Permutation representations arising from simplicial complexes
Let \({\mathcal D}\) be a complete discrete valuation ring with residue field k and let G be a finite group. Let \(\Omega\) (G) be the Burnside ring formed from G-sets, \(A_{{\mathcal D}}(G)\) the Green ring formed from \({\mathcal D}G\)- lattices and \(R_ k(G)\) the Grothendieck ring formed from kG-modules. There are natural homomorphisms \(\Omega\) (G)\(\to^{r}A_{{\mathcal D}}(G)\to^{\pi}R_ k(G).\) Let \(\Delta\) be a G-complex. For each dimension q, the set \(\Delta_ q\) of q-simplexes is a G-set. The Lefschetz invariant \(\Lambda_ G(\Delta)\) is defined to be \(\sum_{q}(-1)\) \(q\Delta_ q\), considered as an element of \(\Omega\) (G). This is the principal object of study in this paper. Set \(L_ G(\Delta)=r(\Omega_ G(D))\) and \(\lambda_ G(\Delta)=\pi (L_ G(\Delta))\). \(\lambda_ G(\Delta)\), called the Lefschetz character of \(\Delta\), has been widely used. The Lefschetz module \(L_ G(\Delta)\) was introduced by P. J. Webb; it contains more information than \(\lambda_ G(\Delta)\), as ker \(\pi\) is large. However \(\Omega_ G(\Delta)\) contains more information again. Other advantages of working in \(\Omega\) (G) are that (a) \(\Omega_ G(\Delta)\) depends only on Euler characteristics \(\chi\) (\(\Delta\) S) of fixed point complexes \(\Delta\) S for certain subgroups S of G and (b) one can take full advantage of explicit formulae for idempotents in \(\Omega\) (G). In section 2, (a) is used to generalize Webb's result that the reduced Lefschetz module \(\tilde L_ G(\Delta)\) is projective, provided sufficiently many Euler characteristics of fixed point complexes vanish. The generalization is to relative projectivity. As a corollary Brown's (and Quillen, etc.) theorem that \({\tilde \chi}\)(S\({}_ p(G))\equiv 0 mod | G_ p|\) and the fact that the number of Sylow p-subgroups \(G_ p\) of G is \(\equiv 1 (mod p)\) both follow. Section 3 looks at \(\Delta\) arising from a G-poset, with chains as simplexes. Bouc's notion of Möbius modules is generalized. Stanley's theorem on the Lefschetz characters of rank-selected posets is extended to \(\Omega\) (G). Section 4 tackles the poset S(G) of nontrivial proper subgroups of G and \(\tilde L_ G(S(G))\) is determined. The lattice of all subsets of a G-set is considered in section 5, with a back application to Euler characteristics of certain subposets of S(G). In section 6, the G-poset T(V) of K-subspaces of a KG-module V, where K is a finite field, is considered and \(\Omega_ G(T(V))\) studied. There arises a connection to the Steinberg character of GL(V).
scientific article; zbMATH DE number 4040155
scientific article