Whitney homology of semipure shellable posets
Group representations on homologies of posets have been the object of intensive research in the past few decades. Usually posets are expected to be Cohen-Macaulay or shellable, since then only the top homology is nontrivial so that the result follows from the routine application of the Hopf trace formula. The situation becomes much more complicated if a poset \(P\) has nonvanishing homology in various dimensions, as it could occur, for example, for nonpure \(P\). In the paper under review the author applies the theory of nonpure shellability (developed by \textit{A. Björner} and \textit{M. L. Wachs} [Trans. Am. Math. Soc. 348, 1299-1327 (1996; Zbl 0857.05102) and ibid. 349, 3945-3975 (1997; Zbl 0886.05126)]) to computing representations of the symmetric group on each homology for a poset of partitions whose block sizes are congruent to \(k \text{mod} d\) for any \(k\) and \(d\). The cases \(k=0\) and \(k=1\) (when the posets are pure shellable) were considered by \textit{A. R. Calderbank, P. Hanlon} and \textit{R. W. Robinson} [Proc. Lond. Math. Soc., III. Ser. 53, 288-320 (1986; Zbl 0602.20017)]. However, for general \(k\) the \(k \text{mod} d\) partition poset is not pure! An application to subspace arrangements is also discussed.
scientific article; zbMATH DE number 1275699
scientific article