Mixed hook-length formula for degenerate affine Hecke algebras

Details Mixed hook-length formula for degenerate affine Hecke algebras Mixed hook-length formula for degenerate affine Hecke algebras

1999-06-22

arxiv.org/abs/math/9906148v4

Lecture Notes in Math. 1815 (2003), 223-236

Mathematics

Representation Theory

AmS-TeX, 12 pages, final version

Scientific paper

Take the degenerate affine Hecke algebra $H_{l+m}$ corresponding to the group $GL_{l+m}$ over a $p$-adic field. Consider the $H_{l+m}$-module $W$ induced from the tensor product of the evaluation modules over the algebras $H_l$ and $H_m$. The module $W$ depends on two partitions $\lambda$ of $l$ and $\mu$ of $m$, and on two complex numbers $z$ and $w$. There is a canonical operator $J$ acting in $W$, it corresponds to the rational Yang $R$-matrix. The algebra $H_{l+m}$ contains the symmetric group $S_{l+m}$, and $J$ commutes with the action of $S_{l+m}$ in $W$. 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 obtain a nice formula for the ratio of two eigenvalues of $J$, corresponding to the "highest" and "lowest" (multiplicity-free) irreducible components of $W$.

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