Shellability of the higher pinched Veronese posets

Tancer, Martin

Journal of Algebraic Combinatorics, Tome 40 (2014) no. 3, pp. 711-742.

Résumé

The pinched Veronese poset $\mathcal V^\bullet_n$ is the poset with ground set consisting of all nonnegative integer vectors of length $n$ such that the sum of their coordinates is divisible by $n$ with exception of the vector $(1,\dots,1)$. For two vectors $\mathbf a$ and $\mathbf b$ in $\mathcal V^\bullet_n$, we have $\mathbf a\preceq\mathbf b$ if and only if $\mathbf b-\mathbf a$ belongs to the ground set of $\mathcal V^\bullet_n$. We show that every interval in $\mathcal V^\bullet_n$ is shellable for $n\geq 4$. 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 $\mathcal V^\bullet_n$ 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 A. Conca et al. [Am. J. Math. 119, No. 4, 859--901 (1997; Zbl 0920.13003)]).

Détail

Classification : 06A07, 13H10
Keywords: shellability, pinched Veronese poset, Cohen-Macaulay, Koszul