Centers of Generic Hecke Algebras

Abstract

ABSTRACT. Let W be a Weyl group and let W' be a parabolic subgroup of W . Define A as follows: ®Q[uj .9f(W) where .9f (W) is the generic algebra of type An over Q [u] , u an indeterminate, associated with the group W, and R is a Q[u]-algebra, possibly of infinite rank, in which u is invertible. Similarly, we define A' associated with W'. Let M be an A-A bimodule, and let b EM. Define the relative normwhere T is the set of distinguished right coset representives for W' in W.In addition, other properties of the relative norm are given and used to develop a theory of induced modules for generic Heeke algebras including a Markey decomposition. This section of the paper is previously unpublished work of P. Hoefsmit and L. L. Scott. with 'W; a k;-cycle of length k; -I in Ski' Then the main result of this paper IS Theorem. The set {b" I rd-n} is a basis for ZA(Sn) (A(Sn)) over Q [u, u-1 ] .Remark. The norms b" in ZA(Sn)(A(Sn)) are analogs of conjugacy class sums in the center of QSn and, in fact, specialization of these norms at u = I gives the standard conjugacy class sum basis of the center of QSn up to coefficients from Q.

The Green correspondence for the representations of Hecke algebras of type $A\sb {r-1}$

“…In this paper we shall generalize some basic results of Green along the lines of the work of L. Jones (see [Jo]). We organize the paper as follows: After recalling some basic results, we shall prove the conjecture (Theorem 2.7) which has been mentioned in [Jo,5.3]. The Brauer homomorphism constructed by Jones will playa key role in proving that conjecture.…”

“…Then, by [Jo,3.35], there exists a parabolic subgroup JV;. of W unique up to conjugation such that M is relatively ~-projective (see the definition in [Jo,Chapter 3]) and such that JV;.…”

“…These results are mostly due to P. Hoefsmit and L. Scott. One can find a complete proof in [Jo,Chapter 3].…”

On the square root of the centre of the Hecke algebra of type A

“…In [5], we determined explicitly, for the centre of the Hecke algebra of type A, how to express each element of the Q[q, ^~']-norm basis in [8] as a linear combination of the elements of the Z[q, ^"'J-minimal basis (see [4]). During our research for [5], we were led naturally to the square of the element of the Hecke algebra corresponding to the longest word in the symmetric group.…”

The fundamental invariant of the Hecke algebra Hn(q) characterizes the representations of Hn(q), S n, SUq(N), and SU(N)

“…For recent expositions of the mathematical structure and of the physical relevance of the Hecke algebra, jointly providing access to many earlier references, see refs. [1][2][3][4][5][6][7][8][9][10][11].…”