By Michel Broué
Weyl teams are specific circumstances of complicated mirrored image teams, i.e. finite subgroups of GLr(C) generated through (pseudo)reflections. those are teams whose polynomial ring of invariants is a polynomial algebra.
It has lately been found that advanced mirrored image teams play a key function within the thought of finite reductive teams, giving upward thrust as they do to braid teams and generalized Hecke algebras which govern the illustration thought of finite reductive teams. it truly is now additionally widely agreed upon that a few of the identified homes of Weyl teams might be generalized to complicated mirrored image teams. the aim of this paintings is to provide a reasonably broad remedy of many easy homes of complicated mirrored image teams (characterization, Steinberg theorem, Gutkin-Opdam matrices, Solomon theorem and purposes, etc.) together with the fundamental findings of Springer concept on eigenspaces. In doing so, we additionally introduce simple definitions and homes of the linked braid teams, in addition to a short advent to Bessis' lifting of Springer idea to braid groups.
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Additional info for Introduction to Complex Reflection Groups and Their Braid Groups
28). (1) Let i such that 1 ≤ i ≤ r. The family (u1 , u2 , . . , ui ) is algebraically free, hence it cannot be contained in k[v1 , v2 , . . , vi−1 ]. Hence there exist j ≥ i and l ≤ i such that vj does appear in ul . It follows that ej ≤ ul , hence ei ≤ ej ≤ dl ≤ di . ei −1 (2) We know that grdimR = ( i=r . Thus it suﬃces to prove i=1 (1 − q )) i=r i=r ei di that i=1 (1 − q ) = i=1 (1 − q ) if and only if ei = di for all i (1 ≤ i ≤ r), which is left as an exercise. 28, we see in particular that the family (e1 , e2 , .
Un ] mudulo M is k, hence k[u1 , u2 , . . , un ] = R again by Nakayama’s lemma. (2) The equivalence between (ii) and (iii) follows from Nakayama’s lemma. If (i) holds, then (u1 , u2 , . . , un ) generates M by (1), and if it contains a proper system of generators of R, say (u1 , u2 , . . , um ) (m < r) then again by (1) we have R = k[u1 , u2 , . . , um ], a contradiction with the hypothesis about the Krull dimension of R. Assume (iii) holds. Since R is a polynomial algebra with Krull dimension r, and since (i)⇒(iii), we see that the dimension of M/M2 is r.
8 that Δp is invariant by all reﬂections in G, hence invariant under G if G is generated by reﬂections. 1 The Coinvariant Algebra We set SG := S/MS and we call that graded k–algebra the coinvariant algebra of G. The algebra SG is a ﬁnite dimensional k–vector space, whose dimension is the minimal cardinality of a set of generators of S as an R–module (by Nakayama’s lemma). Thus there is an integer M such that 1 M ⊕ · · · ⊕ SG , SG = k ⊕ SG and so in particular n>M S n ⊆ MS . 13. The set of ﬁxed points of G in SG is k.
Introduction to Complex Reflection Groups and Their Braid Groups by Michel Broué