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**A mixed hook-length formula for affine Hecke algebras.** / Nazarov, Maxim.

Research output: Contribution to journal › Article › peer-review

Nazarov, M 2004, 'A mixed hook-length formula for affine Hecke algebras', *European Journal of Combinatorics*, vol. 25, no. 8, pp. 1345-1376. https://doi.org/10.1016/j.ejc.2003.10.010

Nazarov, M. (2004). A mixed hook-length formula for affine Hecke algebras. *European Journal of Combinatorics*, *25*(8), 1345-1376. https://doi.org/10.1016/j.ejc.2003.10.010

Nazarov M. A mixed hook-length formula for affine Hecke algebras. European Journal of Combinatorics. 2004 Nov;25(8):1345-1376. https://doi.org/10.1016/j.ejc.2003.10.010

@article{28b28d1950f64fa191291a7dc3c8bc0d,

title = "A mixed hook-length formula for affine Hecke algebras",

abstract = "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 ¿.",

keywords = "Algebra, Pure Mathematics",

author = "Maxim Nazarov",

year = "2004",

month = nov,

doi = "10.1016/j.ejc.2003.10.010",

language = "English",

volume = "25",

pages = "1345--1376",

journal = "European Journal of Combinatorics",

number = "8",

}

TY - JOUR

T1 - A mixed hook-length formula for affine Hecke algebras

AU - Nazarov, Maxim

PY - 2004/11

Y1 - 2004/11

N2 - 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 ¿.

AB - 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 ¿.

KW - Algebra, Pure Mathematics

U2 - 10.1016/j.ejc.2003.10.010

DO - 10.1016/j.ejc.2003.10.010

M3 - Article

VL - 25

SP - 1345

EP - 1376

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

IS - 8

ER -