Homology of Segre powers of boolean and subspace lattices

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