2003/07/31 by Maxim Nazarov · 1 citation
Mathematics · #math.RT #math.CO #math.QA
paper · pdf · doi:10.1016/j.ejc.2003.10.010
published as European J. Combinatorics 25 (2004), 1345-1376 · 36 pages, final version
arxiv created 2004/04/05 · arxiv updated 2009/12/01
Consider the affine Hecke algebra Hl corresponding to the group GLl over a p-adic field with the residue field of cardinality q. Regard Hl as an associative algebra over the field C(q). Consider the Hl+m-module W induced from the tensor product of the evaluation modules over the algebras Hl and Hm. The module W depends on two partitions λ of l and μ of m, and on two non-zero elements of the field C(q). There is a canonical operator J acting on W, it corresponds to the trigonometric R-matrix. The algebra Hl+m contains the finite dimensional Hecke algebra of rank l+m as a subalgebra, and the operator J commutes with the action of this subalgebra on W. Under this action, W decomposes into irreducible subspaces according to the Littlewood-Richardson rule. We compute the eigenvalues of J, corresponding to certain multiplicity-free irreducible components of W. In particular, we give a formula for the ratio of two eigenvalues of J, corresponding to the ``highest'' and the ``lowest'' components. As an application, we derive the well known q-analogue of the hook-length formula for the number of standard tableaux of shape λ.