Embeddings of root systems. II: Permutation root systems.
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 \(t\in W'\cap R\) and \(\varepsilon\in\{\pm 1\}\), where \(\varepsilon_{w,t}\) is 1 if \(\ell(wt)>\ell(w)\) and \(-1\) if \(\ell(wt)<\ell(w)\). Then \(\widehat W\) is a group of permutations of \(\Psi\) containing \(W\) as a subgroup. The paper applies the above main result to the following case: \(R\) is a subset of the set of reflections \(T'\) of a Coxeter system \((W',S')\), and \(G\) is a subgroup of \(\widehat{W'}\) acting on \(R\) by conjugation, with cocycle \(N\colon G\to\mathcal P(R)\) given by \(N(g)=N'(g)\cap R\) where \(N'\) is the reflection cocycle of \(\widehat{W'}\). The author considers a class of the embeddings \(W\to\widehat{W'}\) which naturally generalizes \textit{B. Mühlherr}'s embedding \(W\to W'\) [in Finite geometry and combinatorics. Lond. Math. Soc. Lect. Note Ser. 191, 277-287 (1993; Zbl 0820.20048)]. For embeddings in this class, each simple reflection of \(W\) is a convergent, possible infinite product of commuting simple reflections of \(W'\). The second main result of the paper completely classifies all embeddings of this type, essentially as those induced by ``strong'' admissible maps in the sense of \textit{G. Lusztig}, [Trans. Am. Math. Soc. 277, 623-653 (1983; Zbl 0526.22015)].
scientific article; zbMATH DE number 5549345
scientific article