Frobenius map for the centers of Hecke algebras

We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of Frobenius, we establish an algebra isomorphism from the algebra Z to the ring of symmetric functions. This isomorphism provides an identification between several distinguished bases for the centers (introduced by Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.


Introduction
Frobenius [Fro] constructed a natural isomorphism ψ from the direct sum of the centers of the symmetric group algebras to the space of symmetric functions, which sends the conjugacy class sums to the power-sum symmetric functions (up to some scalars). This gives rise to the characteristic map under which the irreducible characters of symmetric groups are matched with the Schur functions (cf. [Mac,I. 7]). This construction of Frobenius has played a fundamental role in the interaction between the representation theory of symmetric groups and combinatorics of symmetric functions.
The Hecke algebra H n is a v-deformation of the group algebra of the symmetric group S n . The structures of the centers Z(H n ) of the Hecke algebras have been the subject of studies by various authors (cf. [DJ, FrG, FW, GR, Jo, La]). Using the class polynomials for Hecke algebras [GP], Geck and Rouquier [GR] constructed a basis of v-class sums for the center Z(H n ), which specializes at v = 1 to the usual class sums of S n . There is a neat characterization of these v-class sums by Francis [Fra] though no closed formula is known. Extending a basis of Jones [Jo] of the center Z(H n ), Lascoux [La] constructed several more bases of Z(H n ), and moreover, he provided an elegant description of the transition matrices from these bases to the Geck-Rouquier basis in terms of the transition matrices of well-known bases in the ring of symmetric functions. Lascoux stated the identifications of these bases of Z(H n ) with symmetric functions as an open problem.
The goal of this paper is to formulate a Frobenius map Ψ : Z −→ Λ v , which is a v-deformation of the classical Frobenius map ψ, from the direct sum Z of the centers of H n for all n to the C(v)-space Λ v of symmetric functions. We establish various basic properties of Ψ, and in particular, we provide a positive answer to Lascoux's question of identifying bases in Z and Λ v . It is our belief that the Frobenius map Ψ will play a fundamental role in further understanding of the centers of Hecke algebras with powerful tools from the theory of symmetric functions.
Let us explain in some detail. As in the case of symmetric groups, considering the centers of H n for all n simultaneously is again crucial in the formulation of this paper (surprisingly, this view was not taken advantage of in the literature). We first construct a graded linear isomorphism Ψ : Z −→ Λ v by sending the Geck-Rouquier class elements to a variant of monomial symmetric functions in the λ-ring notation. Our Frobenius map Ψ is compatible with Lascoux's earlier remarkable work on transition matrices between bases, and in this way, we are able to identify precisely (see Theorem 4.6) various bases of Z(H n ) in [La] with well-known bases in Λ v of degree n up to some λring notation shifts, that is, {s λ }, {m λ }, {e λ }, {h λ } and {p λ } in Macdonald's notation.
Via a notion of relative norm on Hecke algebras (which goes back to unpublished work of Peter Hoefsmit and Leonard Scott in 1976 according to [Jo]), we show that Z is naturally a commutative associative graded algebra and that Z is isomorphic to a polynomial algebra. Then we show that Ψ : Z → Λ v is actually an isomorphism of graded algebras. Our proof uses the identification under Ψ of one basis of Lascoux for Z with a multiplicative basis for Λ v . Using Ψ, we are able to provide new simple proofs for Lascoux's difficult results on some other bases of Z.
The Frobenius map Ψ is shown to fit well with the character formula for the Hecke algebras established in [Ram]. Moreover, Ψ readily gives rise to a v-characteristic map ch v for the Hecke algebra H n which sends the irreducible characters to the corresponding Schur functions. In our approach, Ψ plays a more fundamental role than ch v just as in the original formulation of Frobenius.
Here is a layout of the paper. In Section 2, we review some basic constructions on Hecke algebras including Geck-Rouquier basis. In Section 3, we show that Z endowed with a product from relative norm is commutative, associative, and isomorphic to a polynomial algebra. The Frobenius map Ψ is defined and shown to be an isomorphism of graded algebras. The images of various bases of the centers under the isomorphism Ψ are determined in Section 4. In Section 5, we show how Ψ gives rise to a v-characteristic map which identifies the irreducible characters with Schur functions.
Acknowledgements. We thank Shun-Jen Cheng and Institute of Mathematics, Academia Sinica, Taipei for support and providing an excellent atmosphere in December 2011, during which this project was initiated. The first author was partially supported by NSFC-11101031, and the second author was partially supported by NSF DMS-1101268.

Centers of Hecke algebras
2.1. The Hecke algebra H n . Let S n be the symmetric group in n letters generated by the simple transpositions s i = (i, i + 1), i = 1, . . . , n − 1. Let v be an indeterminate. The Hecke algebra H n is the unital C(v)-algebra generated by T 1 , . . . , T n−1 satisfying the relations If w = s i 1 · · · s ir is a reduced expression, then we define T w = T i 1 · · · T ir . The Hecke algebra H n is a free C(v)-module with basis {T w |w ∈ S n }, and it can be regarded as a v-deformation of the group algebra CS n of the symmetric group S n .
Denote by P n (resp. CP n ) the set of partitions (resp. compositions) of n. For each λ ∈ P n , there exists an irreducible character χ λ v of H n such that {χ λ v |λ ∈ P n } is a complete set of irreducible characters of H n ; moreover the specialization of χ λ v at v = 1 coincides with the character χ λ of the Specht module of S n associated to λ for λ ∈ P n .
2.2. Geck-Rouquier class elements. For λ ∈ P n , denote by C λ the conjugacy class of S n containing all permutations of cycle type λ, and accordingly let c λ be the class sum of C λ . Clearly, the set {c λ |λ ∈ P n } forms a linear basis of the center of the group algebra CS n .
There exists a canonical symmetrizing trace form τ on H n defined as follows [GP] τ : Then the bilinear form (·, ·) induced by τ on H n satisfies (T w , T σ ) = δ w,σ −1 v ℓ(w) and hence the basis dual to {T w |w ∈ S n } is Then it is easy to check that χ is a trace function on H n , that is, Observe that w λ is a minimal length element in the conjugacy class C λ . It follows from [Ram,Theorem 5.1] (cf. [GP,Section 8.2 and moreover is a trace function. Then we obtain the Geck-Rouquier central elements Observe that f * λ specializes at v = 1 to the class sum c λ . Lemma 2.1. [GR] The set {f * λ |λ ∈ P n } forms a basis of the center Z(H n ) of H n . The definition of the Hecke algebra H n used in this paper is different from the one used in [La], where the parameter q is used. The generators in [La], which we denote by

Central idempotents.
Since H n is semisimple and τ is a symmetrizing trace function, we can write are known as the Schur elements for H n (cf. [GP,Theorem 7.2.6] where the notation c λ is used for Schur elements). Denote by It turns out that d λ (v) is a polynomial in v, and d λ (1) equals the degree d λ of of the irreducible character χ λ . A closed formula for d λ (v) due to Steinberg can be found in [GP,Theorem 10.5.3]. For λ ∈ P n , set Lemma 2.2. [GR,Proposition 4.2] The following holds for λ ∈ P n : Since the specializations of τ and χ λ v at v = 1 coincide with the canonical trace function on CS n and the irreducible character χ λ , respectively, ε λ v specializes to the central primitive idempotent ε λ in CS n given by 3. The Frobenius map Ψ 3.1. Lascoux's results. For a composition α = (α 1 , . . . , α ℓ ) ∈ CP n , denote by H α the Hecke algebra associated to the Young subgroup S α , which can be identified with the subalgebra of H n generated by the elements T i with 1 ≤ i ≤ n − 1 and i = α 1 , α 1 + α 2 , . . . , α 1 + · · · + α ℓ−1 . Denote by D α the set of minimal length coset representatives of S α in S n . For h ∈ H α , define the relative norm [Jo] (3.1) The following fundamental result is due to Jones.
(2) The set {N λ (1) |λ ∈ P n } is a basis of Z(H n ). Moreover the transition matrix from {N λ (1) |λ ∈ P n } to the v-class sum basis {f * λ |λ ∈ P n } is equal to the product M (e, m) · D.
For a one-part partition λ = (n), we shall simplify notation N (n) (1) by N n (1), and so on. Let w 0 n be the longest element in S n . Recall the notation T w from (2.2). Remark 3.3. Part (1) of Proposition 3.2 is a special case of [La,Theorem 6] and (2) follows immediately from (1). Another interesting special case of [La,Theorem 6] states that 3.2. The algebra Z. Consider the direct sum of the centers We define a product • on Z via the relative norm (3.1) as follows: for z 1 ∈ Z(H m ) and where z 1 ⊗ z 2 is viewed as an element in the algebra H m ⊗ H n ∼ = H (m,n) associated to the composition (m, n) of m + n.

The Frobenius map Ψ.
Definition 3.7. Define the (quantum) Frobenius map to be the linear map Proposition 3.8. The following holds for λ ∈ P n : Proof. In the notation of Proposition 3.2, we write e λ (x) = µ b λµ m µ (x). By Proposition 3.2(2) we obtain We compute that The proposition is proved.
Proof. Clearly Ψ is a linear isomorphism since {f * λ |λ ∈ P n } and {m λ (x) |λ ∈ P n } are C(v)-linear basis of Z(H n ) and Λ n v , respectively, for n ≥ 0. It remains to show that Ψ is an algebra homomorphism. Suppose µ = (µ 1 , . . . , µ k ) ∈ P m and λ = (λ 1 , . . . , λ ℓ ) ∈ P n . Then by (3.5) we have where µ ∪ λ is the partition corresponding to the composition obtained by concatenating the parts of µ and λ, and the last equality is due to the commutativity of Z by Theorem 3.5. Then by Proposition 3.8 one deduces that This proves the theorem.
Remark 3.10. We mention that there is another connection in a different direction between the center Z(H n ) and the ring of symmetric functions, which has yet to be developed [FW]. In the v = 1 case, such a connection was developed in [Mac,I. 7,Ex. 25], which was in turn based on a classical work of Farahat-Higman.
4. Identification of bases of Z and Λ v 4.1. Identification of bases. Given a partition λ = (λ 1 , . . . , λ ℓ ), let w 0 λ be the longest element in the Young subgroup S λ , and set Proposition 4.1. Suppose λ is a partition of n. Then Proof. Assume λ = (λ 1 , λ 2 , . . . , λ ℓ ) with ℓ(λ) = ℓ. By (3.3), one deduces that, for n ≥ 1, where the last equality is due to fact that h n = µ∈Pn m µ . Observe that T 2 w 0 λ can be written as T 2 Therefore, by Theorem 3.9 and (4.1), we calculate This proves Part (1) of the proposition. By definition, we have Note that m k = p k for k ≥ 1. Then by Theorem 3.9, we compute that Hence Part (2) of the proposition is proved.
Remark 4.2. T 2 w 0 n is the product of all "multiplicative" Jucys-Murphy elements in H n (cf. [La]). Meanwhile F n = f * n ∈ Z(H n ) is the product of all "additive" Jucys-Murphy elements in H n [Jo] (also see [La]).
Recall the matrix D from (3.2).
Corollary 4.3. The following holds for n ≥ 0: (1) The set {N λ ( T 2 w 0 λ ) |λ ∈ P n } is a basis of Z(H n ) and the transition matrix from this basis to {f * λ |λ ∈ P n } is equal to the product (2) The set {N λ (F λ ) |λ ∈ P n } is a basis of Z(H n ) and the transition matrix from this basis to {f * λ |λ ∈ P n } is equal to the product Proof. Observe that (v − 1) n h λ x v − 1 |λ ∈ P n is a basis of Λ n v . Then by Proposition 4.1(1), the set N λ ( T 2 w 0 λ ) |λ ∈ P n is a basis of Z(H n ). Part (1) follows now by combining (3.7) and Proposition 4.1(1). By a similar argument, Part (2) can be verified via Proposition 4.1(2).
Remark 4.4. Corollary 4.3 was originally proved by Lascoux [La,Theorem 10,Theorem 12], while the basis {N λ (F λ ) |λ ∈ P n } was constructed earlier by Jones [Jo]. Lascoux's proof used ingenious connection to non-commutative symmetric functions which is difficult for us to follow. In his approach, Lascoux established the transition matrices in the three cases as in Proposition 3.2 and Corollary 4.3 separately.
(1) Z is a polynomial algebra with generators N n ( T 2 w 0 n ), for n ≥ 1.
(2) Z is a polynomial algebra with generators N n (F n ), for n ≥ 1.
In the following, we summarize the identification of various bases of Z and Λ v , for the convenience of the reader.
Theorem 4.6. Suppose λ ∈ P n . Then the following holds: Proof. Note that (4.2) follows from the definition of Ψ, (4.4), (4.5) and (4.6) follow from Proposition 3.8 and Proposition 4.1. One the other hand, (4.3) follows from the two identities (5.2) and (5.7) in the subsequent section (This can be viewed as a reformulation of the main result of [Ram]).
Remark 4.7. It follows by (4.3) that the restriction Ψ n : Z(H n ) → Λ n v of Ψ is an algebra isomorphism if we impose an algebra structure on Λ n v by s λ * s µ = δ λµ κ λ (v)s λ . 4.2. Specialization at v = 1. For a partition λ = (1 m 1 2 m 2 · · · ), define It is known that the Frobenius morphism from CS n to Λ n sends w to p λ (x) if w ∈ C λ . This gives rise to an isomorphism (called the classical Frobenius map): For a symmetric function f ∈ Λ v , denote when possible by f | v=1 the specialization of f at v = 1.
By (4.7), Lemma 4.8 and the fact that f * λ specializes to c λ at v = 1, we have the following.
Proposition 4.9. The specialization of the (quantum) Frobenius map Ψ at v = 1 coincides with the classical Frobenius map ψ.

5.
The v-characteristic map 5.1. The v-characteristic map ch v . Denote by R n and R n v the Grothendieck groups of the categories of CS n -modules and H n -modules, respectively. Set It is known that R is a commutative algebra with the multiplication given by Sm⊗Sn M ⊗ N for S m -module M and S n -module N . Similarly, R v also affords an algebra structure via the functor ind H m+n Hm⊗Hn . Below we freely identify elements in C⊗ Z R n and C(v)⊗ Z R n v with trace functions on CS n and H n , respectively. The Frobenius characteristic map (cf. [Mac,I. 7]) ch : R −→ Λ χ → µ∈Pn χ(w µ )ψ(c µ ) for χ ∈ R n sends the irreducible character χ λ to the Schur function s λ for any partition λ. Observe that the characteristic map ch can also be described using ε λ in (2.5) for λ ∈ P n as Recall the generic degrees d λ (v) from (2.4). In light of (5.1), we make the following.
Definition 5.1. Define the Frobenius characteristic map ch v to be the linear map by For r ≥ 1 and for partition µ = (µ 1 , . . . , µ ℓ ) of length ℓ, we set Recall that {χ λ v |λ ∈ P n } is a complete set of irreducible characters of the Hecke algebra H n , and recall the following Frobenius character formula for H n of Ram.
Proposition 5.2. [Ram,Theorem 4.14] Suppose µ ∈ P n . Then The Frobenius character formula admits the following dual reformulation.
Remark 5.5. Fomin and Thibon [FT] took the correspondence in Theorem 5.4 between irreducible characters and Schur functions as a definition of the characteristic map for Hecke algebras, though they did not study the centers.

A commutative diagram.
Lemma 5.6. Suppose µ ∈ P m , λ ∈ P n . Then ind H m+n where c ν µλ are the Littlewood-Richardson coefficients.
Proof. Assume µ ∈ P m , λ ∈ P n . The induced character ind H m+n Hm⊗Hn (χ µ v ⊗ χ λ v ) can be written as ind H m+n for some non-negative integer u ν . Then specializing at v = 1, we obtain ind S m+n Then it is known from the representation theory of symmetric groups that u ν coincides with the Littlewood-Richardson coefficients c ν µλ for ν ∈ P m+n . Then by (5.7) and Lemma 5.6, we have the following.
Theorem 5.7. The characteristic map ch v : R v → Λ v is an algebra isomorphism.
Recall from (2.1) that the map is an isomorphism. This leads to a linear isomorphism Θ = ⊕ n≥0 Θ n : R v −→ Z.