Journal of the
Korean Mathematical Society
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ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2003; 40(1): 1-28

Printed January 1, 2003

Copyright © The Korean Mathematical Society.

Highest weight vectors of irreducible representations of the quantum superalgebra $\mathfrak{U}_q(gl(m,n))$

Dongho Moon

Sejong University

Abstract

The Iwahori-Hecke algebra $\mathcal H_k(q^2)$ of type A acts on the $k$-fold tensor product space of the natural representation of the quantum superalgebra $\mathfrak U_q(gl(m,n))$. We show the Hecke algebra $\mathcal H_k(q^2)$ and the quantum superalgebra $\mathfrak U_q(gl(m,n))$ have commuting actions on the tensor product space, and determine the centralizer of each other. Using this result together with Gyoja's $q$-analogue of the Young symmetrizers, we construct highest weight vectors of irreducible summands of the tensor product space.

Keywords: quantum superalgebra, maximal vector, Hecke algebra, Schur-Weyl duality

MSC numbers: Primary 17B37, 20C08; Secondary 05A17

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