Pattern Recognition on Oriented Matroids: Decompositions of Topes, and Orthogonality Relations

TL;DR

This paper explores decompositions of topes in simple oriented matroids, establishing unique minimal subsets related to symmetric cycles and revealing orthogonality relations under specific conditions.

Contribution

It introduces a novel framework for decomposing topes via symmetric cycles and uncovers orthogonality relations based on subset sizes and parity conditions.

Findings

Unique minimal subsets Q(T,R) exist for topes T in simple oriented matroids.

Orthogonality relations hold for certain decompositions with large subset sizes and opposite parity ground sets.

The results connect to structural properties of tope graphs and symmetric cycles in 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 for decompositions Q(T',R') and Q(T",R") with respect to symmetric cycles R' and R" in the tope graphs of two simple oriented matroids, whose ground sets have the cardinalities of opposite parity, we have |Q(T',R')|>3 and |Q(T",R")|>3, then these decompositions satisfy a certain orthogonality relation.

Equations

$$s<t,$$

$$s\not\equiv t(\mathrm{mod}2).$$

$$\displaystyle\sum_{Q^{\prime}\in\boldsymbol{Q}(T^{\prime},\boldsymbol{R}^{\prime})}Q^{\prime}=T^{\prime}\ \ \$$

$$\displaystyle|\boldsymbol{Q}(T^{\prime},\boldsymbol{R}^{\prime})|\geq 5\ \ \$$

$${\mathbf{\Lambda }}^{\prime \prime }:=\mathbf{\Lambda }(T^{\prime \prime },{\mathbf{R}}^{\prime \prime })$$

$$\bigl{\{}E_{t}-\mathbf{S}(T^{\prime\prime},Q^{\prime\prime})\colon Q^{\prime\prime}\in\boldsymbol{Q}(T^{\prime\prime},\boldsymbol{R}^{\prime\prime})\bigr{\}}\;,$$

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

$$f_j({\mathbf{\Lambda }}^{\prime \prime };t):=\#\{F\in {\mathbf{\Lambda }}^{\prime \prime }:|F|=j\},0\le j\le t;$$

$$\mathbf{\beta }(t;t):$$

$$\mathbf{\beta }(s;t):$$

$$\bigl{(}\boldsymbol{\beta}(t;t)-\boldsymbol{f}(\boldsymbol{\Lambda}^{\prime\prime};t)\bigr{)}\mathbf{U}(t)\in\mathbb{N}^{t+1}$$

$${\mathbf{\Lambda }}^{\prime }:=\mathbf{\Lambda }(T^{\prime },{\mathbf{R}}^{\prime })$$

$$\bigl{\{}E_{s}-\mathbf{S}(T^{\prime},Q^{\prime})\colon Q^{\prime}\in\boldsymbol{Q}(T^{\prime},\boldsymbol{R}^{\prime})\bigr{\}}\;,$$

$$\mathbf{f}({\mathbf{\Lambda }}^{\prime };t)\in {\mathbb{N}}^{t+1}$$

$$\bigl{(}\boldsymbol{\beta}(s;t)-\boldsymbol{f}(\boldsymbol{\Lambda}^{\prime};t)\bigr{)}\mathbf{T}(t)^{t-s}\mathbf{U}(t)\in\mathbb{N}^{t+1}$$

$$\mathbf{S}(t)$$

$$h_k({\mathbf{\Omega }}^{\prime \prime })$$

$$h_k({\mathbf{\Omega }}^{\prime })$$

$$\boldsymbol{h}(\boldsymbol{\Omega}^{\prime\prime};t):=\bigl{(}\boldsymbol{\beta}(t;t)-\boldsymbol{f}(\boldsymbol{\Lambda}^{\prime\prime};t)\bigr{)}\mathbf{U}(t)\mathbf{S}(t)\in\mathbb{Z}^{t+1}$$

$$\boldsymbol{h}(\boldsymbol{\Omega}^{\prime};t):=\bigl{(}\boldsymbol{\beta}(s;t)-\boldsymbol{f}(\boldsymbol{\Lambda}^{\prime};t)\bigr{)}\mathbf{T}(t)^{t-s}\mathbf{U}(t)\mathbf{S}(t)\in\mathbb{Z}^{t+1}$$

$$h_k({\mathbf{\Omega }}^{\prime \prime };t)$$

$$h_k({\mathbf{\Omega }}^{\prime };t)$$

$$\bigl{\langle}\boldsymbol{h}(\boldsymbol{\Omega}^{\prime};t),\boldsymbol{\iota}(t)\bigr{\rangle}=\bigl{\langle}\boldsymbol{h}(\boldsymbol{\Omega}^{\prime\prime};t),\boldsymbol{\iota}(t)\bigr{\rangle}=0\;,$$

$$\mathbf{\iota }(t):=(1,1,\dots ,1)\in {\mathbb{N}}^{t+1},$$

$$\overline{2^{[k]}}:=2^{[k]}-\{[k]\}.$$

$$\operatorname{\mathrm{span}}\Bigl{(}\boldsymbol{h}(\overline{\mathbf{2}^{[1]}};t),\boldsymbol{h}(\overline{\mathbf{2}^{[3]}};t),\ldots,\boldsymbol{h}(\overline{\mathbf{2}^{[s-2]}};t)\Bigr{)}\;,$$

$$\operatorname{\mathrm{span}}\Bigl{(}\boldsymbol{h}(\overline{\mathbf{2}^{[2]}};t),\boldsymbol{h}(\overline{\mathbf{2}^{[4]}};t),\ldots,\boldsymbol{h}(\overline{\mathbf{2}^{[s-2]}};t)\Bigr{)}\;,$$

$$\bigl{\langle}\boldsymbol{h}(\boldsymbol{\Omega}^{\prime};t),\boldsymbol{h}(\boldsymbol{\Omega}^{\prime\prime};t)\bigr{\rangle}=0\;.$$

$$\bigl{(}\boldsymbol{\beta}(s;t)-\boldsymbol{f}(\boldsymbol{\Lambda}^{\prime};t)\bigr{)}\mathbf{T}(t)^{t-s}\mathbf{U}(t)\mathbf{S}(t)\mathbf{S}(t)^{\!\top}\mathbf{U}(t)\bigl{(}\boldsymbol{\beta}(t;t)^{\!\top}-\boldsymbol{f}(\boldsymbol{\Lambda}^{\prime\prime};t)^{\!\top}\bigr{)}=0\;.$$

$$\mathbf{M}(t):=\mathbf{U}(t)\mathbf{S}(t)\mathbf{S}(t{)}^{\top }\mathbf{U}(t)$$

$$s<t,$$

$$s\not\equiv t(\mathrm{mod}2),$$

$$\displaystyle\sum_{Q^{\prime}\in\boldsymbol{Q}(T^{\prime},\boldsymbol{R}^{\prime})}Q^{\prime}=T^{\prime}\ \ \$$

Full text

If for decompositions $\boldsymbol{Q}(T^{\prime},\boldsymbol{R}^{\prime})$ and $\boldsymbol{Q}(T^{\prime\prime},\boldsymbol{R}^{\prime\prime})$ with respect to symmetric cycles $\boldsymbol{R}^{\prime}$ and $\boldsymbol{R}^{\prime\prime}$ in the tope graphs of two simple oriented matroids, whose ground sets have the cardinalities of opposite parity, we have $|\boldsymbol{Q}(T^{\prime},\boldsymbol{R}^{\prime})|\geq 5$ and $|\boldsymbol{Q}(T^{\prime\prime},\boldsymbol{R}^{\prime\prime})|\geq 5$, then these decompositions satisfy a certain orthogonality relation.

1. Introduction

Let $\mathcal{M}^{\prime}:=(E_{s},\mathcal{L}^{\prime})=(E_{s},\mathcal{T}^{\prime})$ and $\mathcal{M}^{\prime\prime}:=(E_{t},\mathcal{L}^{\prime\prime})=(E_{t},\mathcal{T}^{\prime\prime})$ be simple (i.e., with no loops, parallel elements or antiparallel elements) oriented matroids on their ground sets $E_{s}=[s]:=\{1,\ldots,s\}$ and $E_{t}$, with sets of covectors $\mathcal{L}^{\prime}$ and $\mathcal{L}^{\prime\prime}$, and with sets of topes $\mathcal{T}^{\prime}\subseteq\{1,-1\}^{E_{s}}$ and $\mathcal{T}^{\prime\prime}\subseteq\{1,-1\}^{E_{t}}$, respectively; see [1] on oriented matroids. We suppose that

[TABLE]

Let $\boldsymbol{R}^{\prime}$ be a symmetric $2s$-cycle, with its vertex set $\mathrm{V}(\boldsymbol{R}^{\prime})=-\mathrm{V}(\boldsymbol{R}^{\prime})$, in the tope graph $\mathcal{T}^{\prime}(\mathcal{L}^{\prime})$ of $\mathcal{M}^{\prime}$, and let $\boldsymbol{R}^{\prime\prime}$ be a symmetric $2t$-cycle in the tope graph $\mathcal{T}^{\prime\prime}(\mathcal{L}^{\prime\prime})$ of $\mathcal{M}^{\prime\prime}$.

Let $T^{\prime}\in\mathcal{T}^{\prime}$ and $T^{\prime\prime}\in\mathcal{T}^{\prime\prime}$ be any topes of the oriented matroids $\mathcal{M}^{\prime}$ and $\mathcal{M}^{\prime\prime}$ such that for the unique inclusion-minimal subsets $\boldsymbol{Q}(T^{\prime},\boldsymbol{R}^{\prime})\subset\mathrm{V}(\boldsymbol{R}^{\prime})$ and $\boldsymbol{Q}(T^{\prime\prime},\boldsymbol{R}^{\prime\prime})\subset\mathrm{V}(\boldsymbol{R}^{\prime\prime})$ with the properties

[TABLE]

Let

[TABLE]

be the abstract simplicial complex on the vertex set $E_{t}$, with the facet family

[TABLE]

where $\mathbf{S}(T^{\prime\prime},Q^{\prime\prime})$ is the separation set of the topes $T^{\prime\prime}$ and $Q^{\prime\prime}$; see e.g. [5] on such complexes.

Associate with the complex $\boldsymbol{\Lambda}^{\prime\prime}$ its “long” $f$-vector

[TABLE]

defined by

[TABLE]

throughout the paper, the components of all vectors, as well as the rows and columns of matrices are indexed starting with zero.

Define vectors $\boldsymbol{\beta}(t;t)\in\mathbb{P}^{t+1}$ and $\boldsymbol{\beta}(s;t)\in\mathbb{N}^{t+1}$ by

[TABLE]

Let $\mathbf{U}(t)$ denote the square backward identity matrix of order $t+1$, whose $(i,j)$th entry is the Kronecker delta $\delta_{i+j,t}$. We denote by $\mathbf{T}(t)$ the square forward shift matrix of order $t+1$, whose $(i,j)$th entry is $\delta_{j-i,1}$.

According to the argument given in [5], and by [2, Prop. 3.51(a)], the vector

[TABLE]

is the long $f$-vector $\boldsymbol{f}(\boldsymbol{\Omega}^{\prime\prime};t)$ of the boundary complex $\boldsymbol{\Omega}^{\prime\prime}$ of a $(t-3)$-dimensional simplicial convex polytope with $t$ vertices.

Associate with the abstract simplicial complex

[TABLE]

on the vertex set $E_{s}$, with the facet family

[TABLE]

its long $f$-vector

[TABLE]

defined by $f_{j}(\boldsymbol{\Lambda}^{\prime};t):=\#\{F\in\boldsymbol{\Lambda}^{\prime}\colon|F|=j\}$, $0\leq j\leq t$.

The vector

[TABLE]

is the long $f$-vector $\boldsymbol{f}(\boldsymbol{\Omega}^{\prime};t)$ of the boundary complex $\boldsymbol{\Omega}^{\prime}$ of an $(s-3)$-dimensional simplicial convex polytope with $s$ vertices.

Define the $(i,j)$th entry of a square matrix

[TABLE]

of order $t+1$ to be $(-1)^{j-i}\tbinom{t-i}{j-i}$.

Recall that the standard $h$-vectors $\boldsymbol{h}(\boldsymbol{\Omega}^{\prime\prime})\in\mathbb{N}^{t-2}$ and $\boldsymbol{h}(\boldsymbol{\Omega}^{\prime})\in\mathbb{N}^{s-2}$ of the boundary complexes $\boldsymbol{\Omega}^{\prime\prime}$ and $\boldsymbol{\Omega}^{\prime}$ of simplicial polytopes both satisfy the Dehn–Sommerville relations [6, Sect. 8.3]

[TABLE]

As a consequence, the “long” $h$-vectors

[TABLE]

and

[TABLE]

of the complexes $\boldsymbol{\Omega}^{\prime\prime}$ and $\boldsymbol{\Omega}^{\prime}$ satisfy the Dehn–Sommerville type relations

[TABLE]

see e.g. [4, Sect. 2.3].

2. Orthogonality relations for decompositions of topes with respect to symmetric cycles in the tope graphs

Since the maximal face $[t]$ of the simplex $\mathbf{2}^{[t]}$ does not belong to the complexes $\boldsymbol{\Omega}^{\prime}$ and $\boldsymbol{\Omega}^{\prime\prime}$, by [4, Eq. (2.3) of Prop. 2.1] we have

[TABLE]

for the vector

[TABLE]

where $\langle\cdot,\cdot\rangle$ means the standard scalar product.

For the positive integers $k$, define abstract simplicial complexes $\overline{\mathbf{2}^{[k]}}$ with their long $h$-vectors $\boldsymbol{h}(\overline{\mathbf{2}^{[k]}};t):=\boldsymbol{f}(\overline{\mathbf{2}^{[k]}};t)\mathbf{S}(t)$ to be the boundary complexes of the simplices $\mathbf{2}^{[k]}$ by

[TABLE]

In view of (1.7), the long $h$-vector $\boldsymbol{h}(\boldsymbol{\Omega}^{\prime})$ lies either in the linear span

[TABLE]

when $t$ is even, or in the linear span

[TABLE]

when $t$ is odd; see [4, Sect. 3.1].

The Dehn–Sommerville type relations (1.6) and (1.7) imply that $\boldsymbol{h}(\boldsymbol{\Omega}^{\prime};t)$ is a left eigenvector of the backward identity matrix $\mathbf{U}(t)$ that corresponds to its eigenvalue $1$, while $\boldsymbol{h}(\boldsymbol{\Omega}^{\prime\prime};t)$ is a right eigenvector of $\mathbf{U}(t)$ that corresponds to the other eigenvalue $-1$. By the principle of biorthogonality [3, Th. 1.4.7(a)] we have

[TABLE]

In other words, together with the relation (2.3), the definitions (1.4) and (1.5) yield

[TABLE]

Note that the $(i,j)$th entry of the square matrix

[TABLE]

of order $t+1$ is $(-1)^{i+j}\tbinom{i+j}{i}$.

Let us sum up our conclusions:

Proposition 2.1**.**

Let $\mathcal{M}^{\prime}:=(E_{s},\mathcal{L}^{\prime})=(E_{s},\mathcal{T}^{\prime})$ and $\mathcal{M}^{\prime\prime}:=(E_{t},\mathcal{L}^{\prime\prime})=(E_{t},\mathcal{T}^{\prime\prime})$ be simple oriented matroids on their ground sets $E_{s}$ and $E_{t}$ such that

[TABLE]

with sets of covectors $\mathcal{L}^{\prime}$ and $\mathcal{L}^{\prime\prime}$, and with sets of topes $\mathcal{T}^{\prime}\subseteq\{1,-1\}^{E_{s}}$ and $\mathcal{T}^{\prime\prime}\subseteq\{1,-1\}^{E_{t}}$, respectively.

Let $\boldsymbol{R}^{\prime}$ be a symmetric cycle in the tope graph $\mathcal{T}^{\prime}(\mathcal{L}^{\prime})$ of $\mathcal{M}^{\prime}$, and let $\boldsymbol{R}^{\prime\prime}$ be a symmetric cycle in the tope graph $\mathcal{T}^{\prime\prime}(\mathcal{L}^{\prime\prime})$ of $\mathcal{M}^{\prime\prime}$.

Let $T^{\prime}\in\mathcal{T}^{\prime}$ and $T^{\prime\prime}\in\mathcal{T}^{\prime\prime}$ be any topes of the oriented matroids $\mathcal{M}^{\prime}$ and $\mathcal{M}^{\prime\prime}$ such that for the unique inclusion-minimal subsets $\boldsymbol{Q}(T^{\prime},\boldsymbol{R}^{\prime})\subset\mathrm{V}(\boldsymbol{R}^{\prime})$ and $\boldsymbol{Q}(T^{\prime\prime},\boldsymbol{R}^{\prime\prime})\subset\mathrm{V}(\boldsymbol{R}^{\prime\prime})$ with the properties

[TABLE]

The long $f$-vectors $\boldsymbol{f}(\boldsymbol{\Lambda}^{\prime\prime};t)$ and $\boldsymbol{f}(\boldsymbol{\Lambda}^{\prime};t)$ of the complexes $\boldsymbol{\Lambda}^{\prime\prime}$ and $\boldsymbol{\Lambda}^{\prime}$ whose families of facets are defined by (1.2) and (1.3), respectively, satisfy the orthogonality relation

[TABLE]