Complex reflection groups and their associated braid groups and Hecke algebras Michel Brou´e Institut Henri{Poincar´e January 2009 Michel Brou´e Reflection groups, braids, Hecke algebras Each choice of a basis (v1; v2;:::; vr ) of V determines an identification of S with a graded polynomial algebra S ' K[v1; v2;:::; vr ] with deg vi = 1 : Let G be a finite subgroup of GL(V ). The group G acts on the algebra S, and we let R := SG denote the subalgebra of G–fixed polynomials. Let K be a characteristic zero field and let V be an r{dimensional K{vector space. Let S be the symmetric algebra of V . Polynomial invariants of finite linear groups Michel Brou´e Reflection groups, braids, Hecke algebras Let G be a finite subgroup of GL(V ). The group G acts on the algebra S, and we let R := SG denote the subalgebra of G–fixed polynomials. Each choice of a basis (v1; v2;:::; vr ) of V determines an identification of S with a graded polynomial algebra S ' K[v1; v2;:::; vr ] with deg vi = 1 : Polynomial invariants of finite linear groups Let K be a characteristic zero field and let V be an r{dimensional K{vector space. Let S be the symmetric algebra of V . Michel Brou´e Reflection groups, braids, Hecke algebras Let G be a finite subgroup of GL(V ). The group G acts on the algebra S, and we let R := SG denote the subalgebra of G–fixed polynomials. Polynomial invariants of finite linear groups Let K be a characteristic zero field and let V be an r{dimensional K{vector space. Let S be the symmetric algebra of V . Each choice of a basis (v1; v2;:::; vr ) of V determines an identification of S with a graded polynomial algebra S ' K[v1; v2;:::; vr ] with deg vi = 1 : Michel Brou´e Reflection groups, braids, Hecke algebras Polynomial invariants of finite linear groups Let K be a characteristic zero field and let V be an r{dimensional K{vector space. Let S be the symmetric algebra of V . Each choice of a basis (v1; v2;:::; vr ) of V determines an identification of S with a graded polynomial algebra S ' K[v1; v2;:::; vr ] with deg vi = 1 : Let G be a finite subgroup of GL(V ). The group G acts on the algebra S, and we let R := SG denote the subalgebra of G–fixed polynomials. Michel Brou´e Reflection groups, braids, Hecke algebras S = K[v1; v2;:::; vr ] MMM MMMnot free unless... MMM MMM free of rank mjGj R = SG o not a polynomial algebra unless... qqq qqq qqqfree of rank m qqq P = K[u1; u2;:::; ur ] but there exists a graded polynomial algebra P := K[u1; u2;:::; ur ] with deg ui = di and an integer m, such that In general R is NOT a polynomial algebra, Michel Brou´e Reflection groups, braids, Hecke algebras S = K[v1; v2;:::; vr ] MMM MMMnot free unless... MMM MMM free of rank mjGj R = SG o not a polynomial algebra unless... qqq qqq qqqfree of rank m qqq P = K[u1; u2;:::; ur ] and an integer m, such that In general R is NOT a polynomial algebra, but there exists a graded polynomial algebra P := K[u1; u2;:::; ur ] with deg ui = di Michel Brou´e Reflection groups, braids, Hecke algebras S = K[v1; v2;:::; vr ] MMM MMMnot free unless... MMM MMM free of rank mjGj R = SG o not a polynomial algebra unless... qqq qqq qqqfree of rank m qqq P = K[u1; u2;:::; ur ] In general R is NOT a polynomial algebra, but there exists a graded polynomial algebra P := K[u1; u2;:::; ur ] with deg ui = di and an integer m, such that Michel Brou´e Reflection groups, braids, Hecke algebras S = K[v1; v2;:::; vr ] MMM MMMnot free unless... MMM MMM free of rank mjGj R = SG o not a polynomial algebra unless... qqq qqq qqqfree of rank m qqq P = K[u1; u2;:::; ur ] In general R is NOT a polynomial algebra, but there exists a graded polynomial algebra P := K[u1; u2;:::; ur ] with deg ui = di and an integer m, such that S = K[v1; v2;:::; vr ] MMM MMM MMM MMM free of rank mjGj R = SG qqq qqq qqqfree of rank m qqq P = K[u1; u2;:::; ur ] Michel Brou´e Reflection groups, braids, Hecke algebras In general R is NOT a polynomial algebra, but there exists a graded polynomial algebra P := K[u1; u2;:::; ur ] with deg ui = di and an integer m, such that S = K[v1; v2;:::; vr ] MMM MMMnot free unless... MMM MMM free of rank mjGj R = SG o not a polynomial algebra unless... qqq qqq qqqfree of rank m qqq P = K[u1; u2;:::; ur ] Michel Brou´e Reflection groups, braids, Hecke algebras Then S = K[x; y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2; y 2] ⊕ K[x2; y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2; y 2] 1 mjGj = d1d2 ··· dr m 2 As a PG{module, we have S ' (PG) : Example. 1 0 −1 0 Consider G = ; ⊂ GL (K) : 0 1 0 −1 2 Denote by (x; y) the canonical basis of V = K 2. Moreover, Michel Brou´e Reflection groups, braids, Hecke algebras Then S = K[x; y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2; y 2] ⊕ K[x2; y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2; y 2] m 2 As a PG{module, we have S ' (PG) : Example. 1 0 −1 0 Consider G = ; ⊂ GL (K) : 0 1 0 −1 2 Denote by (x; y) the canonical basis of V = K 2. Moreover, 1 mjGj = d1d2 ··· dr Michel Brou´e Reflection groups, braids, Hecke algebras Then S = K[x; y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2; y 2] ⊕ K[x2; y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2; y 2] Example. 1 0 −1 0 Consider G = ; ⊂ GL (K) : 0 1 0 −1 2 Denote by (x; y) the canonical basis of V = K 2. Moreover, 1 mjGj = d1d2 ··· dr m 2 As a PG{module, we have S ' (PG) : Michel Brou´e Reflection groups, braids, Hecke algebras Then S = K[x; y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2; y 2] ⊕ K[x2; y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2; y 2] 1 0 −1 0 Consider G = ; ⊂ GL (K) : 0 1 0 −1 2 Denote by (x; y) the canonical basis of V = K 2. Moreover, 1 mjGj = d1d2 ··· dr m 2 As a PG{module, we have S ' (PG) : Example. Michel Brou´e Reflection groups, braids, Hecke algebras Then S = K[x; y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2; y 2] ⊕ K[x2; y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2; y 2] Denote by (x; y) the canonical basis of V = K 2. Moreover, 1 mjGj = d1d2 ··· dr m 2 As a PG{module, we have S ' (PG) : Example. 1 0 −1 0 Consider G = ; ⊂ GL (K) : 0 1 0 −1 2 Michel Brou´e Reflection groups, braids, Hecke algebras Then S = K[x; y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2; y 2] ⊕ K[x2; y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2; y 2] Moreover, 1 mjGj = d1d2 ··· dr m 2 As a PG{module, we have S ' (PG) : Example. 1 0 −1 0 Consider G = ; ⊂ GL (K) : 0 1 0 −1 2 Denote by (x; y) the canonical basis of V = K 2. Michel Brou´e Reflection groups, braids, Hecke algebras S = K[x; y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2; y 2] ⊕ K[x2; y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2; y 2] Moreover, 1 mjGj = d1d2 ··· dr m 2 As a PG{module, we have S ' (PG) : Example. 1 0 −1 0 Consider G = ; ⊂ GL (K) : 0 1 0 −1 2 Denote by (x; y) the canonical basis of V = K 2. Then Michel Brou´e Reflection groups, braids, Hecke algebras Moreover, 1 mjGj = d1d2 ··· dr m 2 As a PG{module, we have S ' (PG) : Example. 1 0 −1 0 Consider G = ; ⊂ GL (K) : 0 1 0 −1 2 Denote by (x; y) the canonical basis of V = K 2. Then S = K[x; y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2; y 2] ⊕ K[x2; y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2; y 2] Michel Brou´e Reflection groups, braids, Hecke algebras A finite reflection group (abbreviated frg) on K is a finite subgroup of GLK (V )(V a finite dimensional K{vector space) generated by reflections, i.e., linear maps represented by 0ζ 0 ··· 01 B0 1 ··· 0C B C B. .. .C @. .A 0 0 ··· 1 A finite reflection group on R is called a Coxeter group. A finite reflection group on Q is called a Weyl group. Unless... Michel Brou´e Reflection groups, braids, Hecke algebras A finite reflection group on R is called a Coxeter group. A finite reflection group on Q is called a Weyl group.