90 research outputs found
A mixed hook-length formula for affine Hecke algebras
Consider the affine Hecke algebra corresponding to the group
over a -adic field with the residue field of cardinality . Regard
as an associative algebra over the field . Consider the -module
induced from the tensor product of the evaluation modules over the algebras
and . The module depends on two partitions of and
of , and on two non-zero elements of the field . There is a
canonical operator acting on , it corresponds to the trigonometric
-matrix. The algebra contains the finite dimensional Hecke algebra
of rank as a subalgebra, and the operator commutes with the action of
this subalgebra on . Under this action, decomposes into irreducible
subspaces according to the Littlewood-Richardson rule. We compute the
eigenvalues of , corresponding to certain multiplicity-free irreducible
components of . In particular, we give a formula for the ratio of two
eigenvalues of , corresponding to the ``highest'' and the ``lowest''
components. As an application, we derive the well known -analogue of the
hook-length formula for the number of standard tableaux of shape .Comment: 36 pages, final versio
Combinatorial Representation Theory
We attempt to survey the field of combinatorial representation theory,
describe the main results and main questions and give an update of its current
status. We give a personal viewpoint on the field, while remaining aware that
there is much important and beautiful work that we have not been able to
mention
The hook fusion procedure for Hecke algebras
We derive a new expression for the q-analogue of the Young symmetrizer which
generate irreducible representations of the Hecke algebra. We obtain this new
expression using Cherednik's fusion procedure. However, instead of splitting
Young diagrams into their rows or columns, we consider their principal hooks.
This minimises the number of auxiliary parameters needed in the fusion
procedure.Comment: 19 page
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