Embeddings of root systems. II: Permutation root systems.
type: Item
instance of: scholarly article
title: Embeddings of root systems. II: Permutation root systems.
zbMATH Open document ID: 1193.20047
author: Matthew Dyer
published in: Journal of Algebra
publication date: 2009-04-29T00:00:00Z
review text: In part I [\textit{M. J. Dyer}, J. Algebra 321, No. 11, 3226-3248 (2009; Zbl 1181.20036)] the author studied the embeddings of linear root systems over commutative rings. The present paper studies a class of embeddings of permutation root systems, with properties abstracting those of examples in part I. Let \(G\) be a group and \(\{\pm 1\}\) the sign group. The group \(G\times\{\pm 1\}\) acts on a set of the form \(R\times\{\pm 1\}\), where the action of \(\{\pm 1\}\) is the natural one. Such an action is equivalent to an action of \(G\) on \(R\) together with a cocycle \(N\colon G\to\mathcal P(R)\) of \(G\) in the power set \(\mathcal P(R)\) (with symmetric difference as addition and natural \(G\)-action). An expression \(g_1\cdots g_n\) in \(G\) is defined to be compatible if \(N(g_1\cdots g_n)\) is the disjoint union of the sets \(g_1\cdots g_{i-1}N(g_i)\). A cocycle \(N\colon W\to\mathcal P(R)\) of a Coxeter system \((W,S)\) is called a root cocycle if any expression \(xy\), \(x,y\in W\), with \(\ell(xy)=\ell(x)+\ell(y)\) is compatible for \(N\), where \(\ell\) is the length function of \((W,S)\). The author defines, by some rank-two conditions, a notion of an admissible subset \(S\) of \(G\) with respect to a cocycle \(N\colon G\to\mathcal P(R)\). Then the first main result of the paper asserts that for an admissible subset \(S\) generating a subgroup \(W\) of \(G\), the pair \((W,S)\) is a Coxeter system and the restriction of \(N\) to \(W\) is a non-degenerate root cocycle for \((W,S)\); further, for \(w_i\in W\), an expression \(w=w_1\cdots w_n\) is compatible for \(N\) iff \(\ell(w)=\ell(w_1)+\cdots+\ell(w_n)\). For a Coxeter system \((W,S)\) with \(T\) its reflection set, define \(\widehat W\) to be the set of all permutations \(\theta\) of \(\Psi:=T\times\{\pm 1\}\) such that for any finitely generated reflection subgroup \(W'\) of \(W\), there is some element \(w\in W\) such that \(\theta(t,\varepsilon)=(wtw^{-1},\varepsilon_{w,t}\varepsilon)\) for all \…
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 17B22
zbMATH DE Number: 5549345
zbMATH Keywords: Coxeter groups
zbMATH Keywords: based permutation root systems
zbMATH Keywords: cocycles
zbMATH Keywords: Coxeter systems
zbMATH Keywords: group actions
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.11.001
OpenAlex ID: W2028535090
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DOI: 10.1016/J.JALGEBRA.2008.11.001
description: scientific article; zbMATH DE number 5549345