Skew shape representations are irreducible

Details Skew shape representations are irreducible Skew shape representations are irreducible

2004-01-23

arxiv.org/abs/math/0401326v2

Published in Combinatorial and Geometric representation theory, S.-J. Kang and K.-H. Lee eds., Contemp. Math. 325 Amer. Math.

Mathematics

Representation Theory

Scientific paper

In this paper all of the classical constructions of A. Young are generalized to affine Hecke algebras of type A. It is proved that the calibrated irreducible representations of the affine Hecke algebra are indexed by placed skew shapes and that these representations can be constructed explicitly with a generalization of Young's seminormal construction of the irreducible representations of the symmetric group. The seminormal construction of an irreducible calibrated module does not produce a basis on which the affine Hecke algebra acts integrally but using it one is able to pick out a different basis, an analogue of Young's natural basis, which does generate an integral lattice in the module. Analogues of the "Garnir relations" play an important role in the proof. The Littlewood-Richardson coefficients arise naturally as the decomposition multiplicities for the restriction of an irreducible representation of the affine Hecke algebra to the Iwahori-Hecke algebra.

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