Schur-Weyl Reciprocity for the Hecke Algebra of (\Bbb Z/r\Bbb Z)\wr \frak S_n

1993/10/14 by Susumu Ariki, Ariki, Susumu, Tomohide Terasoma +3
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th

paper · pdf · doi:10.48550/arxiv.hep-th/9310078

15

arxiv created 1993/10/14 · arxiv updated 2009/11/30

Abstract

The purpose of this paper is to give a reciprocity between U_q(h) and \Cal H_n,r, the Hecke algebra of (\Bbb Z / r\Bbb Z)\wr \frak S_n introduced by Ariki and Koike. Let K=\Bbb Q(q,u₁,… ,u_r) be the field of rational funcitons in variables q,u₁,… ,u_r. We adopt K as the base field for both the quantized universal enveloping algebra U_q(gl_r) and the Hecke algebra \Cal H_n. We denote by U_q(h) the K-subalgebra of U_q(gl_r) generated by q^E_ii 's (1≤ i ≤ r). In this paper, we show that the commutant of U_q(h) in End((K^r)^{⊗ n}) is isomorphic to a quotient of \Cal H_n,r. We also determine the irreducible decomposition of (K^r)^{⊗ n} under the action of \Cal H_n,r. As a consequence, we obtain the reciprocity for U_q(h) and \Cal H_n,r.

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