The Lefschetz property for barycentric subdivisions of shellable complexes

Abstract

Abstract. We show that an 'almost strong Lefschetz' property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the h-vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its g-vector is an M -sequence. In particular, the (combinatorial) g-conjecture is verified for barycentric subdivisions of homology spheres. In addition, using the above algebraic result, we derive new inequalities on a refinement of the Eulerian statistics on permutations, where permutations are grouped by the number of descents and the image of 1.

Face enumeration on simplicial complexes

“…In general, very little is known about either version of the g-conjecture. Kubitzke and Nevo [44], building on work of Brenti and Welker [13], proved that the barycentric subdivision of a simplicial homology sphere satisfies the g-conjecture for spheres. Murai [62] extended this result to show that the barycentric subdivision of a simplicial homology manifold satisfies the manifold g-conjecture.…”

Squarefree P-modules and the cd-index

“…Indeed, we prove that the order complex of a Cohen-Macaulay CW-poset of polyhedral type has the WLP over R if its rank is even (Corollary 6.3). This result gives a partial solution to the conjecture of Kubitzke and Nevo [KN,Conjecture 4.12] who conjectured that the barycentric subdivision of a Cohen-Macaulay simplicial complex has the WLP.…”

Face numbers of barycentric subdivisions of cubical complexes

“…Being real-rooted, h(sd(∆), x) is unimodal for every n-dimensional simplicial complex ∆ with nonnegative h-vector. Kubitzke and Nevo showed [15,Corolalry 4.7] that the corresponding h-vector (h i (sd(∆)) 0≤i≤n+1 has a peak at i = (n+1)/2, if n is odd, and at i = n/2 or i = n/2+1, if n is even. The analogous statement for cubical complexes follows from Theorem 3.2 since, as in the simplicial setting, the unimodal polynomial p B n,k (x) has a peak at i = (n + 1)/2, if n is odd, at i = n/2 if n is even and k ≤ n/2, and at i = n/2 + 1, if n is even and k ≥ n/2 + 1.…”