A faithful unitary representation of the 2-compact group \(DI(4)\)
type: Item
instance of: scholarly article
title: A faithful unitary representation of the 2-compact group \(DI(4)\)
zbMATH Open document ID: 1169.55009
author: Krzysztof Ziemiański
published in: Journal of Pure and Applied Algebra
publication date: 2009-04-16T00:00:00Z
review text: A \(p\)-compact group, as introduced by \textit{W. G. Dwyer} and \textit{C. W. Wilkerson} in [Ann. Math. (2) 139, No.~2, 395--442 (1994; Zbl 0801.55007)], is the homotopical analogue at a prime \(p\) of a compact Lie group. In fact, the \(p\)-completion of a compact Lie group with a \(p\)-group of components yields such an object and basically the only other, exotic, \(2\)-compact group is \(DI(4)\) (a complete classification has been obtained by \textit{K. K. S. Andersen} and \textit{J. Grodal} after this article has been written [``The classification of \(2\)-compact groups'', J. Am. Math. Soc. 22, No.~2, 387--436 (2009)]). The article under review is concerned with finding a faithful unitary homotopy representation of \(DI(4)\), i.e. a map \(BDI(4) \rightarrow B\text{U}(n)^{\wedge}_2\) with mod \(2\) finite homotopy fibre, which makes therefore \(H^*(BDI(4);\mathbb{F}_2)\) a finitely generated module over \(H^*(B\text{U}(n);\mathbb{F}_2)\). The main result is the construction of such a representation for \(n=2^{46}\). Together with previous work of \textit{N. Castellana} at odd primes [``Representacions homotòpiques de grups \(p\)-compactes'', Ph.D. Thesis, Univ. Autón. Barcelona, Bellatera (2000) and Trans. Am. Math. Soc. 358, No.~7, 2799--2819 (2006; Zbl 1145.55014)] this shows by a case-by-case investigation based on the classification that every \(p\)-compact group admits a faithful unitary representation. The strategy is to consider the centralizer decomposition diagram used by \textit{W. G. Dwyer} and \textit{C. W. Wilkerson} in [J. Am. Math. Soc. 6, No.~1, 37--64 (1993; Zbl 0769.55007)] to construct \(BDI(4)\). The largest centralizer is \(\text{Spin}(7)\) and the author carefully chooses a representation of the (discrete approximation of the) maximal torus normalizer which he shows extends to a homotopy compatible family of maps from the \(p\)-stubborn subgroup decomposition diagram of \(\text{Spin}(7)\). Such a map was proven by the author in [J. Pure Ap…
reviewed by: Jérôme Scherer
Mathematics Subject Classification ID: 55R37
Mathematics Subject Classification ID: 55S35
zbMATH DE Number: 5544378
zbMATH Keywords: homotopy representation
zbMATH Keywords: \(p\)-compact group
zbMATH Keywords: centralizer decomposition
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.11.042
OpenAlex ID: W2006624485
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DOI: 10.1016/J.JPAA.2008.11.042
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description: scientific article; zbMATH DE number 5544378