Pattern Recognition on Oriented Matroids: Decompositions of Topes, and Dehn-Sommerville Type Relations

TL;DR

This paper explores the decomposition of topes in oriented matroids using symmetric cycles, revealing unique minimal subsets and establishing Dehn-Sommerville type relations for complex decompositions.

Contribution

It introduces a novel decomposition method for topes in oriented matroids and connects these decompositions to Dehn-Sommerville type relations, advancing theoretical understanding.

Findings

Unique minimal subsets Q(T;R) exist for topes T

Decompositions with more than three elements satisfy specific relations

Provides new insights into the structure of oriented matroids

Abstract

If V(R) is the vertex set of a symmetric cycle R in the tope graph of a simple oriented matroid M, then for any tope T of M there exists a unique inclusion-minimal subset Q(T;R) of V(R) such that T is the sum of the topes of Q(T;R). If |Q(T;R)|>3, then the decomposition Q(T;R) of the tope T with respect to the symmetric cycle R satisfies certain Dehn-Sommerville type relations.

Equations

$$\{\mathbf{x}\in {\mathbb{R}}^2:⟨{\mathbf{a}}_e,\mathbf{x}⟩=0,e\in E_t\}$$

$$\mathbf{A}:=\{{\mathbf{a}}_e:e\in E_t\},$$

$$\sum _{Q\in \mathbf{Q}(T,\mathbf{R})}Q=T,$$

$$\mathbf{Q}({\mathrm{T}}^{(+)},\mathbf{R})={\max }^{+}\mathrm{V}(\mathbf{R}).$$

$$|\mathbf{Q}({\mathrm{T}}^{(+)},\mathbf{R})|\ge 5,$$

$$|\{\mathbf{a}\in \mathbf{A}:\mathbf{a}\in {\mathbf{C}}_{>}\}|\ge 2,$$

$$\bigl{\{}\langle\boldsymbol{a}_{e},\mathbf{x}\rangle>0\colon\mathbf{x}\in\mathbb{R}^{2},\;\boldsymbol{a}_{e}\in\boldsymbol{A}\bigr{\}}$$

$$\{\begin{array}{l}{\nu }_j=\left(\genfrac{}{}{0ex}{}{t}{j}\right),\text{ if 0≤j≤2},\\ {\nu }_{t-1}={\nu }_t=0,\\ {\sum }_{j=3}^t\left(\left(\genfrac{}{}{0ex}{}{t}{j}\right)-{\nu }_j\right)(\mathrm{x}-1{)}^{t-j}=-{\sum }_{j=3}^t(-1{)}^j\left(\left(\genfrac{}{}{0ex}{}{t}{j}\right)-{\nu }_j\right){\mathrm{x}}^{t-j}\end{array}$$

$${\nu }_3=\left(\genfrac{}{}{0ex}{}{t}{2}\right)-t;$$

$${\nu }_3$$

$${\nu }_4$$

$${\nu }_5$$

$$\tbinom{t}{j}\!-\!\nu_{j}=-\sum_{i=3}^{j}(-1)^{i}\tbinom{t-i}{j-i}\Bigl{(}\tbinom{t}{i}\!-\!\nu_{i}\Bigr{)}\;.$$

$$\sum _{j=1}^{t-2}(-1{)}^j{\nu }_j=0.$$

$$T=\sum _{Q\in \mathbf{Q}(T,\mathbf{R})}Q$$

$$|\mathbf{Q}(T,\mathbf{R})|\ge 5,$$

$${\mathcal{Y}}_{T,\mathbf{R}}:={}_{-T^{-}}({\mathcal{M}}_{\mathbf{R}})$$

$$\Delta :={\Delta }_{\mathrm{acyclic}}({\mathcal{Y}}_{T,\mathbf{R}})$$

$$\boldsymbol{f}(\Delta;t):=\bigl{(}f_{0}(\Delta;t),f_{1}(\Delta;t),\ldots,f_{t}(\Delta;t)\bigr{)}\in\mathbb{N}^{t+1}$$

$$f_j(\Delta ;t):=\#\{F\in \Delta :|F|=j\},0\le j\le t.$$

$$\{\begin{array}{l}{f}_j(\Delta ;t)=\left(\genfrac{}{}{0ex}{}{t}{j}\right),\text{ if 0≤j≤2},\\ f_{t-1}(\Delta ;t)=f_t(\Delta ;t)=0,\\ {\sum }_{j=3}^t\left(\left(\genfrac{}{}{0ex}{}{t}{j}\right)-f_j(\Delta ;t)\right)(\mathrm{x}-1{)}^{t-j}=-{\sum }_{j=3}^t(-1{)}^j\left(\left(\genfrac{}{}{0ex}{}{t}{j}\right)-f_j(\Delta ;t)\right){\mathrm{x}}^{t-j}.\end{array}$$

$$\{E_t-\mathbf{S}(T,Q):Q\in \mathbf{Q}(T,\mathbf{R})\},$$

$$\sum _{Q\in \mathbf{Q}(T,\mathbf{R})}Q=T$$

$$|\mathbf{Q}(T,\mathbf{R})|\ge 5.$$

$$\begin{array}{rl}\mathbf{\Lambda }:& =\mathbf{\Lambda }(T,\mathbf{R})\\ & =\mathbf{\Lambda }(-T,\mathbf{R})\end{array}$$

$$\{\begin{array}{l}{f}_j(\mathbf{\Lambda };t)=\left(\genfrac{}{}{0ex}{}{t}{j}\right),\text{ if 0≤j≤2},\\ f_{t-1}(\mathbf{\Lambda };t)=f_t(\mathbf{\Lambda };t)=0,\\ {\sum }_{j=3}^t\left(\left(\genfrac{}{}{0ex}{}{t}{j}\right)-f_j(\mathbf{\Lambda };t)\right)(\mathrm{x}-1{)}^{t-j}=-{\sum }_{j=3}^t(-1{)}^j\left(\left(\genfrac{}{}{0ex}{}{t}{j}\right)-f_j(\mathbf{\Lambda };t)\right){\mathrm{x}}^{t-j}.\end{array}$$

$$f_3(\mathbf{\Lambda };t)=\left(\genfrac{}{}{0ex}{}{t}{2}\right)-t=5;$$

$$f_3(\mathbf{\Lambda };t)$$

$$f_4(\mathbf{\Lambda };t)$$

$$\begin{array}[]{cclcl}f_{3}(\boldsymbol{\Lambda};t)&=&\tfrac{1}{2}f_{4}(\boldsymbol{\Lambda};t)+\tbinom{t}{4}-\tfrac{1}{2}\tbinom{t}{3}&=&\tfrac{1}{2}\bigl{(}f_{4}(\boldsymbol{\Lambda};t)+35\bigr{)}\\ &=&f_{5}(\boldsymbol{\Lambda};t)+\tbinom{t}{4}-\tbinom{t}{2}+t&=&f_{5}(\boldsymbol{\Lambda};t)+21\;,\\ f_{4}(\boldsymbol{\Lambda};t)&=&2f_{3}(\boldsymbol{\Lambda};t)-2\tbinom{t}{4}+\tbinom{t}{3}&=&2f_{3}(\boldsymbol{\Lambda};t)-35\\ &=&2f_{5}(\boldsymbol{\Lambda};t)+\tbinom{t}{3}-2\tbinom{t}{2}+2t&=&2f_{5}(\boldsymbol{\Lambda};t)+7\;,\\ f_{5}(\boldsymbol{\Lambda};t)&=&f_{3}(\boldsymbol{\Lambda};t)-\tbinom{t}{4}+\tbinom{t}{2}-t&=&f_{3}(\boldsymbol{\Lambda};t)-21\\ &=&\tfrac{1}{2}f_{4}(\boldsymbol{\Lambda};t)-\tfrac{1}{2}\tbinom{t}{3}+\tbinom{t}{2}-t&=&\tfrac{1}{2}\bigl{(}f_{4}(\boldsymbol{\Lambda};t)-35\bigr{)}+14\;;\end{array}$$

$$\tbinom{t}{j}\!-\!f_{j}(\boldsymbol{\Lambda};t)=-\sum_{i=3}^{j}(-1)^{i}\tbinom{t-i}{j-i}\Bigl{(}\tbinom{t}{i}\!-\!f_{i}(\boldsymbol{\Lambda};t)\Bigr{)}\;.$$

Full text

If $|\boldsymbol{Q}(T,\boldsymbol{R})|\geq 5$, then the decomposition $\boldsymbol{Q}(T,\boldsymbol{R})$ of the tope $T$ with respect to the symmetric cycle $\boldsymbol{R}$ satisfies certain Dehn–Sommerville type relations.

1. Introduction

Let $\mathcal{A}:=(E_{t},\mathcal{L})=(E_{t},\mathcal{T})$ be a simple (i.e., with no loops, parallel elements or antiparallel elements) acyclic oriented matroid on the ground set $E_{t}:=\{1,2,\ldots,t\}$, with set of covectors $\mathcal{L}$ and with set of topes $\mathcal{T}\subseteq\{1,-1\}^{E_{t}}$, of rank $r\geq 3$. Let $\boldsymbol{R}$ be a symmetric $2t$-cycle, with its vertex set $\mathrm{V}(\boldsymbol{R})=-\mathrm{V}(\boldsymbol{R})$, in the tope graph $\mathcal{T}(\mathcal{L}(\mathcal{A}))$ of $\mathcal{A}$ such that $(1,1,\ldots,1)=:\mathrm{T}^{(+)}\not\in\mathrm{V}(\boldsymbol{R})$.

Recall that the vertex set $\mathrm{V}(\boldsymbol{R})$ of the cycle $\boldsymbol{R}$ is the set of topes of a rank $2$ oriented matroid denoted $\mathcal{A}_{\boldsymbol{R}}$, see e.g. [3, Rem. 1.7], and thus by [1, Example 7.1.7] we can consider the representation of $\mathcal{A}_{\boldsymbol{R}}$ by a certain central line arrangement

[TABLE]

in the plane $\mathbb{R}^{2}$, with the set of corresponding normal vectors

[TABLE]

and with the standard scalar product $\langle\cdot,\cdot\rangle$.

For any tope $T$ of the oriented matroid $\mathcal{A}$ there exists a unique inclusion-minimal subset $\boldsymbol{Q}(T,\boldsymbol{R})\subset\mathrm{V}(\boldsymbol{R})$ such that

[TABLE]

see [3, Sect. 11.1].

If the tope graph $\mathcal{T}(\mathcal{L}(\mathcal{A}))$ is the hypercube graph $\boldsymbol{H}(t,2)$ on $2^{t}$ vertices, then for any odd integer $j$, $1\leq j\leq t$, by [3, Th. 13.6] there are precisely $2\tbinom{t}{j}$ vertices $T$ of $\boldsymbol{H}(t,2)$ such that $|\boldsymbol{Q}(T,\boldsymbol{R})|=j$.

Let $\operatorname{\mathbf{max}}^{+}\mathrm{V}(\boldsymbol{R})$ denote the subset of topes of $\mathrm{V}(\boldsymbol{R})$ with inclusion-maximal positive parts. For the positive tope $\mathrm{T}^{(+)}$ of $\mathcal{A}$, by [3, Prop. 1.12(ii)] we have

[TABLE]

If

[TABLE]

then for the set of vectors (1.2) we have

[TABLE]

for any open half-space $\mathbf{C}_{>}\subset\mathbb{R}^{2}$ bounded by a $1$-dimensional subspace of $\mathbb{R}^{2}$; see e.g. [2, Proof of Prop. 2.33].

Denote by $\nu_{j}$ the number of feasible subsystems, of cardinality $j$, of the infeasible system of homogeneous strict linear inequalities

[TABLE]

associated with the arrangement (1.1). Since the condition (1.3) holds, the quantities $\nu_{j}$ satisfy the relations (where $\mathrm{x}$ is a formal variable)

[TABLE]

called the Dehn–Sommerville equations for the feasible subsystems of the system (1.4); see e.g. [2, Prop. 3.53].

In particular,

• $\bullet$

if $t=5$, then

[TABLE]

• $\bullet$

if $t=6$, then

[TABLE]

• $\bullet$

if $t=7$, then

[TABLE]

For any $j$, $3\leq j\leq t-2$, by [2, Cor. 3.55(i)] we have

[TABLE]

Recall also that

[TABLE]

2. Decompositions of topes with respect to symmetric cycles in the tope graph, and Dehn–Sommerville type relations

Let $\mathcal{M}:=(E_{t},\mathcal{L})=(E_{t},\mathcal{T})$ be an arbitrary simple oriented matroid of rank $r\geq 3$, and let $\boldsymbol{R}$ be a symmetric $2t$-cycle in the tope graph $\mathcal{T}(\mathcal{L}(\mathcal{M}))$ of $\mathcal{M}$.

Given a tope $T\in\mathcal{T}$ of $\mathcal{M}$, consider the corresponding decomposition

[TABLE]

of $T$ with respect to the cycle $\boldsymbol{R}$, for a unique inclusion-minimal subset $\boldsymbol{Q}(T,\boldsymbol{R})$ of the vertex set $\mathrm{V}(\boldsymbol{R})$ of the cycle $\boldsymbol{R}$. Suppose that

[TABLE]

and consider the totally cyclic oriented matroid

[TABLE]

obtained from the rank $2$ oriented matroid $\mathcal{M}_{\boldsymbol{R}}:=(E_{t},\mathrm{V}(\boldsymbol{R}))$ with the set of topes $\mathrm{V}(\boldsymbol{R})$ by reorientation of $\mathcal{M}_{\boldsymbol{R}}$ on the negative part $T^{-}$ of the tope $T$.

Associate with the abstract simplicial complex

[TABLE]

of acyclic subsets of the ground set of the oriented matroid $\mathcal{Y}_{T,\boldsymbol{R}}$ its “long” $f$-vector

[TABLE]

defined by

[TABLE]

In view of (1.5), we have

[TABLE]

Let us return to the symmetric cycle $\boldsymbol{R}$, and associate with the tope $T$ the abstract simplicial complex $\boldsymbol{\Lambda}:=\boldsymbol{\Lambda}(T,\boldsymbol{R})$ with the facet family

[TABLE]

where $\mathbf{S}(T,Q):=\{e\in E_{t}\colon T(e)\neq Q(e)\}$ is the separation set of the topes $T$ and $Q$. By construction, the complex $\boldsymbol{\Lambda}$ and the complex $\Delta$ defined by (2.1) coincide, and we come to the following result:

Proposition 2.1**.**

Let $\boldsymbol{R}$ be a symmetric cycle in the tope graph of a simple oriented matroid $\mathcal{M}:=(E_{t},\mathcal{T})$.

Let $T\in\mathcal{T}$ be a tope of $\mathcal{M}$ such that for the unique inclusion-minimal subset of topes $\boldsymbol{Q}(T,\boldsymbol{R})\subset\mathrm{V}(\boldsymbol{R})$ with the property

[TABLE]

we have

[TABLE]

(i)* The components of the long $f$-vector $\boldsymbol{f}(\boldsymbol{\Lambda};t)$ of the complex*

[TABLE]

whose family of facets is defined by (2.2) satisfy the Dehn–Sommerville type relations

[TABLE]

In particular,

• $\bullet$

if $t=5$, then

[TABLE]

• $\bullet$

if $t=6$, then

[TABLE]

• $\bullet$

if $t=7$, then

[TABLE]

the quantity $f_{4}(\boldsymbol{\Lambda};7)$ is odd.

(ii)* For any $j$, $3\leq j\leq t-2$, we have*

[TABLE]

(iii)* We have*

[TABLE]