to be _extensible_ if there is a set $I\subset[n]$
such that $J\cup I$ and $J\cup-I$ are both ideals of $P$. An ideal which is
not extensible is called _non-extensible_.

For example, in $P^{\star}$ (Figure 10), the ideal $\\{1,2\\}$ is extensible
because both $\\{1,2,3\\}$ and $\\{1,2,-3\\}$ are ideals of $P^{*}$. However,
it is not obligatory that $J\cup I$ and $J\cup-I$ be connected, nor that $J$
be connected. If one considers instead
$P=\\{+e_{1}-e_{2},+e_{1}+e_{2},+e_{1}-e_{3},+e_{2}-e_{3},+e_{1}\\}$ (see
Figure 13 for $\widehat{G}(P)$ and recall that when thinking about ideals, one
can always refer to $\widehat{G}_{C}({P^{\scriptscriptstyle\vee}})$), the
ideal $\\{1,2\\}$ is extensible, as $\\{1,2,3\\}$ and $\\{1,2,-3\\}$ are
ideals, though $\\{1,2,-3\\}$ is not connected. Similarly, $\\{1,3\\}$ is
extensible and disconnected.

$1$$-3$$2$$-2$$3$$-1$ Figure 13: $\widehat{G}(P)$ for
$P=\\{+e_{1}-e_{2},+e_{1}+e_{2},+e_{1}-e_{3},+e_{2}-e_{3},+e_{1}\\}$

######  Proposition 12.5.

Suppose $P$ is a signed poset. Its weight cone $K_{P}^{\mathrm{wt}}$ is an
affine polyhedral cone whose extreme rays correspond to the connected, non-
extensible ideals of $P$.

Proposition 12.5 generalizes [9, Proposition 5.1(ii)].

######  Proposition 12.6.

Suppose $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) is a type A signed poset.
Let $P^{\prime}=P\cap\Phi_{A_{n-1}}$. Then
$K_{P}^{\mathrm{wt}}=K_{P^{\prime}}^{\mathrm{wt}}$.

###### Proof.

It suffices to show that the ideals of $P^{\prime}$ are precisely the ideals
of $P$. (If one is considering
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$, let $P$ be the dual poset.)
Certainly ideals of $P$ are ideals of $P^{\prime}$ since $P^{\prime}\subset
P$. Suppose $f$ is an ideal of $P^{\prime}$. meaning $\langle
f,\alpha\rangle\geq 0$ for all $\alpha\in\Phi_{B_{n}}^{+}$. Since
$P=P^{\prime}\cup\Phi_{B_{n}}^{+}$, one then has that $f$ is an ideal of $P$.
Thus $K_{P}^{\mathrm{wt}}=K_{P^{\prime}}^{\mathrm{wt}}$. ∎

Figure 14 shows the root and weight cones of
$\\{+e_{1}-e_{2},+e_{1}+e_{2},+e_{1}-e_{3},+e_{2}-e_{3},+e_{1}\\}$.

$y$$z$$(1,-1,0)$$(0,1,-1)$$(1,1,0)$$x$ (a) $K_{P}^{\mathrm{rt}}$

$y$$z$$x$$(1,1,-1)$$(1,1,1)$$(1,-1,-1)$ (b) $K_{P}^{\mathrm{wt}}$

Figure 14: The root cone and weight cone of
$\\{+e_{1}-e_{2},+e_{1}+e_{2},+e_{1}-e_{3},+e_{2}-e_{3},+e_{1}\\}$

The proof of Proposition 12.5 requires the following lemma.

######  Lemma 12.7.

Suppose $P\subset\Phi_{B_{n}}$ is a signed poset and
$f=(f_{1},f_{2},\ldots,f_{n})\in K_{P}^{\mathrm{wt}}$. Let $c_{1}\leq
c_{2}\leq\cdots\leq c_{k}$ be the _nonzero_ $|f_{i}|$ arranged in increasing
order. Then

$J_{k}=\\{\mathrm{sgn}(f_{i})i\colon i\in[n],|f_{i}|\geq c_{k}\\}$

is an ideal.

As an example, consider the signed poset in Figure 13. One has $c_{1}=1$,
$c_{2}=2$ and $c_{3}=3$. Writing $J_{1},J_{2},J_{3}$ in a tower as shown in
[48], one has

$\begin{array}[]{lr}J_{3}&(1,0,\phantom{-}0)\\\ J_{2}&(1,1,\phantom{0}0)\\\
J_{1}&+(1,1,-1)\\\ \hline\cr f&(3,2,-1)\end{array}$

######  Definition 12.8.

The ideal $J_{1}$ from Lemma 12.7 is the _signed support_ of $f$, denoted
$\mathop{\mathrm{supp}_{\pm}}(f)$.

###### Proof.

Suppose $J_{k}$ is not an ideal, so it is does not correspond to an order
ideal in $\widehat{G}(P)$ (it is isotropic by construction). Then there is a
$\pm k$ such that $\pm i\in J_{k}$ with $\pm k<\pm i$. Since $\pm k\notin
J_{k}$, $|f_{i}|\geq c_{k}>|f_{k}|$. One must consider several cases, one for
each of the possible combinations of signs of $\pm i$ and $\pm k$.

  * •

Suppose $+k<+i$. Then $e_{k}-e_{i}\in P$. Then $\langle
f,e_{k}-e_{i}\rangle=f_{k}-f_{i}\leq|f_{k}|-f_{i}<0$, a contradiction.

  * •

Suppose $-k<+i$. Then $-e_{k}-e_{i}\in P$. Then $\langle
f,-e_{k}-e_{i}\rangle=-f_{k}-f_{i}<0$, since $0<|f_{k}|<f_{i}$, a
contradiction.

  * •

Suppose $+k<-i$. Then $e_{k}+e_{i}\in P$. In this case, $f_{i}<0$ and
$|f_{i}|>|f_{k}|$. Then $\langle
f,e_{k}+e_{i}\rangle=f_{k}+f_{i}=f_{k}-|f_{i}|<0$, a contradiction.

  * •

Suppose $-k<-i$. Then $e_{i}-e_{k}\in P$. Then $\langle
f,e_{i}-e_{k}\rangle=f_{i}-f_{k}\leq-|f_{i}|+|f_{k}|<0$, a contradiction.

Thus, $J_{k}$ must be closed under going down in $\widehat{G}(P)$. Recall
that, it is isotropic by construction, so $J_{k}$ is an ideal of $P$. ∎

###### Proof of Proposition 12.5.

First, one shows that $K_{P}^{\mathrm{wt}}$ is spanned by the ideals of $P$,
then that the connected ideals suffice to span $K_{P}^{\mathrm{wt}}$ and,
finally, that the connected nonextensible ideals suffice.

Suppose $f=(f_{1},\ldots,f_{n})\in K_{P}^{\mathrm{wt}}$ and $c_{i}$ and
$J_{i}$ are as in Lemma 12.7. Then

$f=\sum_{k=1}^{n}(c_{k}-c_{k-1})\chi_{J_{k}},$

taking $c_{0}=0$. The $J_{k}$ can be decomposed into their connected
components, showing $K_{P}^{\mathrm{wt}}$ is spanned by the connected ideals
of $P$. Since there are only finitely many ideals, the definition of
extensibility means every extensible ideal can be written as a positive linear
combination of connected nonextensible ideals. Thus, the nonextensible
connected ideals span $K_{P}^{\mathrm{wt}}$.

To show that the nonextensible connected ideals of $P$ are the extreme rays of
$K_{P}^{\mathrm{wt}}$, one can show that every such ideal lies in the
intersection of $n-1$ (linearly independent) hyperplanes, each supporting
$K_{P}^{\mathrm{wt}}$. Let $J$ be a connected nonextensible ideal and consider
a spanning tree $T$ of its Hasse diagram (as a subposet of $\widehat{G}(P)$).
Then $J$ lies in the intersection of the following $n-1$ hyperplanes:

$\begin{array}[]{ll}x_{i}=0&\text{for }\pm i\notin J\\\ x_{i}=x_{j}&\text{for
}+i\lessdot+j\text{ or }-j\lessdot-i\in T\ (+e_{i}-e_{j}\in P)\\\
x_{i}=-x_{j}&\text{for }+i\lessdot-j\text{ or }-i\lessdot+j\in T\
(\pm(e_{i}+e_{j})\in P)\end{array}$

The above could, _a priori_ , fail to specify $n-1$ hyperplanes in one of two
ways. First, it may specify both $x_{i}=x_{j}$ and $x_{i}=-x_{j}$ as
hyperplanes, which really would specify only the hyperplane $x_{i}=0$.
However, this situation cannot arise since $J$ is isotropic. The second
possible problem is that a single hyperplane is specified by two different
edges in $T$. Again, this is not possible since $J$ is isotropic, so for each
pair $(i,j)$ there can only be one edge in $T$ involving $\pm i$ and $\pm j$.
Thus the above truly specifies $n-1$ linearly independent hyperplanes
supporting $K_{P}^{\mathrm{wt}}$ such that $J$ lies in their intersection,
meaning $J$ is an extreme ray of $K_{P}^{\mathrm{wt}}$, as claimed. ∎

####  12.1 Dual Characterizations

Since the root cone and weight cone are dual to one another, one should have
that a characterization of when $K_{P}^{\mathrm{rt}}$ is pointed gives a
characterization of when $K_{P}^{\mathrm{wt}}$ is full-dimensional and vice
versa. Equally, characterizing when $K_{P}^{\mathrm{rt}}$ is simplicial gives
a characterization of when $K_{P}^{\mathrm{wt}}$ is simplicial and vice versa.

As a consequence of the antisymmetry condition in the definition of signed
poset, $K_{P}^{\mathrm{rt}}$ is always pointed, meaning $K_{P}^{\mathrm{wt}}$
is always full-dimensional. One can also exhibit $n$ linearly independent
ideals of a signed poset $P$.

######  Lemma 12.9.

Suppose $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) is a signed poset. For each
$i\in[n]$, at least one of $I_{\leq}(i)=\\{j\in\widehat{G}(P)\colon j\leq
i\\}$ and $I_{\leq}(-i)=\\{j\in\widehat{G}(P)\colon j\leq-i\\}$ is isotropic.

This lemma introduces some notation that will be used again. In addition to
$I_{\leq}(i)$, let $I_{<}(i)=\\{i\in\widehat{G}(P)\colon j<i\\}$.

###### Proof of Lemma 3.4.4.

Suppose not and neither $I_{\leq}(i)$ nor $I_{\leq}(-i)$ is isotropic. Then
there are $j$ and $k$, not necessarily distinct, such that $j,-j<i$ and
$k,-k<-i$. Then, since $j<i$, one has that $-i<-j$ and since $-j<-i$ one has
that $-i<j$. Transitivity means that $-i<i$. On the other hand, since $k<-i$,
one has that $i<-k$ and since $-k<-i$ one has that $i<k$. Then $i<-i$. It is
impossible that both $i<-i$ and $-i<i$, so at least one of $I_{\leq}(i)$ and
$I_{\leq}(-i)$ is isotropic. ∎

######  Proposition 12.10.

Suppose $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) is a signed poset. Then
$K_{P}^{\mathrm{wt}}$ is full-dimensional.

###### Proof.

Fix a linear extension, $\prec$, of $\widehat{G}(P)$. For each $i$, pick an
ideal $J_{i}$ as follows:

$J_{i}=\begin{cases}I_{\leq}(i)&\text{if }I_{\leq}(-i)\text{ is not
isotropic}\\\ I_{\leq}(-i)&\text{if }I_{\leq}(i)\text{ is not isotropic}\\\
I_{\leq}(i)&\text{if both are isotropic and }i\prec-i\\\ I_{\leq}(-i)&\text{if
both are isotropic and }-i\prec i\\\ \end{cases}$

The key observation is that $J_{i}$ contains only elements of $\pm[n]$ that
precede $i$ in the $\prec$ order. Writing the $J_{i}$ as the rows of a matrix
(ordered by $\prec$), one has a lower triangular matrix with $\pm 1$ on the
diagonal, meaning the $J_{i}$ are linearly independent. Since
$K_{P}^{\mathrm{wt}}$ contains $n$ linearly independent vectors, it must be
full-dimensional. ∎

####  12.2 Matroids

In contrast to Proposition 12.10, the root cone is not always full-dimensional
and the weight cone is not always pointed. Understanding when the root cone is
full-dimensional, the weight cone is pointed and when both the root cone and
weight cone are simplicial requires some facts from the theory of signed
graphic matroids developed by [64]. This section will review the two
equivalent definitions of a matroid that will be useful.

######  Definition 12.11.

A _matroid_ is a pair $(E,\mathcal{I})$ where $E$ is a finite set called the
_ground set_ and $\mathcal{I}$ is a collection of subsets called the
_independent sets_ of $E$ such that

  1. (a)

$\mathcal{I}$ is nonempty.

  2. (b)

If $I\in\mathcal{I}$ and $J\subset I$, then $J\in\mathcal{I}$.

  3. (c)

If $I,J\in\mathcal{I}$ and $|I|=|J|+1$, there is an $x\in I\smallsetminus J$
such that $J\cup\\{x\\}\in\mathcal{I}$.

Subsets of $E$ which are not independent are said to be _dependent_.

That the elements of $\mathcal{I}$ are called the independent sets is not a
coincidence—these are the properties defining the collection of sets of
linearly independent columns of a matrix.

Whitney introduced matroids in [62] and gave a number of equivalent
definitions. Rather than defining a matroid as a pair $(E,\mathcal{I})$, one
can define a matroid as a pair $(E,\mathcal{C})$, where $\mathcal{C}$ is the
collection of _circuits_ , a collection of subsets of $E$ such that

  1. (a)

If $I\in\mathcal{C}$ and $J$ is a proper subset of $I$. Then
$J\notin\mathcal{C}$.

  2. (b)

If $C_{1},C_{2}\in\mathcal{C}$, $x\in C_{1}\cap C_{2}$ and $y\in
C_{1}\smallsetminus C_{2}$, then there is a $C_{3}\in\mathcal{C}$ with $y\in
C_{3}$ and $x\notin C_{3}$.

One can show that these two definitions are equivalent by taking the circuits
to be the minimal dependent sets. (See [1, (6.13) ].)

####  12.3 When the root cone is full-dimensional and simplicial

Characterization of when the root cone is full-dimensional as well as when it
is simplicial relies on the notion of balance in a signed graph. (Signed
graphs were discussed in Section 8.) Recall that the Hasse diagram of a signed
poset is an oriented signed graph. Denote the underlying signed graph by
$\Sigma_{P}$.

######  Definition 12.12.

A cycle in a signed graph is said to be _balanced_ if it has an even number of
edges whose sign is $-$. A cycle containing an odd number of edges with sign
$-$ is said to be _unbalanced_. If all cycles in a signed graph are balanced,
the graph itself is said to be balanced.

Like unsigned graphs, signed graphs have an associated matroid, introduced by
[64].

######  Definition 12.13.

Suppose $\Sigma$ is a signed graph. The _signed graphic matroid_
$\Gamma(\Sigma)$ is the matroid whose circuits are the balanced cycles of
$\Sigma$ and pairs of unbalanced cycles joined by a (possibly empty) path.

Figure 15 shows the Hasse diagram of
$P=\\{+e_{1}-e_{2},+e_{3}-e_{2},+e_{1}+e_{3},e_{3}-e_{5},e_{3}-e_{4},-e_{4}-e_{5},+e_{6}-e_{4},+e_{6}-e_{5}\\}$
and the signed graph it orients, $\Sigma_{P}$. There is one balanced cycle
$3-4-6-5$ and two pairs of unbalanced cycles joined by a path: $1-2-3$ and
$3-4-5$ are joined by the empty path and $1-2-3$ and $4-5-6$ are joined either
by $3-4$ or $3-5$.

$1$$2$$3$$4$$5$$6$$+$$-$$+$$+$$-$$+$$+$$-$$+$$-$$-$$-$$-$$+$$-$$+$ (a) Hasse
diagram

$1$$2$$3$$4$$5$$6$$+$$-$$+$$+$$+$$-$$+$$+$ (b) Associated signed graph

Figure 15: Hasse diagram of
$P=\\{+e_{1}-e_{2},+e_{3}-e_{2},+e_{1}+e_{3},e_{3}-e_{5},e_{3}-e_{4},-e_{4}-e_{5},+e_{6}-e_{4},+e_{6}-e_{5}\\}$
and associated signed graph

On the other hand, any set of vectors can be used to define a matroid by
taking the independent sets to be the linearly independent subsets, so the
extreme rays of $K_{P}^{\mathrm{rt}}$, i.e. the edges of the Hasse diagram of
$P$, define a matroid, call it $\mathcal{M}_{P}$. Thus, one has two matroids
arising from the edges of the Hasse diagram of $P$. However, it is a
consequence of a result of Zaslavsky [64] that these two matroids are the
same.

Zaslavsky also defines the _incidence matrix_ $M(\Sigma)$ of a signed graph.

######  Definition 12.14.

Suppose $\Sigma$ is a signed graph. The _incidence matrix_ of $\Sigma$ is the
matrix $M(\Sigma)$ whose columns $M_{e}$ are indexed by the edges of $\Sigma$
and

if $e=(v,w)$ is an edge, |  $m_{ve}=\pm 1$ and $m_{we}=-\sigma(e)m_{ve}$,
---|---
if $e=(v,v)$ is a loop, |  $m_{ve}=0$ if $\sigma(e)=+$ and $m_{ve}=\pm 2$ if $\sigma(e)=-$,
if $e$ and $v$ not incident, |  $m_{ve}=0$.

Note that these $M_{e}$ are the elements of a signed poset
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$ corresponding to the edges
of the Hasse diagram. (Dividing $m_{ve}$ by two for a loop in type B also does
not alter the matroid.) For the poset $P$ in Figure 15, one has

$M(\Sigma_{P})=\leavevmode\hbox to148.94pt{\vbox
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That the matroid formed by the columns of $M(\Sigma_{P})$ and the signed
graphic matroid of $\Sigma_{P}$ are the same is a consequence of the
observation that the $M_{e}$ correspond to elements of the signed poset and
the following result of Zaslavsky.

######  Theorem 12.15 ([64, Theorem 8B.1]).

Let $\Sigma$ be a signed graph and $f\colon E\to{\mathbb{R}}^{n}$ be the
mapping $f(e)=M_{e}$. The matroid structure induced on $E$ by the dependencies
among the vectors $f(e)$ is precisely that of the signed graphic matroid
$\Gamma(\Sigma)$.

In other words, the circuits of $\Gamma(\Sigma)$, i.e. balanced cycles and
pairs of unbalanced cycles joined by a path in $\Sigma_{P}$, correspond
linearly dependent subsets of $P$ which are minimal with respect to inclusion.
This allows one to characterize when the root cone is full-dimensional and
simplicial.

######  Proposition 12.16.

Suppose $P\subset\Phi_{B_{n}}$ (resp.
${P^{\scriptscriptstyle\vee}}\subset\Phi_{C_{n}}$) is a signed poset. Its root
cone is full-dimensional if and only if each connected component of
$\Sigma_{P}$ contains an unbalanced cycle.

The proof requires an alternate characterization of balance due to Harary
[33].

######  Theorem 12.17 ([33, Theorem 3]).

A signed graph is balanced if and only if its vertices can be partitioned into
two sets $V_{+}$ and $V_{-}$ such that every edge of the graph labeled $+$