Permutation representations arising from simplicial complexes
type: Item
instance of: scholarly article
title: Permutation representations arising from simplicial complexes
zbMATH Open document ID: 0638.20006
DOI: 10.1016/0097-3165(87)90078-1
author: Jacques Thévenaz
published in: Journal of Combinatorial Theory. Series A
publication date: 1987-01-01T00:00:00Z
review text: 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 the…
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C11
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 18G30
zbMATH DE Number: 4040155
zbMATH Keywords: finite group
zbMATH Keywords: Burnside ring
zbMATH Keywords: G-sets
zbMATH Keywords: Green ring
zbMATH Keywords: Grothendieck ring
zbMATH Keywords: Lefschetz invariant
zbMATH Keywords: Lefschetz character
zbMATH Keywords: Lefschetz module
zbMATH Keywords: Euler characteristics
zbMATH Keywords: fixed point complexes
zbMATH Keywords: relative projectivity
zbMATH Keywords: G-poset
zbMATH Keywords: Steinberg character
MaRDI profile type: Publication
full work available at URL: 0097-3165(87)90078-1
OpenAlex ID: W2091236822
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description: scientific article; zbMATH DE number 4040155