Reflection Groups

Reflection Groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 20 October 2010

Structure theorems for reflection groups

  1. (Chevalley, Shephard-Todd) A finite group WGL𝔥* is generated by reflections if and only if S 𝔥* W = I1,,Ir where I1,,Ir are algebraiclly independent and homogenous.
  2. (Solomon) Let W be a finite reflection group. Then S𝔥* Λ𝔥 W = I1,,Ir Λ dI1,,dIr (see Benson page 86).
  3. S𝔥* S 𝔥*W.

Some additional remarks:
  1. det 𝜕Ij 𝜕xj =λp, where p= αR+ α and λ, λ0.
  2. has basis hw= Δw* wW where Δw are the BGG operators and deg hw = lw.

(Molien theorems)

  1. P S𝔥* Λ𝔥 W ; q,t = 1 W wW det1+wq det1-wt
  2. P S 𝔥* W ;t = 1 W wW 1 det 1-wt

Proof.
j0 qj Tr w, Λj𝔥 = i=1 r 1+ λi-1 q = det 1+w-1q, 𝔥* . j0 tj Tr w,Sj𝔥 = 1 det 1-wt,𝔥* = i=1 r 1 1-λit . Now apply 1 W wW w to S𝔥*Λ𝔥: P S𝔥* Λ𝔥 W ; q,t = 1 W Trq,t wW w, S𝔥* Λ𝔥 = 1 W wW Trq,t w, S𝔥* Λ𝔥 .

P S𝔥* ;t = i=1 r 1 1-t P S𝔥*W ;t = i=1 r 1 1-tdi P ;t = i=1 r 1-tdi 1-t and P S𝔥* Λ𝔥 W ;q,t = i=1 r 1+qtdi-1 1-tdi .

Proof.
(a) is clear. (b) follows from Chevalley's theorem. (c) follows from part (c) of the structure theorem since it implies P S𝔥* ;t = P;t P S ( 𝔥* ) W ;t . (d) follows from Solomon's theorem.

Let dw = dimVw= multiplicity of 1 as an eigenvalue of w, dmw = multiplicity of  e2πi/m  as an eigenvalue of  w, Χmdi = 1, if m divides di, 0, if m does not divide di.

  1. wW t dm(w) = i=1 r t χmdi + di -1 .
  2. wW t dw = i=1 r t+di-1 .
  3. The number of reflections in W is i=1 r di-1 .
  4. W= i=1 r di .

Proof.
  1. follows from (a) by putting m=1.
  2. follows from taking the coefficient of tr-1 on both sides of the identity in (b).
  3. follows by putting t=1 in (b).
  4. (following Macdonald) Replace q and t with t/ξ, where ξ=2πi/m. Then 1 W wW detξ+qw detξ-tw = i=1 r ξdi +q t di-1 ξ di - t di from 1 W wW det 1+qw det 1-tw = i=1 r 1+q tdi-1 1- tdi . Now let q=1-tX-1. Then det ξ-qw det ξ-tw = i=1 r ξ+λi 1-t X -1 ξ-λit . So, now take the limit as t1. When we do this we get the result we want but we will need i=1 r di= W . To get this set q=0 and multiply by 1-tr in the Molien formula to get 1 W wW 1-tr det 1-tw = i=1 r 1-t 1-tdi . Now set t=1. Then LHS= 1 W 1+ w1 0 = 1 W and RHS= i=1 r 1 di .

References

[AMR] S. Ariki, A. Mathas, and H. Rui, Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J. 182 (2006), 47-134. MR2235339 (2007d:20005)

[BB] A. Beliakova and C. Blanchet, Skein construction of idempotents in Birman-Murakami-Wenzl algebras, Math. Ann. 321 (2001), 347-373. MR1866492 (2002h:57018)

[Bou] N. Bourbaki, Groupes et Alg├Ębres de Lie, Masson, Paris, 1990.

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.

[GH1] F. Goodman and H. Hauschild Mosley, Cyclotomic Birman-Wenzl-Murakami algebras. I. Freeness and realization as tangle algebras, J. Knot Theory Ramifications 18 (2009), 1089-1127. MR2554337 (2010j:57014)

[Naz] M. Nazarov, Young's orthogonal form for Brauer's centralizer algebra, J. Algebra 182 (1996), no. 3, 664-693. MR1398116 (97m:20057)

[OR] R. Orellana and A. Ram, Affine braids, Markov traces and the category 𝒪, Algebraic groups and homogeneous spaces, 423-473, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007. MR2348913 (2008m:17034)

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