Elementary symmetric polynomials in Stanley--Reisner face ring

TL;DR

This paper investigates the algebraic properties of elementary symmetric polynomials in Stanley--Reisner face rings of simple polytopes and their implications for the combinatorics, topology, and geometry of associated toric spaces.

Contribution

It introduces algebraic criteria for decomposability and colorability of polytopes using elementary symmetric polynomials and defines a non-commutative Stanley--Reisner exterior face ring to study toric space invariants.

Findings

Decomposability of the n-th elementary symmetric polynomial relates to polytope combinatorics.

Criteria established for the Buchstaber invariant in terms of elementary symmetric polynomials.

Connections made between polynomial decomposability and existence of almost complex structures.

Abstract

Let $P$ be a simple polytope of dimension $n$ with $m$ facets. In this paper we pay our attention on those elementary symmetric polynomials in the Stanley--Reisner face ring of $P$ and study how the decomposability of the $n$-th elementary symmetric polynomial influences on the combinatorics of $P$ and the topology and geometry of toric spaces over $P$. We give algebraic criterions of detecting the decomposability of $P$ and determining when $P$ is $n$-colorable in terms of the $n$-th elementary symmetric polynomial. In addition, we define the Stanley--Reisner {\em exterior} face ring $\mathcal{E}(K_P)$ of $P$, which is non-commutative in the case of ${\Bbb Z}$ coefficients, where $K_P$ is the boundary complex of dual of $P$. Then we obtain a criterion for the (real) Buchstaber invariant of $P$ to be $m-n$ in terms of the $n$-th elementary symmetric polynomial in $\mathcal{E}(K_P)$. Our results as above can directly associate with the topology and geometry of toric spaces over $P$. In particular, we show that the decomposability of the $n$-th elementary symmetric polynomial in $\mathcal{E}(K_P)$ with ${\Bbb Z}$ coefficients can detect the existence of the almost complex structures of quasitoric manifolds over $P$, and if the (real) Buchstaber invariant of $P$ is $m-n$, then there exists an essential relation between the $n$-th equivariant characteristic class of the (real) moment-angle manifold over $P$ in $\mathcal{E}(K_P)$ and the characteristic functions of $P$.