Iwahori–Hecke Algebras of TypeAat Roots of Unity

Abstract

Let H n q be the Iwahori-Hecke algebra of type A n−1 over a field K of characteristic 0. For each Young diagram λ, an H n q module S λ , called a Specht module, was defined in [DJ]. The dimension of S λ does not depend on the choice of q, and for q not a root of unity, the S λ 's provide a complete set of irreducible H n q -modules, up to isomorphism. If q is a primitive lth root of unity, a complete set of simple H n q -modules D µ has been constructed in [DJ], where µ runs through all Young diagrams with at most l − 1 rows of equal length. These modules are not well understood, but their dimensions can be computed if one knows the multiplicities d λµ of the D µ in a composition series of S λ , for all possible µ's and λ's. A result known as the Nakayama conjecture, or equivalently, in the context of quantum groups, as the linkage principle, gives some information about the d λµ 's. It says that d λµ = 0 only if µ + ρ is in the orbit of λ + ρ with respect to an affine reflection group. Here ρ = k − 1 k − 2 1 0 for some k ≥ n, and the reflection group is that generated by the affine reflections in the hyperplanes y i − y j = ml, 1 ≤ i < j ≤ k and m ∈ . * We thank the referee for an exceptionally careful reading of a previous version of this paper and for pointing the way towards several improvements in the exposition. H.

Untitled

“…It is possible to give a simple combinatorial proof, adapting arguments from [GW1]. Here we will make use of Ariki's theorem, and the positivity result of Varagnolo and Vasserot, in the interest of brevity.…”

“…It is well known that the decomposition numbers for the Hecke algebra H n (q; C) satisfy d λ,µ = 0 for all Young diagrams λ = µ with no more than k rows; see, for example, [GW1]. It follows from d λ,µ (v) ∈ N[v] and d λ,µ (1) = d λ,µ that d λ,µ (v) = 0 as well for all Young diagrams λ = µ with no more than k rows.…”

“…This construction was suggested by our approach to the decomposition numbers in [GW1]. For µ an interior diagram, the definition of the elementÃ(µ) is a "qversion" of the construction in Section 5 of [GW1]. There we defined N(λ, µ) to be the number of paths from µ c to λ which are conjugate to the skew tableau T, and…”

“…(The equality can also be proved using Schur-Weyl duality; see [Du] and [DPS] for Schur-Weyl duality and [GW1,Section 6] for a proof using Schur-Weyl duality.) Our proof of the equality of the polynomial ana- tableaux.…”

Hecke algebras with independent parameters

“…Tits showed that, if W is finite, F = C is the field of complex numbers, and q ∈ C is neither zero nor a root of unity, then the Hecke algebra H S (q) is semisimple and isomorphic to the group algebra CW. The representation theory of Hecke algebras at roots of unity has been studied to some extent, with connections to other topics found (see Geck and Jacon [12]), but has not been completely determined yet even in type A (see Goodman and Wenzl [13]).…”