Homology of Segre powers of Boolean and subspace lattices

Paper

Abstract

Segre products of posets were defined by Björner and Welker (2005). We investigate the homology representations of the

-fold Segre power

of the Boolean lattice

. The direct product

of the symmetric group

acts on the homology of rank-selected subposets of

. We give an explicit formula for the decomposition into

-irreducibles of the homology of the full poset, as well as formulas for the diagonal action of the symmetric group

. For the rank-selected homology, we show that the stable principal specialisation of the product Frobenius characteristic of the

-module coincides with the corresponding rank-selected invariant of the

-fold Segre power of the subspace lattice.