A mixed hook-length formula for affine Hecke algebras

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Publication details

JournalEuropean Journal of Combinatorics
DatePublished - Nov 2004
Issue number8
Number of pages32
Pages (from-to)1345-1376
Original languageEnglish


Let Image be the affine Hecke algebra corresponding to the group GLl over a p-adic field with residue field of cardinality q. We will regard Image as an associative algebra over the field Image . Consider the Image -module W induced from the tensor product of the evaluation modules over the algebras Image and Image . The module W depends on two partitions ¿ of l and µ of m, and on two non-zero elements of the field Image . There is a canonical operator J acting on W; it corresponds to the trigonometric R-matrix. The algebra Image contains the finite dimensional Hecke algebra Hl+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 ¿.

    Research areas

  • Algebra, Pure Mathematics

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