Homology of Segre powers of Boolean and subspace lattices

2024/08/15 by Yifei Li, Li, Yifei, Sheila Sundaram +1 · 1 citation
Computer Science · Mathematics · #05E05 #05E10 #20C30 #55P99 #Advanced Algebra and Logic #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #Rough Sets and Fuzzy Logic

paper · pdf · doi:10.48550/arxiv.2408.08421

openalex publication_date 2024/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Segre products of posets were defined by Björner and Welker (2005). We investigate the homology representations of the t-fold Segre power B_n^(t) of the Boolean lattice B_n. The direct product \sym_n^{× t} of the symmetric group \sym_n acts on the homology of rank-selected subposets of B_n^(t). We give an explicit formula for the decomposition into \sym_n^{× t}-irreducibles of the homology of the full poset, as well as formulas for the diagonal action of the symmetric group \sym_n. For the rank-selected homology, we show that the stable principal specialisation of the product Frobenius characteristic of the \sym_n^{× t}-module coincides with the corresponding rank-selected invariant of the t-fold Segre power of the subspace lattice.

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