Séminaire Lotharingien de Combinatoire, 80B.93 (2018), 12 pp.
David Jordan and Monica Vazirani
A Schur-Weyl Like Construction of the Rectangular Representation for the Double Affine Hecke Algebra
Abstract.
Let G = GLN and V be its N-dimensional defining representation.
Given a module M for the algebra of quantum differential operators on
G, and a positive integer n, we may equip the space Fn(M) of
invariant tensors in $V^{\otimes n} \otimes M$, with an action of the double affine
Hecke algebra of type GLn.
In this paper we take M to be the basic module, i.e. the quantized
coordinate algebra M = Oq(G). We describe a weight basis for
Fn(Oq(G)) combinatorially in terms of walks in the type A weight
lattice; these are equivalent to standard periodic tableaux,
and subsequently we identify
Fn(Oq(G)) with the irreducible "rectangular representation" of
height N of the double affine Hecke algebra.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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