Shellability of the higher pinched Veronese posets

TL;DR

This paper proves that all intervals in the higher pinched Veronese posets are shellable for n≥4, introducing a new method for shellability, and shows this implies the Koszul property of the associated ring.

Contribution

It introduces a novel approach to establish shellability of posets, specifically applied to the higher pinched Veronese posets, with implications in algebra.

Findings

All intervals in V*_n are shellable for n≥4

Develops a new method for proving shellability

Shows the pinched Veronese ring is Koszul for n≥4

Abstract

The pinched Veronese poset $V^*_n$ is the poset with ground set consisting of all non-negative integer vectors of length n such that the sum of their coordinates is divisible by $n$ with exception of the vector $(1,...,1)$. For two vectors $a$ and $b$ in $V^*_n$ we have $a \leq b$ if and only if $b - a$ belongs to the ground set of $V^*_n$. We show that every interval in $V^*_n$ is shellable for $n$ at least 4. In order to obtain the result, we develop a new method for showing that a poset is shellable. This method differs from classical lexicographic shellability. Shellability of intervals in $V^*_n$ has consequences in commutative algebra. As a corollary we obtain a combinatorial proof of the fact that the pinched Veronese ring is Koszul for $n \geq 4$. (This also follows from a result by Conca, Herzog, Trung and Valla.)