2013/05/14 by Martin Tancer, Tancer, Martin
Mathematics · #05E40 #06A11 #16S37 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #O6A07 #math.AC #math.CO #msc:05E40 #msc:06A11 #msc:16S37 #msc:O6A07
paper · pdf · doi:10.48550/arxiv.1305.3159
41 pages, 15 figures (Version 4: minor improvements of the proof of Lemma 5.3 and other minor fixes)
arxiv created 2014/02/24 · arxiv updated 2014/02/25
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 ≤ 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 ≥ 4. (This also follows from a result by Conca, Herzog, Trung and Valla.)