Crystals and affine Hecke algebras of type D

The Lascoux-Leclerc-Thibon-Ariki theory asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra $g$ where $g$ is $gl_\infty$ or the affine Lie algebra $A^{(1)}_\ell$, and the irreducible representations correspond to the upper global bases. Recently, N. Enomoto and the first author presented the notion of symmetric crystals and formulated analogous conjectures for the affine Hecke algebras of type B. In this note, we present similar conjectures for certain classes of irreducible representations of affine Hecke algebras of type D. The crystal for type D is a double cover of the one for type B.Comment: 8 page

Reduced Kronecker products which are multiplicity free or contain only few components

It is known that the Kronecker coefficient of three partitions is a bounded and weakly increasing sequence if one increases the first part of all three partitions. Furthermore if the first parts of partitions \lambda,\mu are big enough then the coefficients of the Kronecker product [\lambda][\mu]=\sum_\n g(\l,\m,\n)[\nu] do not depend on the first part but only on the other parts. The reduced Kronecker product [\lambda]_\bullet \star[\mu]_\bullet can be viewed (roughly) as the Kronecker product [(n-|\lambda|,\lambda)][(n-|\mu|,\m)] for n big enough. In this paper we classify the reduced Kronecker products which are multiplicity free and those which contain less than 10 components.We furthermore give general lower bounds for the number of constituents and components of a given reduced Kronecker product. We also give a lower bound for the number of pairs of components whose corresponding partitions differ by one box. Finally we argue that equality of two reduced Kronecker products is only possible in the trivial case that the factors of the product are the same.Comment: 11 pages, final version. appears in European J. Combi