Standard multipartitions and a combinatorial affine Schur-Weyl duality. (English) Zbl 1523.20015
Summary: We introduce the notion of standard (Kleshchev) multipartitions and establish a one-to-one correspondence between standard multipartitions and irreducible representations with integral weights for the affine Hecke algebra of type \(A\) with a parameter \(q \in \mathbb{C}^{\times}\) which is not a root of unity. We then extend the correspondence to all Kleshchev multipartitions for Ariki-Koike algebras of integral type. By the affine Schur-Weyl duality, we further extend this to a correspondence between standard multipartitions and Drinfeld multipolynomials of integral type whose associated irreducible polynomial representations completely determine all irreducible polynomial representations for the quantum loop algebra \(\mathrm{U}_q (\widehat{\mathfrak{gl}}_n)\). We will see, in particular, the notion of standard multipartitions gives rise to a combinatorial description of the affine Schur-Weyl duality in terms of a column-reading vs. row-reading of residues of a multipartition.
MSC:
| 20C08 | Hecke algebras and their representations |
| 20C32 | Representations of infinite symmetric groups |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 20B30 | Symmetric groups |
| 20G43 | Schur and \(q\)-Schur algebras |
Keywords:
affine Hecke algebra; Ariki-Koike algebra; quantum loop algebra; affine Schur-Weyl duality; standard multipartition; Drinfeld polynomialReferences:
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