311 research outputs found
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 where is or the affine Lie algebra
, 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
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