A basis for the homology of the \(d\)-divisible partition lattice
Let \(\Pi^d_n\) (for \(d|n\)) denote the lattice of all those partitions of \(\{1, 2,\dots, n\}\) whose block sizes are divisible by \(d\). These interesting lattices are shellable, but not geometric for \(d> 1\) and \(n\geq 3d\). Their homology \(H_{n/d- 2}(\Pi^d_n)\) is at the core of \textit{A. R. Calderbank}, \textit{P. Hanlon} and \textit{R. W. Robinson's} work [Proc. Lond. Math. Soc., III. Ser. 53, 288-320 (1986; Zbl 0602.20017)] on characters of the symmetric group, since it supports especially interesting representations by the permutation action of \({\mathcal S}_n\) on \(\{1, 2,\dots, n\}\). In this paper, an explicit basis of spherical cycles is constructed for \(\Delta(\overline{\Pi^d_n})\), which serves to explain the homology action of \({\mathcal S}_n\) as a permutation representation, and provides an isomorphism of the restriction to \({\mathcal S}_{n- 1}\) with a particular Specht module (corresponding to a skew hook shape). Interestingly, this very natural cycle basis does not, in the special case of \(d= 1\), restrict to the ``broken circuit'' cycle basis of the partition lattice constructed by \textit{A. Björner} [On the homology of geometric lattices, Algebra Univers. 14, 107-128 (1982; Zbl 0484.06014)]. Also, it is not the cycle basis that is induced by the explicit EL-shelling of \(\Pi^d_n\) that is also constructed in the present paper.
scientific article; zbMATH DE number 859771
scientific article