Homology of Segre powers of Boolean and subspace lattices

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 ${\text{sym}}_n^{\times t}$ of the symmetric group ${\text{sym}}_n$ acts on the homology of rank-selected subposets of $B_n^{(t)}$. We give an explicit formula for the decomposition into ${\text{sym}}_n^{\times t}$-irreducibles of the homology of the full poset, as well as formulas for the diagonal action of the symmetric group ${\text{sym}}_n$. For the rank-selected homology, we show that the stable principal specialisation of the product Frobenius characteristic of the ${\text{sym}}_n^{\times t}$-module coincides with the corresponding rank-selected invariant of the $t$-fold Segre power of the subspace lattice.