Segre and Rees products of posets, with ring-theoretic applications

The Rees product of posets

We determine how the flag f-vector of any graded poset changes under the Rees product with the chain, and more generally, any t-ary tree. As a corollary, the Möbius function of the Rees product of any graded poset with the chain, and more generally, the t-ary tree, is exactly the same as the Rees product of its dual with the chain, respectively, t-ary chain. We then study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give a bijective proof that the Möbius function of this poset can be expressed as n times a signed derangement number. From …

Some applications of Rees products of posets to equivariant gamma-positivity

The Rees product of partially ordered sets was introduced by Björner and Welker. Using the theory of lexicographic shellability, Linusson, Shareshian and Wachs proved formulas, of significance in the theory of gamma-positivity, for the dimension of the homology of the Rees product of a graded poset $P$ with a certain $t$-analogue of the chain of the same length as $P$. Equivariant generalizations of these formulas are proven in this paper, when a group of automorphisms acts on $P$, and are applied to establish the Schur gamma-positivity of certain symmetric functions arising in algebraic and g…

Poset homology of Rees products, and $q$-Eulerian polynomials

The notion of Rees product of posets was introduced by Björner and Welker, where they study connections between poset topology and commutative algebra. Björner and Welker conjectured and Jonsson proved that the dimension of the top homology of the Rees product of the truncated Boolean algebra $B_n \setminus \{0\}$ and the $n$-chain $C_n$ is equal to the number of derangements in the symmetric group $\mathfrak S_n$. Here we prove a refinement of this result, which involves the Eulerian numbers, and a $q$-analog of both the refinement and the original conjecture, which comes from replacing the B…

格雷厄姆幂保持EL壳性

Segre powers of posets preserve EL-shellability

本文研究了格 poset 的格雷厄姆幂保持 EL-shellable 性，并推导了其等级选择不变量的公式。

布尔格和子空间格的Segre幂的同调

Homology of Segre powers of Boolean and subspace lattices

研究布尔格和子空间格的Segre幂的同调表示，揭示对称群作用下的分解公式及秩选同调的不变量关系。