Equivariant K-theory of real vector spaces and real projective spaces
type: Item
instance of: scholarly article
title: Equivariant K-theory of real vector spaces and real projective spaces
zbMATH Open document ID: 1011.19005
DOI: 10.1016/S0166-8641(01)00190-0
author: Max Karoubi
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: Let \(G\) be a finite group and \(V\) a real vector space provided with a linear action of \(G\). When \(V=\mathbb{R}^{8n}\) the equivariant \(K\)-theory \(K^*_G(V)\) can be described in terms of representations of \(G\) under some spinoriality conditions. One purpose of the author is to fill the gap in this result due to \textit{M. F. Atiyah} [Q. J. Math., Oxf., II. Ser. 19, 113-140 (1968; Zbl 0159.53501)], and another one is to study the problems which interest the author in such a computation of \(K^*_G(V)\). The paper consists of fifteen theorems, four corollaries and also five remarks. In this review we consider only complex \(K\)-theory and confine ourself to the case when \(G=S_n\) the symmetric group and \(V=\mathbb{R}^n\) on which \(G\) acts naturally. Then one of the main results is for example: (Corollary 1.9) \(K^0_G(V)\cong \mathbb{Z}^{p_n}\) and \(K^1_G(V) \cong\mathbb{Z}^{i_n}\) where \(p_n\) and \(i_n\) denote the number of partitions of \(n\) of the type \(n=\lambda_1+ \cdots+\lambda_{2k}\) with \(1\leq \lambda_1< \cdots< \lambda_{2k}\) and of the type \(n=\lambda_1+ \cdots+ \lambda_{2 k+1}\) with \(1\leq\lambda_1 <\cdots< \lambda_{2k+1}\), respectively. The key to the author's approach is using the \(K\)-theories of Banach categories built up in his book [\textit{M. Karoubi}, ``\(K\)-theory'', Grundlehren 226 (1978; Zbl 0382.55002)] for instance and the method developed by himself as well. In fact the proof of the freeness of the group \(K^*_G(V)\) above is referred to the author's earlier results [\textit{H. Karoubi}, Lect. Notes Math. 136, 187-253 (1970; Zbl 0197.19902)]. On the one hand, the computation of its rank is done using an isomorphism \(K^*_G(V) \otimes\mathbb{C} \cong\oplus H^*_c(V^{g_i} /C_{g_i}; \mathbb{C})\) induced by the equivariant Chern character introduced by \textit{J. Slominska} et al. Here \(g_i\)'s denote representatives of the conjugacy classes of elements of \(G\) and \(C_{g_i}\) the centralizer of \(g_i\). Let \(P(V)\) d…
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 19L47
Mathematics Subject Classification ID: 20B30
Mathematics Subject Classification ID: 19L64
zbMATH DE Number: 1787583
zbMATH Keywords: equivariant \(K\)-theory
zbMATH Keywords: symmetric groups
zbMATH Keywords: real projective spaces
zbMATH Keywords: Banach categories
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1787583