Séminaire Lotharingien de Combinatoire, 93B.151 (2025), 12 pp.

Yifei Li and Sheila Sundaram

Rank-Selected Segre Powers of the Boolean Lattice

Abstract. Segre products of posets were defined by Björner and Welker [J. Pure Appl. Algebra (2005)]. We determine the rank-selected homology representations of the t-fold Segre power B_n^(t) of the Boolean lattice B_n, which carries an action of the t-fold direct product S_n^×t of the symmetric group S_n. We give formulas for the decomposition into S_n^×t-irreducibles of the homology of the full poset, as well as for the diagonal action of S_n. We show that the stable principal specialisation of the product Frobenius characteristic coincides with the corresponding rank-selected invariant of the t-fold Segre power of the subspace lattice.

Received: November 15, 2024. Accepted: February 15, 2025. Final version: April 1, 2025.

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