DEGREE AND VALUATION OF THE SCHUR ELEMENTS OF CYCLOTOMIC HECKE ALGEBRAS MARIA CHLOUVERAKI

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1 DEGREE AND VALUATION OF THE SCHUR ELEMENTS OF CYCLOTOMIC HECKE ALGEBRAS MARIA CHLOUVERAKI arxiv: v1 [math.rt] 28 Feb 2008 ABSTRACT. Following the definition of Rouquier for the families of characters of Weyl groups and its generalization to the case of complex reflection groups, we show that the degree and the valuation of the Schur elements (functions A and a) remain constant on the families of the cyclotomic Hecke algebras of the exceptional complex reflection groups. The same result has been obtained by Broué-Kim for the groups of the infinite series and by Malle-Rouquier for some special cases of exceptional groups. Aknowledgements. I would like to thank Jean Michel for making my algorithm look better and run faster. Introduction The work of G.Lusztig on the irreducible characters of reductive groups over finite fields has displayed the important role of the character families of the Weyl groups concerned. More recent results of Gyoja [9] and Rouquier [17] have made possible the definition of a substitute for families of characters which can be applied to all complex reflection groups. Rouquier has shown that the families of characters of a Weyl group W are exactly the blocks of characters of the Iwahori-Hecke algebra of W over a suitable coefficient ring, the Rouquier ring. This definition generalizes without problem to all cyclotomic Hecke algebras of complex reflection groups. Since the character families of the Weyl group play an essential role in the definition of the families of unipotent characters of the corresponding finite reductive group (cf.[11]), we can hope that the character families of the cyclotomic Hecke algebras play a key role in the organization of families of unipotent characters more generally. Moreover, the determination of these families is crucial for the program Spets (cf.[2]), whose ambition is to give to complex reflection groups the role of Weyl groups of as yet mysterious objects. In the case of the Weyl groups and their usual Hecke algebra, the families of characters can be defined using the existence of Kazhdan-Lusztig bases. Lusztig attaches to every irreducible character two integers, denoted by a 1

2 and A, and shows (cf.[12], 3.3 and 3.4) that they are constant on the families. In an analogue way, we can define integers a and A attached to every irreducible character of a cyclotomic Hecke algebra of a complex reflection group. For the groups of the infinite series, it has been shown in [1] that a and A are constant on the Rouquier blocks. Moreover, a and A are constant on the Rouquier blocks of the spetsial cyclotomic Hecke algebra of the spetsial exceptional complex reflection groups by theorem 5.1 of [16]. The aim of this paper is the proof of the same result for all cyclotomic Hecke algebras of all exceptional complex reflection groups. In [4], we show that the Rouquier blocks of a cyclotomic Hecke algebra of any complex reflection group W depend on some numerical data of the group, its essential hyperplanes. These hyperplanes are defined by the factorization of the Schur elements of the generic Hecke algebra H associated to W. We can associate a partition of the set Irr(W) of irreducible characters of W to every essential hyperplane H, which we call Rouquier blocks associated with the hyperplane H (see definition 4.5). Following theorem 4.4 and corollary 4.6, these partitions generate the partition of Irr(W) into Rouquier blocks. They have been determined for all exceptional complex reflection groups in [4]. Let φ be a cyclotomic specialization and H φ the corresponding cyclotomic Hecke algebra. For every irreducible character, we define a and A to be, respectively, the valuation and the degree of the corresponding Schur element in H φ. In order to show that a and A are constant on the Rouquier blocks, we introduce the notions of generic valuation and generic degree (definition 5.8). Then corollary 5.10 in combination with corollary 4.6 imply that it is enough to check whether they remain constant on the Rouquier blocks associated with each essential hyperplane. We have created a GAP program which verifies that the generic valuation and the generic degree remain constant on the Rouquier blocks for the groups G 7, G 11, G 19, G 26, G 28 and G 32. We provide the algorithm in section 6.1. Then Clifford theory allows us to extend this result to the groups G 4,...,G 22 and G 25. Finally, in section 6.2, we explain why it is trivial to verify that the functions a and A remain constant on the Rouquier blocks of the cyclotomic Hecke algebras of the remaining exceptional complex reflection groups. 1 Generalities on blocks Let O be a Noetherian and integrally closed domain with field of fractions F. Let A be an O-algebra free and finitely generated as an O-module. Definition 1.1 The blocks of A are the central primitive idempotents of A. Let K be a finite Galois extension of F such that the algebra KA := K O A is split semisimple. Then there exists a bijection between the set 2

3 Irr(KA) of irreducible characters of KA and the set Bl(KA) of blocks of KA which sends every irreducible character χ to the central primitive idempotent e χ. Theorem We have 1 = χ Irr(KA) e χ and the set {e χ } χ Irr(KA) is the set of all the blocks of the algebra KA. 2. There exists a unique partition Bl(A) of Irr(KA) such that (a) For all B Bl(A), the idempotent e B := χ B e χ is a block of A. (b) We have 1 = B Bl(A) e B and for every central idempotent e of A, there exists a subset Bl(A,e) of Bl(A) such that e = e B. B Bl(A,e) In particular the set {e B } B Bl(A) is the set of all the blocks of A. If χ B for some B Bl(A), we say that χ belongs to the block e B. Now let us suppose that there exists a symmetrizing form for A, i.e., a linear map t : A O such that t(aa ) = t(a a) for all a,a A, the map ˆt : A Hom O (A, O) a (x t(ax)) is an isomorphism of A-modules-A. Then we have the following result due to Geck (cf.[6]). Proposition We have t = χ Irr(KA) 1 s χ χ, where s χ is the Schur element associated to χ. 2. For all χ Irr(KA), the central primitive idempotent associated to χ is e χ = ˆt 1 (χ) s χ. 3

4 2 Generic Hecke algebras Let µ be the group of all the roots of unity in C and K a number field contained in Q(µ ). We denote by µ(k) the group of all the roots of unity of K. For every integer d > 1, we set ζ d := exp(2πi/d) and denote by µ d the group of all the d-th roots of unity. Let V be a K-vector space of finite dimension r. Let W be a finite subgroup of GL(V ) generated by (pseudo-)reflections acting irreducibly on V. Let us denote by A the set of the reflecting hyperplanes of W. We set M := C V H A C H. For x 0 M, let P := Π 1 (M,x 0 ) and B := Π 1 (M/W,x 0 ). Then there exists a short exact sequence (cf.[3]): {1} P B W {1}. We denote by τ the central element of P defined by the loop [0,1] M, t exp(2πit)x 0. For every orbit C of W on A, we denote by e C the common order of the subgroups W H, where H is any element of C and W H the subgroup formed by id V and all the reflections fixing the hyperplane H. We choose a set of indeterminates u = (u C,j ) (C A/W)(0 j ec 1) and we denote by Z[u,u 1 ] the Laurent polynomial ring in all the indeterminates u. We define the generic Hecke algebra H of W to be the quotient of the group algebra Z[u,u 1 ]B by the ideal generated by the elements of the form (s u C,0 )(s u C,1 )...(s u C,eC 1), where C runs over the set A/W and s runs over the set of monodromy generators around the images in M/W of the elements of the hyperplane orbit C. Example 2.1 Let W := G 4 =< s, t sts = tst, s 3 = t 3 = 1 >. Then s and t are conjugate in W and their reflecting hyperplanes belong to the same orbit in A/W. The generic Hecke algebra of W can be presented as follows H(G 4 ) =< S, T STS = TST, (S u 0 )(S u 1 )(S u 2 ) = 0, (T u 0 )(T u 1 )(T u 2 ) = 0 >. We make some assumptions for the algebra H. Note that they have been verified for all but a finite number of irreducible complex reflection groups ([2], remarks before 1.17, 2; [8]). Assumptions 2.2 The algebra H is a free Z[u,u 1 ]-module of rank W. Moreover, there exists a linear form t : H Z[u,u 1 ] with the following properties: 4

5 1. t is a symmetrizing form for H, i.e., t(hh ) = t(h h) for all h,h H and the map ˆt : H Hom(H, Z[u,u 1 ]) h (h t(hh )) is an isomorphism. 2. Via the specialization u C,j ζ j e C, the form t becomes the canonical symmetrizing form on the group algebra ZW. 3. If we denote by α α the automorphism of Z[u,u 1 ] consisting of the simultaneous inversion of the indeterminates, then for all b B, we have t(b 1 ) = t(bτ) t(τ). We know that the form t is unique ([2], 2.1). From now on, let us suppose that the assumptions 2.2 are satisfied. Then we have the following result by G.Malle ([14], 5.2). Theorem 2.3 Let v = (v C,j ) (C A/W)(0 j ec 1) be a set of C A/W e C indeterminates such that, for every C,j, we have v µ(k) C,j = ζe j C u C,j. Then the K(v)-algebra K(v)H is split semisimple. By Tits deformation theorem (cf., for example, [2], 7.2), it follows that the specialization v C,j 1 induces a bijection χ χv from the set Irr(K(v)H) of absolutely irreducible characters of K(v)H to the set Irr(W) of absolutely irreducible characters of W, such that the following diagram is commutative χv : H Z K [v,v 1 ] χ : Z K W Z K. The following result concerning the form of the Schur elements associated with the irreducible characters of K(v)H is proved in [4], Thm Theorem 2.4 The Schur element s χ (v) associated with the character χv of K(v)H is an element of Z K [v,v 1 ] of the form s χ (v) = ξ χ N χ i I χ Ψ χ,i (M χ,i ) nχ,i where ξ χ is an element of Z K, 5

6 N χ = C,j vb C,j C,j is a monomial in Z K [v,v 1 ] such that e C 1 j=0 b C,j = 0 for all C A/W, I χ is an index set, (Ψ χ,i ) i Iχ is a family of K-cyclotomic polynomials in one variable (i.e., minimal polynomials of the roots of unity over K), (M χ,i ) i Iχ is a family of monomials in Z K [v,v 1 ] and if M χ,i = C,j va C,j C,j, then gcd(a C,j) = 1 and e C 1 j=0 a C,j = 0 for all C A/W, (n χ,i ) i Iχ is a family of positive integers. This factorization is unique in K[v,v 1 ]. Moreover, the monomials (M χ,i ) i Iχ are unique up to inversion, whereas the coefficient ξ χ is unique up to multiplication by a root of unity. Remark: The bijection Irr(K(v)H) Irr(W), χv χ implies that the specialization v C,j 1 sends s χv to W /χ(1) (which is the Schur element of χ in the group algebra with respect to the canonical symmetrizing form). Let A := Z K [v,v 1 ] and p be a prime ideal of Z K. Definition 2.5 Let M = C,j va C,j C,j be a monomial in A such that gcd(a C,j ) = 1. We say that M is p-essential for a character χ Irr(W), if there exists a K-cyclotomic polynomial Ψ such that Ψ(M) divides s χ (v). Ψ(1) p. We say that M is p-essential for W, if there exists a character χ Irr(W) such that M is p-essential for χ. The following proposition ([4], Prop.3.2.6) gives a characterization of p- essential monomials, which plays an essential role in the proof of theorem 4.4. Proposition 2.6 Let M = C,j va C,j C,j be a monomial in A such that gcd(a C,j ) = 1. We set q M := (M 1)A + pa. Then 1. The ideal q M is a prime ideal of A. 2. M is p-essential for χ Irr(W) if and only if s χ (v)/ξ χ q M. 6

7 3 Cyclotomic Hecke algebras Let y be an indeterminate. We set x := y µ(k). Definition 3.1 A cyclotomic specialization of H is a Z K -algebra morphism φ : Z K [v,v 1 ] Z K [y,y 1 ] with the following properties: φ : v C,j y n C,j where n C,j Z for all C and j. For all C A/W, and if z is another indeterminate, the element of Z K [y,y 1,z] defined by Γ C (y,z) := e C 1 (z ζe j C y n C,j ) j=0 is invariant by the action of Gal(K(y)/K(x)). If φ is a cyclotomic specialization of H, the corresponding cyclotomic Hecke algebra is the Z K [y,y 1 ]-algebra, denoted by H φ, which is obtained as the specialization of the Z K [v,v 1 ]-algebra H via the morphism φ. It also has a symmetrizing form t φ defined as the specialization of the canonical form t. Example 3.2 The spetsial Hecke algebra H s (W) is the cyclotomic algebra obtained by the specialization v C,0 y, v C,j 1 for 1 j e C 1, for all C A/W. The following result is proved in [4] (remarks following Thm.3.3.3): Proposition 3.3 The algebra K(y)H φ is split semisimple. For y = 1 this algebra specializes to the group algebra KW (the form t φ becoming the canonical form on the group algebra). Thus, by Tits deformation theorem, the specialization v C,j 1 defines the following bijections Irr(K(v)H) Irr(K(y)H φ ) Irr(W) χv χ φ χ. The following result is an immediate consequence of Theorem 2.4. Proposition 3.4 The Schur element s χφ (y) associated with the irreducible character χ φ of K(y)H φ is a Laurent polynomial in y of the form s χφ (y) = ψ χ,φ y a χ,φ Φ C K Φ(y) nχ,φ where ψ χ,φ Z K, a χ,φ Z, n χ,φ N and C K is a set of K-cyclotomic polynomials. 7

8 Definition 3.5 A prime ideal p of Z K lying over a prime number p is φ-bad for W, if there exists χ φ Irr(K(y)H φ ) with ψ χ,φ p. If p is φ-bad for W, we say that p is a φ-bad prime number for W. Remark: If W is a Weyl group and φ is the spetsial cyclotomic specialization, then the φ-bad prime ideals are the ideals generated by the bad prime numbers (in the usual sense) for W (see [7], 5.2). Note that if p is φ-bad for W, then p must divide the order of the group (since s χφ (1) = W /χ(1)). 4 Rouquier blocks Definition 4.1 We call Rouquier ring of K and denote by R the Z K - subalgebra of K(y) R := Z K [y,y 1,(y n 1) 1 n 1 ] Let φ : v C,j y n C,j be a cyclotomic specialization and H φ the corresponding cyclotomic Hecke algebra. The Rouquier blocks of H φ are the blocks of the algebra RH φ. Remark: It has been shown by Rouquier (cf.[17]), that if W is a Weyl group and H φ is obtained via the spetsial cyclotomic specialization (see example 3.2), then its Rouquier blocks coincide with the families of characters defined by Lusztig. Thus, the Rouquier blocks play an essential role in the program Spets (see [2]) whose ambition is to give to complex reflection groups the role of Weyl groups of as yet mysterious structures. Proposition 4.2 (Some properties of the Rouquier ring) 1. The group of units R of the Rouquier ring R consists of the elements of the form uy n Φ(y) n φ, Φ Cycl(K) where u Z K, n,n φ Z, Cycl(K) is the set of K-cyclotomic polynomials and n φ = 0 for all but a finite number of Φ. 2. The prime ideals of R are the zero ideal {0}, the ideals of the form pr, where p is a prime ideal of Z K, the ideals of the form P(y)R, where P(y) is an irreducible element of Z K [y] of degree at least 1, prime to y and to Φ(y) for all Φ Cycl(K). 8

9 3. The Rouquier ring R is a Dedekind ring. For the proof of the above result, the reader may refer, for example, to [4], Prop Due to the form of the cyclotomic Schur elements, the form of the prime ideals of the Rouquier ring and an elementary result of blocks theory (see, for example, [1], Prop.1.13), we obtain the following description of the Rouquier blocks: Proposition 4.3 Two characters χ,ψ Irr(W) are in the same Rouquier block of H φ if and only if there exists a finite sequence χ 0,χ 1,...,χ n Irr(W) and a finite sequence p 1,...,p n of φ-bad prime ideals for W such that χ 0 = χ and χ n = ψ, for all j (1 j n), the characters χ j 1 and χ j belong to the same block of R pj RH φ. The above proposition implies that if we know the blocks of the algebra R pr H φ for every φ-bad prime ideal p for W, then we know the Rouquier blocks of H φ. In order to determine the former, we can use the following theorem ([4], Thm ) Theorem 4.4 Let A := Z K [v,v 1 ] and p be a prime ideal of Z K. Let M 1,...,M k be all the p-essential monomials for W such that φ(m j ) = 1 for all j = 1,...,k. Set q 0 := pa, q j := pa + (M j 1)A for j = 1,...,k and Q := {q 0,q 1,...,q k }. Two irreducible characters χ,ψ Irr(W) are in the same block of R pr H ϕ if and only if there exist a finite sequence χ 0,χ 1,...,χ n Irr(W) and a finite sequence q j1,...,q jn Q such that χ 0 = χ and χ n = ψ, for all i (1 i n), the characters χ i 1 and χ i are in the same block of A qji H. Let p be a prime ideal of Z K and φ : v C,j y n C,j a cyclotomic specialization. If M = C,j va C,j C,j is a p-essential monomial for W, then φ(m) = 1 C,j a C,j n C,j = 0. Set m := C A/W e C. The hyperplane defined in C m by the relation a C,j t C,j = 0, C,j 9

10 where (t C,j ) C,j is a set of m indeterminates, is called p-essential hyperplane for W. A hyperplane in C m is called essential for W, if it is p-essential for some prime ideal p of Z K. Let H be an essential hyperplane corresponding to the monomial M. We denote by B H p the partition of Irr(W) into blocks of A q M H, where q M := (M 1)A + pa. Moreover, we denote by B p the partition of Irr(W) into blocks of A pa H. Definition 4.5 Let H be an essential hyperplane for W. We call Rouquier blocks associated with the hyperplane H (resp. with no essential hyperplane), and denote by B H (resp. B ), the partition of Irr(W) generated by the partitions B H p (resp. B p), where p runs over the set of prime ideals of Z K. With the help of the above definition and thanks to proposition 4.3 and theorem 4.4, we obtain the following characterization for the Rouquier blocks of a cyclotomic Hecke algebra: Corollary 4.6 Let φ : v C,j y n C,j be a cyclotomic specialization. The Rouquier blocks of the cyclotomic Hecke algebra H φ correspond to the partition of Irr(W) generated by the partitions B H, where H runs over the set of all essential hyperplanes the integers n C,j belong to. If the n C,j belong to no essential hyperplane, then the Rouquier blocks of H φ coincide with the partition B. In the fourth chapter of [4], we explain how we have obtained the partitions B and B H for every essential hyperplane H for every exceptional complex reflection group. We have stored these data in a computer file and created the function AllBlocks(W) which displays them. We have also created the GAP function RouquierBlocks which, given a cyclotomic specialization φ : v C,j y n C,j, checks to which essential hyperplanes the n C,j belong and calculates the Rouquier blocks of the corresponding cyclotomic Hecke algebra using corollary 4.6. Both functions, along with explanations for their use, can be found on my website: 5 Functions a and A chlouveraki Following the notations in [2], 6B, for every element P(y) C(y), we call valuation of P(y) at y and denote by val y (P) the order of P(y) at 0 (we have val y (P) < 0 if 0 is a pole of P(y) and val y (P) > 0 if 0 is a zero of P(y)), 10

11 degree of P(y) at y and denote by deg y (P) the opposite of the valuation of P(1/y). For χ Irr(W), we define a χφ := val y (s χφ (y)) and A χφ := deg y (s χφ (y)). The following result is proven in [1], Prop.2.9. Proposition 5.1 Let χ,ψ Irr(W). If χ φ and ψ φ belong to the same Rouquier block, then a χφ + A χφ = a ψφ + A ψφ. In the case of the Weyl groups and their usual Hecke algebra, the families of characters can be defined using the existence of Kazhdan-Lusztig bases. Lusztig has shown (cf.[12], 3.3 and 3.4) that the integers a and A are constant on the families. Inspired by this fact, Broué-Kim prove that the integers a and A are constant on the Rouquier blocks of the cyclotomic Hecke algebras of the complex reflection groups of the infinite series (cf.[1], 3.18 and 4.5). We are going to prove the same result for the exceptional complex reflection groups, using the classification of their Rouquier blocks obtained in [4]. In order to do that, let us first introduce the symbols (y n ) + and (y n ). Definition 5.2 Let n Z. { n, if n > 0; (y n ) + = 0, if n 0. { n, if n < 0; (y n ) = 0, if n 0. Now let us fix χ Irr(W). Following Theorem 2.4, the generic Schur element s χ (v) associated to χ is an element of Z K [v,v 1 ] of the form s χ (v) = ξ χ N χ i I χ Ψ χ,i (M χ,i ) nχ,i, ( ) where ξ χ is an element of Z K, N χ = C,j vb C,j C,j is a monomial in Z K [v,v 1 ], I χ is an index set, (Ψ χ,i ) i Iχ is a family of K-cyclotomic polynomials in one variable, (M χ,i ) i Iχ is a family of monomials in Z K [v,v 1 ] and if M χ,i = C,j va C,j C,j, then gcd(a C,j ) = 1, 11

12 (n χ,i ) i Iχ is a family of positive integers. We fix the factorization ( ) for s χ (v). Proposition 5.3 Let φ : v C,j y n C,j be a cyclotomic specialization. Then a χφ = C,j b C,jn C,j + i I χ n χ,i deg(ψ χ,i )(φ(m χ,i )). A χφ = C,j b C,jn C,j + i I χ n χ,i deg(ψ χ,i )(φ(m χ,i )) +. Remark: Together with Jean Michel, we have programmed into the GAP package CHEVIE the generic Schur elements of the exceptional complex reflection groups in a form compatible with Theorem 2.4 (functions SchurM odels and SchurData). We have used the formulas of proposition 5.3 in order to obtain the valuation and the degree of the cyclotomic Schur elements for the algorithm of section 4.2 in [4]. Definition 5.4 Let M = C,j va C,j C,j be a monomial with gcd(a C,j ) = 1 and Ψ a K-cyclotomic polynomial such that Ψ(M) appears in ( ). The factor degree of Ψ(M) for χ with respect to ( ) is defined as the product f Ψ(M) (t) = deg(ψ) ( C,j a C,j t C,j ), where t = (t C,j ) C,j is a set of C A/W e C indeterminates. If n is the greatest positive integer such that Ψ(M) n appears in ( ), then n is called the coefficient of the factor degree f Ψ(M) and it is denoted by c(f Ψ(M) ). Then we can define an equivalence relation on the set F χ of all factor degrees for χ with respect to ( ): Definition 5.5 Two factor degrees f 1,f 2 are equivalent, if there exists a positive number q Q such that f 1 = qf 2. We write f 1 f 2. Before we proceed, let us introduce the notion of a sign map for any non-empty finite set: Definition 5.6 Let E be a non-empty finite set. A sign map for E is a map E { 1,1}. Remark: If E is a non-empty finite set, there exist 2 E sign maps for E. The set F χ of all factor degrees for the character χ with respect to ( ) is a non-empty finite set. Therefore, we can define sign maps for F χ. Definition 5.7 Let F χ be the set of all factor degrees for χ with respect to ( ) and let ǫ : F χ { 1,1} be a sign map for F χ. We say that ǫ is a good sign map for F χ if it satisfies the following conditions: 12

13 1. If f 1,f 2 F χ with f 1 f 2, then ǫ(f 1 ) = ǫ(f 2 ). 2. If f 1,f 2 F χ with f 1 f 2, then ǫ(f 1 ) = ǫ(f 2 ). In order to obtain the main result, we need to introduce the notions of generic valuation and generic degree of the Schur element s χ (v). Definition 5.8 Let F χ be the set of all factor degrees for χ with respect to ( ) and let ǫ : F χ { 1,1} be a good sign map for F χ. Then the generic valuation a χ,ǫ (t) of s χ (v) with respect to ǫ is a χ,ǫ (t) := b C,j t C,j + c(f) f. C,j {f F χ ǫ(f)= 1} the generic degree A χ,ǫ (t) of s χ (v) with respect to ǫ is A χ,ǫ (t) := b C,j t C,j + c(f) f. C,j {f F χ ǫ(f)=1} The following result is a consequence of the above definitions and proposition 5.3. Proposition 5.9 Let φ : v C,j y n C,j be a cyclotomic specialization and χ,ψ Irr(W) with sets of factor degrees F χ, F ψ respectively. If a χ,ǫ (t) = a ψ,ǫ (t) (resp. A χ,ǫ (t) = A ψ,ǫ (t)) with respect to every good sign map ǫ for F χ F ψ, then a χφ = a ψφ (resp. A χφ = A ψφ ). Proof: such that Let n := (n C,j ) C,j. There exists a good sign map ǫ for F χ F ψ ǫ(f) = 1 f(n) 0. Then, by proposition 5.3, we have that a χφ = a χ,ǫ (n) = a ψ,ǫ (n) = a ψφ and A χφ = A χ,ǫ (n) = A ψ,ǫ (n) = A ψφ. Corollary 5.10 Let φ : v C,j y n C,j be a cyclotomic specialization such that the integers n C,j belong to the essential hyperplane H : C,j a C,jt C,j = 0. Let C 0 A/W and j 0 {0,...,e C0 1} such that a C0,j 0 0. We condider the Q-algebra morphism ϑ : Q[t] Q[t {t C0,j 0 }] t C0,j 0 C,j ( a C,j/a C0,j 0 )t C,j t C,j t C,j for (C,j) (C 0,j 0 ). 13

14 Let χ,ψ Irr(W) with sets of factor degrees F χ, F ψ respectively. If ϑ(a χ,ǫ (t)) = ϑ(a ψ,ǫ (t)) (resp. ϑ(a χ,ǫ (t)) = ϑ(a ψ,ǫ (t))) with respect to every good sign map ǫ for F χ F ψ, then a χφ = a ψφ (resp. A χφ = A ψφ ). 6 Exceptional complex reflection groups In this section we will prove the following result Theorem 6.1 Let W be an exceptional irreducible complex reflection group. Let φ : v C,j y n C,j be a cyclotomic specialization and χ,ψ Irr(W). If χ φ and ψ φ belong to the same Rouquier block, then a χφ = a ψφ and A χφ = A ψφ. 6.1 The groups G 4,...,G 22, G 25, G 26, G 28, G 32 Let W := G m, where m {7,11,19,26,28,32}. We have created the following algorithm which verifies that the assumptions of corollary 5.10 are satisfied on the Rouquier blocks associated with each essential hyperplane. This algorithm requires the GAP package CHEVIE (which can be downloaded at: jmichel) and the GAP function AllBlocks contained in the file RouquierBlocks.g (which can be downloaded from my webpage: chlouveraki). A program implementing this algorithm can be also found at my webpage. Algorithm 1. We assume that there exists a function ismultiple(g,f) which takes two polynomials f,g and returns 1, if there exists a rational q > 0 such that g = q f, 1, if there exists a rational q < 0 such that g = q f, 0, otherwise. 2. We define a function FactorDegrees(H,χ), where H is either the list of the coefficients a C,j of the indeterminates in an essential hyperplane for W or the empty list in the case of no essential hyperplane, χ Irr(W) is represented by its position in the list of parameters of W. In the GAP package CHEVIE, the functions SchurData and Schur- Models provide us with the irreducible factors and the coefficients of the generic Schur elements of W. Using definition 5.4, it is easy to create a list of the factor degrees of the Schur element of χ. The function FactorDegrees(H,χ) returns a pair [F,C] such that 14

15 if H is the empty set, then F is the list of factor degrees of the Schur element of χ (a list of polynomials) and C is the term of the generic valuation (and generic degree) induced by the monomial factor N χ, if H is not the empty set, then F is the list of factor degrees of the Schur element of χ (a list of polynomials) and C is the term of the generic valuation (and generic degree) induced by the monomial factor N χ, after applying the morphism ϑ of corollary 5.10 associated with H. 3. We assume that there exists a function SymmetricDifferenceWithMultiplicities(l 1,l 2 ), where l 1, l 2 are two lists, which returns a sublist l of l 1 l 2 such that: x l if and only if the multiplicity of x in l 1 is different than the multiplicity of x in l2. 4. The function compare(a, b) will check the assumptions of corollary 5.10 for two irreducible characters χ, ψ. It returns true, if they are satisfied. In order to do that, it takes the corresponding outputs of the function FactorDegrees, a := [F χ,c χ ] and b := [F ψ,c ψ ], and sets l :=SymmetricDifferenceWithMultiplicities(F χ,f ψ ). If l is empty, then the function returns true. If not, then we have to generate all good sign maps only for l, since the common factors don t affect the result: Step 1: We create a sublist k of l such that: (a) every element of l is a multiple by a non-zero rational number of an element of k, (b) if f,g k then ismultiple(f,g) = 0. We do that by running over l and removing any element f such that ismultiple(f,g) 0 for some previous element g in the list. Step 2: We create a list a 1 as follows: For all f F χ, we set p :=the position of the g in k such that ismultiple(f.g) 0 (there is at most one such position!). If p false, then we add to a 1 the triplet [f,p,ismultiple(f,k[p])]. We create a similar list b 1 for F ψ. Step 3: We create all good sign maps for l which consists of creating all the lists of signs of the same length as k. Let M be such a matrix and f l. Then there exists a triplet of the form [f,p,ismultiple(k[p],f)] in a 1 or b 1. The corresponding good sign map ǫ is given by ǫ(f) := ismultiple(f,k[p]) M[p]. 15

16 Step 4: We compare ϑ(a χ,ǫ ) with ϑ(a ψ,ǫ ) and ϑ(a χ,ǫ ) with ϑ(a ψ,ǫ ) (considering only the non-common terms) with respect to every good sign map ǫ for l. 5. We create a function compareblock(h,b), where H is a list representing one or no essential hyperplane as in FactorDegrees and B is a block represented as a list of integers, each of which is the position of a character in the list of parameters of W. If Length(B) = 1, then it returns true. If not, then it applies FactorDegrees(H,χ) to all the elements χ of B, creating thus the list Sch, and then returns true if compare(sch[1],sch[j]) = true for all j {2,...,Length(B)}. 6. Finally, we create a function CheckT heorem(m) which generates the group G m and applies compareblock(h,b) to every B B H (see definition 4.5), where H runs over the set {, essential hyperplanes for W }. The function CheckTheorem(m) has returned true for all m {7, 11, 19, 26, 28, 32}. Then corollary 5.10 in combination with corollary 4.6, imply that Proposition 6.2 The assertion of Theorem 6.1 holds for W. Now let W := G m, m {4,5,6,8,9,10,12, 13, 14,15, 16, 17,18,21, 22,25}. Then the use of Clifford theory for the determination of the Rouquier blocks of the cyclotomic Hecke algebras associated to W in [4], the use of Clifford theory for the description of the Schur elements of these algebras (see also Appendix) and proposition 6.2, imply that Proposition 6.3 The assertion of Theorem 6.1 holds for W. 6.2 The other exceptional groups Let W be one of the remaining exceptional irreducible complex reflection groups: G 23, G 24, G 27, G 29, G 30, G 31, G 33, G 34, G 35, G 36, G 37. Then W is generated by reflections of order 2 whose reflecting hyperplanes belong to one single orbit under the action of W. Its generic Hecke algebra is defined over a Laurent polynomial ring in two indeterminates, v 0 and v 1, and the only essential monomial is v 0 v1 1. Therefore, its generic Schur elements can be expressed as products of K-cyclotomic polynomials in the one variable v := v 0 v1 1, i.e., the generic Schur element s χ(v) associated with the irreducible character χ is an element of Z K [v,v 1 ] of the form s χ (v) = ξ χ v bχ Ψ χ,i (v) nχ,i, i I χ 16

17 where ξ χ is an element of Z K, b χ is an integer, I χ is an index set, (Ψ χ,i ) i Iχ is a set of K-cyclotomic polynomials, (n χ,i ) i Iχ is a family of positive integers. Let φ : v y n (n Z) be a cyclotomic specialization. Then a χφ = n val v (s χ (v)). A χφ = n deg v (s χ (v)). Therefore, in order to verify theorem 6.1 for W, it suffices to check whether the degree and the valuation of the generic Schur elements remain constant on the Rouquier blocks associated with no essential hyperplane. Note that the generic Schur elements coincide with the Schur elements of the spetsial cyclotomic Hecke algebra and the Rouquier blocks associated with no essential hyperplane coincide with its Rouquier blocks. We can easily create an algorithm which returns true if the degree and the valuation of the Schur elements of the spetsial cyclotomic Hecke algebra remain constant on its Rouquier blocks. A program realizing this algorithm can be found at my webpage: chlouveraki. It requires the GAP package CHEVIE and the GAP function RouquierBlocks contained in the file RouquierBlocks.g. Since this algorithm has returned true for all m {23,24,27,29,30,31, 33,34,35, 36, 37}, we obtain that Proposition 6.4 The assertion of Theorem 6.1 holds for W. 17

18 Appendix For the proofs of the results that follow, the reader may refer to Chapter 2 and the Appendix of [4]. Let us assume that O is a Noetherian and integrally closed domain with field of fractions F, A is an O-algebra free and finitely generated as an O-module, K is a finite Galois extension of F such that the algebra KA is split semisimple t is a symmetrizing form for A. Following proposition 1.3, we have t = χ Irr(KA) 1 s χ χ, where s χ is the Schur element of the irreducible character χ Irr(KA). The Schur elements belong to the integral closure of O in K. Definition 1 Let Ā be a subalgebra of A free and of finite rank as an O- module. We say that Ā is a symmetric subalgebra of A, if it satisfies the following two conditions: 1. Ā is free (of finite rank) as an O-module and the restriction Res Ā A (t) of the form t to Ā is a symmetrizing form for Ā, Ā-module for the action of left multi- 2. A is free (of finite rank) as an plication by the elements of Ā. From now on, let us suppose that Ā is a symmetric subalgebra of A. We denote by Ind Ā A :Ā mod A mod and Res Ā A : A mod Ā mod the functors defined as usual by and Ind Ā A := A Ā where A is viewed as an A-module-Ā Res Ā A := A A where A is viewed as an Ā-module-A. Moreover, let K be a finite Galois extension F such that the algebras KA and KĀ are both split semisimple. 18

19 In this paper, we work on the Hecke algebras of complex reflection groups, which, under the assumptions 2.2, are symmetric. Sometimes the Hecke algebra of a group W appears as a symmetric subalgebra of the Hecke algebra of another group W, which contains W. Therefore, it would be helpful, if we could obtain the Schur elements (resp. the blocks) of the former from the Schur elements (resp. the blocks) of the latter. This is possible with the use of a generalization of some classical results, known as Clifford theory (see, for example, [5]), to the twisted symmetric algebras of finite groups and more precisely of finite cyclic groups. Definition 2 We say that a symmetric O-algebra (A, t) is the twisted symmetric algebra of a finite group G over the subalgebra Ā, if the following conditions are satisfied: Ā is a symmetric subalgebra of A, There exists a family {A g g G} of O-submodules of A such that (a) A = g G A g, (b) A 1 = Ā, (c) A g A h = A gh for all g,h G, (d) t(a g ) = 0 for all g G,g 1, (e) A g A for all g G (where A is the set of units of A). Lemma 1 Let a g A g such that a g is a unit in A. Then A g = a g Ā = Āa g. Proof: Since a g A g, property (b) implies that a 1 g A g 1. If a A g, then a 1 g a A 1 = Ā. We have a = a ga 1 g a a g Ā and thus A g a g Ā. Property (b) implies the inverse inclusion. In the same way, we show that A g = Āa g. Let W be a complex reflection group and let us denote by H(W) its generic Hecke algebra. Suppose that the assumptions 2.2 are satisfied. Let H(W) sp be the algebra obtained from H(W) by specializing some of the parameters. Let W be another complex reflection group such that H(W) sp is the twisted symmetric algebra of a finite cyclic group G over the symmetric subalgebra H(W ). Then, if we know the Schur elements and the blocks of H(W) sp, we can use propositions and of [4] in order to calculate the Schur elements and the blocks of H(W ). In all the cases that will be studied below, these two propositions imply that 19

20 1. if we denote by χ the (irreducible) restriction to H(W ) of an irreducible character χ Irr(H(W) sp ), then their Schur elements verify s χ = W : W s χ, 2. the block-idempotents of the algebras H(W) sp and H(W ) coincide. The groups G 4, G 5, G 6, G 7 The following table gives the specializations of the parameters of the generic Hecke algebra H(G 7 ), (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2 ), which give the generic Hecke algebras of the groups G 4, G 5 and G 6 ([13], Table 4.6). Lemma 2 Group Index S T U G 7 1 x 0,x 1 y 0,y 1,y 2 z 0,z 1,z 2 G 5 2 1, 1 y 0,y 1,y 2 z 0,z 1,z 2 G 6 3 x 0,x 1 1,ζ 3,ζ3 2 z 0,z 1,z 2 G 4 6 1, 1 1,ζ 3,ζ3 2 z 0,z 1,z 2 Specializations of the parameters for H(G 7 ) The algebra H(G 7 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2 ) (1, 1;y 0,y 1,y 2 ;z 0,z 1,z 2 ) is the twisted symmetric algebra of the cyclic group C 2 over the symmetric subalgebra H(G 5 ) with parameters (y 0,y 1,y 2 ;z 0,z 1,z 2 ). The algebra H(G 7 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2 ) (x 0,x 1 ;1,ζ 3,ζ 2 3 ;z 0,z 1,z 2 ) is the twisted symmetric algebra of the cyclic group C 3 over the symmetric subalgebra H(G 6 ) with parameters (x 0,x 1 ;z 0,z 1,z 2 ). The algebra H(G 6 ) specialized via (x 0,x 1 ;z 0,z 1,z 2 ) (1, 1;z 0,z 1,z 2 ) is the twisted symmetric algebra of the cyclic group C 2 over the symmetric subalgebra H(G 4 ) with parameters (z 0,z 1,z 2 ). 20

21 The groups G 8, G 9, G 10, G 11, G 12, G 13, G 14, G 15 The following table gives the specializations of the parameters of the generic Hecke algebra H(G 11 ), (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2,z 3 ), which give the generic Hecke algebras of the groups G 8,...,G 15 ([13], Table 4.9). Group Index S T U G 11 1 x 0,x 1 y 0,y 1,y 2 z 0,z 1,z 2,z 3 G , 1 y 0,y 1,y 2 G 15 2 x 0,x 1 y 0,y 1,y 2 z 1,z 1,z 2,z 3 u0, u 1, u 0, u 1 G 9 3 x 0,x 1 1,ζ 3,ζ3 2 z 0,z 1,z 2,z 3 G 14 4 x 0,x 1 y 0,y 1,y 2 1,i, 1, i G 8 6 1, 1 1,ζ 3,ζ3 2 z 0,z 1,z 2,z 3 G 13 6 x 0,x 1 1,ζ 3,ζ3 2 u0, u 1, u 0, u 1 G x 0,x 1 1,ζ 3,ζ3 2 1,i, 1, i Lemma 3 Specializations of the parameters for H(G 11 ) The algebra H(G 11 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2,z 3 ) (1, 1;y 0,y 1,y 2 ;z 0,z 1,z 2,z 3 ) is the twisted symmetric algebra of the cyclic group C 2 over the symmetric subalgebra H(G 10 ) with parameters (y 0,y 1,y 2 ;z 0,z 1,z 2,z 3 ). The algebra H(G 11 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2,z 3 ) (x 0,x 1 ;1,ζ 3,ζ 2 3 ;z 0,z 1,z 2,z 3 ) is the twisted symmetric algebra of the cyclic group C 3 over the symmetric subalgebra H(G 9 ) with parameters (x 0,x 1 ;z 0,z 1,z 2,z 3 ). The algebra H(G 9 ) specialized via (x 0,x 1 ;z 0,z 1,z 2,z 3 ) (1, 1;z 0,z 1,z 2,z 3 ) is the twisted symmetric algebra of the cyclic group C 2 over the symmetric subalgebra H(G 8 ) with parameters (z 0,z 1,z 2,z 3 ). The algebra H(G 11 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2,z 3 ) (x 0,x 1 ;y 0,y 1,y 2 ;1,i, 1, i) is the twisted symmetric algebra of the cyclic group C 4 over the symmetric subalgebra H(G 14 ) with parameters (x 0,x 1 ;y 0,y 1,y 2 ). 21

22 The algebra H(G 14 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ) (x 0,x 1 ;1,ζ 3,ζ 2 3 ) is the twisted symmetric algebra of the cyclic group C 3 over the symmetric subalgebra H(G 12 ) with parameters (x 0,x 1 ). The algebra H(G 11 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2,z 3 ) (x 0,x 1 ;y 0,y 1,y 2 ; u 0, u 1, u 0, u 1 ) is the twisted symmetric algebra of the cyclic group C 2 over the symmetric subalgebra H(G 15 ) with parameters (x 0,x 1 ;y 0,y 1,y 2 ;u 0,u 1 ). The algebra H(G 15 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ;u 0,u 1 ) (x 0,x 1 ;1,ζ 3,ζ 2 3 ;u 0,u 1 ) is the twisted symmetric algebra of the cyclic group C 3 over the symmetric subalgebra H(G 13 ) with parameters (x 0,x 1 ;u 0,u 1 ). The groups G 16, G 17, G 18, G 19, G 20, G 21, G 22 The following table gives the specializations of the parameters of the generic Hecke algebra H(G 19 ), (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2,z 3,z 4 ), which give the generic Hecke algebras of the groups G 16,...,G 22 ([13], Table 4.12). Group Index S T U G 19 1 x 0,x 1 y 0,y 1,y 2 z 0,z 1,z 2,z 3,z 4 G , 1 y 0,y 1,y 2 z 0,z 1,z 2,z 3,z 4 G 17 3 x 0,x 1 1,ζ 3,ζ3 2 z 0,z 1,z 2,z 3,z 4 G 21 5 x 0,x 1 y 0,y 1,y 2 1,ζ 5,ζ5 2,ζ3 5,ζ4 5 G , 1 1,ζ 3,ζ3 2 z 0,z 1,z 2,z 3,z 4 G , 1 y 0,y 1,y 2 1,ζ 5,ζ5 2,ζ3 5,ζ4 5 G x 0,x 1 1,ζ 3,ζ3 2 1,ζ 5,ζ5 2,ζ3 5,ζ4 5 Lemma 4 Specializations of the parameters for H(G 19 ) The algebra H(G 19 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2,z 3,z 4 ) (1, 1;y 0,y 1,y 2 ;z 0,z 1,z 2,z 3,z 4 ) is the twisted symmetric algebra of the cyclic group C 2 over the symmetric subalgebra H(G 18 ) with parameters (y 0,y 1,y 2 ;z 0,z 1,z 2,z 3,z 4 ). 22

23 The algebra H(G 19 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2,z 3,z 4 ) (x 0,x 1 ;1,ζ 3,ζ 2 3 ;z 0,z 1,z 2,z 3,z 4 ) is the twisted symmetric algebra of the cyclic group C 3 over the symmetric subalgebra H(G 17 ) with parameters (x 0,x 1 ;z 0,z 1,z 2,z 3,z 4 ). The algebra H(G 17 ) specialized via (x 0,x 1 ;z 0,z 1,z 2,z 3,z 4 ) (1, 1;z 0,z 1,z 2,z 3,z 4 ) is the twisted symmetric algebra of the cyclic group C 2 over the symmetric subalgebra H(G 16 ) with parameters (z 0,z 1,z 2,z 3,z 4 ). The algebra H(G 19 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ;z 0,z 1,z 2,z 3,z 4 ) (x 0,x 1 ;y 0,y 1,y 2 ;1,ζ 5,ζ 2 5,ζ 3 5,ζ 4 5) is the twisted symmetric algebra of the cyclic group C 5 over the symmetric subalgebra H(G 21 ) with parameters (x 0,x 1 ;y 0,y 1,y 2 ). The algebra H(G 21 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ) (1, 1;y 0,y 1,y 2 ) is the twisted symmetric algebra of the cyclic group C 2 over the symmetric subalgebra H(G 20 ) with parameters (y 0,y 1,y 2 ). The algebra H(G 21 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ) (x 0,x 1 ;1,ζ 3,ζ 2 3) is the twisted symmetric algebra of the cyclic group C 3 over the symmetric subalgebra H(G 22 ) with parameters (x 0,x 1 ). The groups G 25, G 26 The following table gives the specialization of the parameters of the generic Hecke algebra H(G 26 ), (x 0,x 1 ;y 0,y 1,y 2 ), which give the generic Hecke algebra of the group G 25 ([15], Theorem 6.3). Group Index S T G 26 1 x 0,x 1 y 0,y 1,y 2 G , 1 y 0,y 1,y 2 Specialization of the parameters for H(G 26 ) Lemma 5 The algebra H(G 26 ) specialized via (x 0,x 1 ;y 0,y 1,y 2 ) (1, 1;y 0,y 1,y 2 ) is the twisted symmetric algebra of the cyclic group C 2 over the symmetric subalgebra H(G 25 ) with parameters (y 0,y 1,y 2 ). 23

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