Reflection groups. Arun Ram Department of Mathematics University of Wisconsin Madison, WI s f(α),λ = fs α,λ f 1.

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1 Reflection groups Arun Ram Department of Mathematics University of Wisconsin Madison, WI 576 Definition of a reflection group Let V be a finite dimensional complex vector space of dimension n > Let, : V V C be a Hermitian form on V, ie such that ax + by, z = a x, z + b y, z x, y = y, x, for all x, y V, x, ay + bz = ā x, y + b x, z, where ā is the complex conjugate of a Let α V and let s α,λ : V V be the reflection in the hyperplane with eigenvalue λ Then x, α () s α,λ (x) = x + (λ ) α, α α () s α,λ s α,µ = s α,λµ, H α = {x V x, α = } () s α,λ x, s α,λ y = x, y for all x, y V if and only if λ λ = () Let f : V V be such that fx, fy = x, y for all x, y V Then s f(α),λ = fs α,λ f Proof () Write x = x α + x with x C and x (Cα) Then and s α,λ (x) = λx + x x, α x + (λ ) α, α α = x x, α α + x + (λ ) α, α α = x α + x + λx α x α = λx α + x () s α,λ s α,µ (x) = s α,λ (µx α + x ) = λµx α + x = s α,λµ (x) () λx α+x, λy α+y = λ λ α, α x y + x, y and x α+x, y α+y = x y α, α + x, y If x = y = and x y = we get λ λ = () fs α,λ f = f ( f (x) + (λ ) f x,α α,α α ) = x + (λ ) f x,f fα fα,fα fα = s fα,λ

2 Let h be a vector space over a field F and let n = dim(h ) s α GL(h ) such that A reflection is an element dim ( (h ) sα) = n, where (h ) sα = {x h s α x = x} A reflection group is a finite subgroup W of GL(h ) generated by reflections If W is a reflection group the set {H α s α is a reflection in W } is the hyperplane arrangement corresponding to G Since H s is codimension H α = ker α where α: V C is a linear form on V The form α C is determined up to constant multiples A linear form α: V C determines a hyperplane and a reflection s α : V V by H α = {v V α(v) = } v, α s α,ξ (v) = v + (ξ ) α, α α Let V be a complex vector space of dimension n A reflection is an element s GL(V ) such that codim(v s ) = Let M = (q ij ) be such that (i) q ij Z, (ii) M is symmetric, (iii) if q ij is odd then q ii = q jj Let D be the graph with vertices indexed by,,, n with edges labeled q ij and the label q ii are vertex i q ij q ii q jj If q ij = we do not draw the edge between vertex i and vertex j The Cartan matrix (a ij is given by setting a ii = sin ( π ), aij = if q ij =, q ii a ij = cos ( π ) sin ( π π ), if qij > q ij q ii q ij Since cos (π/) = / and sin (π/) = / we have that cos ( π ) sin ( π π ) q ij q ii q ij if q ij >

3 So A is a real symmetric matrix The matrix A is irreducible if we cannot partition the index set {,,, n} into two proper subsets I and J such that a ij = for all i I, j J, I J ( ) (also figure out how to get TeX to label the columns I and J) A is irreducible if and only if D is connected The matrix A is of affine type if there is a vector of positive real numbers m = m m n such that Am = A subdiagram of D is any diagram obtained from D by (a) deleting some vertices (and the edges issuing from them), (b) decreasing the labels on some of the edges (q ij q ij ) (c) decreasing the labels on some of the vertices (q ii q ii ) such that if q ij > then q ii q jj The graph D is of affine type if the corresponding Cartan matrix is of affine type Lemma Let d be a diagram of affine type Then no proper subdiagram of D is of affine type Proof (a) Reducing numbers on edges of D corresponds to increasing off diagonal entries a ij of A (decreasing (a ij )) (b) Reducing numbers in vertices of D corresponds to increasing diagonal entries a ii of A (c) Deleting vertices of D corresponds to passing to a principal submatrix Hence the Cartan matrix of D is of the form B I where B A (take B = A outside I) By Lemma, B I is nonsingular, hence D is not of affine type We can use the graph D (or the matrix M) to define a group W by generators r,, r n and relations r q ii i =, r i r j r i }{{} q ij mathrmfactors = r j r i r j }{{} q ij mathrmfactors We will define a representation of W on a space V by reflections Let V = span{α,, α n } so that the symbols α,, α n are a basis of V Define a Hermitian form on V by α i, α i = a ii = sin( π ), q ii α i, α j = a ij = q ii cos ( π q ij ) sin ( π q ii q jj π ) q ij

4 In general the formula s α : V V, defines the reflection in the hyperplane x, α s α,λ (x) = x + (λ ) α, α α, α V, λ C, H α = {x V x, α = } with eigenvalue λ The endomorphism s α,λ is an isometry if and only if λ λ = Define a representation of W on V by Φ: W GL(V ) r j s αj,λ j where λ j = e πi/q jj This representation has, as a W -invariant form and it is faithful if Φ(W ) is finite This happens exactly when the form, is positive definite (The proof of this in Koster refers to Coxeter s classifications and presentations for one direction This is unpleasant) Classification of diagrams of affine type Let D be an affine diagram () Suppose D contains a cycle (with vertices) Then D has a subdiagram of the form (A n ) (n + vertices, n ), () D has a branch of order Then D has a subdiagram of the form (D ) () Suppose that D has or more branch points of order The D has a subdiagram of the form (D n, n 5) () Suppose that D has one branch point of order and at least one multiple bond Then D

5 has a subdiagram of the form sin(π/p) p sin(π/p) a a ( B p n, n ) where sin(π/p) a = (5) Suppose D has one branch point or order and no mulitple bonds Then D has a subdiagram of the form E 6 E E 8 (6) We have exhausted the possibilities where D has a branch point Assume now that D is a 5

6 chain If D has at least two multiple bonds it will contain one of the following diagrams sin(π/p) p p sin(π/p) a a where a = sin(π/p) sin(π/p and b = ) sin(π/p) p p sin(π/p ) sin(π/p ) b b sin(π/p ) a a sin(π/p) a (C p, p ) a sin(π/p) where a = (7) Assume now that D is a chain with just one multiple bond (C p,p n, n ) (a) strength 6 (b) strength ` / / 8 ( / / / / / ) ( ) / / / (c) strength ( ( ) ) 6

7 (d) strength / / / / / / / / / / (F ) / 6 ( / / ) (8) Assume that D has no multiple bonds Then D has a subdiagram of the form sin(π/) cos(π/) cos(π/) sin(π/) cos(π/) = cos(π/) sin(π/) 7

8 ( 6 6 sin(π/6) cos(π/) cos(π/) sin(π/6) (note that all the numbers on the vertices must be equal in this case) ) ( = ) The Chevalley-Shephard-Todd theorem Theorem Let h be a vector space and let W be a finite subgroup of GL(h ) The following are equivalent (a) W is a reflection group, W = s α s α W is a reflection (b) S(h ) W is a polynomial ring, S(h ) W = C[f, f,, f n ] (c) S(h ) is a free S(h ) W -module Let R be a local regular ring, fm a maximal ideal, and K = R/m the residue field The R G is a local ring with maximal ideal Assume that m G = m R G (a) R G is noetherian and R is a finite type R G module, and (b) The composition R G R k is surjective Define V = m/m (a k vector space: the tangent space) The action of G on R define a homomorphism ε: G GL(V ) (a) Let p be a prime ideal of height in R and let s G be such that s(p) = p and s operates trivially on R/p Show that ε(s) is a pseudoreflection in V (Remark taht the image of p in m/m is of dimension or Structure theorems Theorem (a) (Chevalley, Shephard-Todd) A finite group W GL(h ) is generated by reflections if and only if S(h ) W = C[I,, I r ] where I,, I r are algebraically independent and homogeneous (b) (Solomon) Let W be a finite reflection group Then (see Benson page 86) (c) S(h ) = H S(h ) W (S(h ) Λ(h)) W = C[I,, I r ] Λ(dI,, di r ) 8

9 Some additional remarks: (a) and λ R, λ det ( I j x j ) = λp, where p = α R + α (b) H has basis {h w = w() w W } where w are the BGG operators and deg(h w ) = l(w) Theorem (Molien theorems) (a) (b) Proof Now apply to S(h ) Λ(h): P ((S(h ) Λ(h)) W ; q, t) = W P (S(h ) W ; t) = W j Z q j Tr(w, Λ j h) = j Z t j Tr(w, S j h) = det( + wq) det( wt) det( wt) ( + λ i q) = det( + w q, h ) i= det( wt, h ) = ( λ i t) W w P ((S(h ) Λ(h)) W ; q, t) = W Tr ( q,t w, S(h ) Λ(h)) ) = W i= Tr q,t (w, S(h ) Λ(h)) Theorem P (S(h ); t) = i= t P (S(h ) W ; t) = t d i i= P (H; t) = i= t d i t and P ((S(h ) Λ(h)) W ; q, t) = i= + qt d i t d i Proof (a) is clear (b) follows from Chevalley s theorem (c) follows from part (c) of the structure theorem since it implies (d) follows from Solomon s theorem P (S(h ); t) = P (H; t)p (S(h ) W ; t) 9

10 Let Corollary 5 (a) (b) d(w) = dim(v w ) = multiplicity of as an eigenvalue of w, d m (w) = multiplicity of e πi/m as an eigenvalue of w, {, if m divides d i, χ(m d i ) =, if m does not divide d i t dm(w) = (t χ(m di) + d i ) i= t d(w) = (t + d i ) i= (c) The number of reflections in W is r i= (d i ) (d) W = r i= d i Proof (b) follows from (a) by putting m = (c) follows from taking the coefficient of t r on both sides of the identity in (b) (d) follows by putting t = in (b) (a) (following Macdonald) Replace q and t with q/ξ and t/ξ, where ξ = e πi/m Then det(ξ + qw) ( ξ d W det(ξ tw) = i + qt d ) i ξ d i t d i from W Now let q = ( t)x Then i= det( + qw) ( + qt d det( tw) = i ) t d i det(ξ + qw) det(ξ tw) = i= i= (ξ + λ i (( t)x )) (ξ λ i t) So, now take the limit as t When we do this we get the result we want but we will need d i = W To get this set q = and multiply by ( t) r in the Molien formula to get Now set t = Then W i= LHS = W ( + ) = W w ( t) r det( tw) = t t d i i= and RHS = i= d i

11 Nice formulas Symmetric and determinantal functions Let S(h ) be the symmetric algebra of h Then u + S(h ) = symmetric polynomials u S(h ) = determinant symmetric polynomials and, as vector spaces u + S(h ) sim u S(h ) f fa ρ Weyl denominators Theorem w t α eα e α = t R(w), α R + where {t α α R + } are indeterminates t E = α E t α, for E R +, and R(w) = {α R + wα R } Corollary () w teα e α = t l(w) α R + () t+ht(α) t ht(α) = α R + t l(w) where ht(α) = ρ, α () If h i is the number of roots of height i then (h, h, ) = (d, d, ) t

12 A n n n Ã n n n B n n n Bn n n n C n n n Cn n n D n n n Dn n n n 5 E 6 Ẽ E 7 Ẽ E 8 Ẽ F F G G References [Dr] G Drinfel d, A new realization of Yangians and quantized affine algebras, Soviet Math Dokl 6 No, (998), 6

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f