, Volume 106, Issue 1, pp 461-488

A Frobenius formula for the characters of the Hecke algebras

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Summary

This paper uses the theory of quantum groups and the quantum Yang-Baxter equation as a guide in order to produce a method of computing the irreducible characters of the Hecke algebra. This approach is motivated by an observation of M. Jimbo giving a representation of the Hecke algebra on tensor space which generates the full centralizer of a tensor power of the “standard” representation of the quantum group \(U_q (\mathfrak{s}l(n))\) . By rewriting the solutions of the quantum Yang-Baxter equation for \(U_q (\mathfrak{s}l(n))\) in a different form one can avoid the quantum group completely and produce a “Frobenius” formula for the characters of the Hecke algebra by elementary methods. Using this formula we derive a combinatorial rule for computing the irreducible characters of the Hecke algebra. This combinatorial rule is aq-extension of the Murnaghan-Nakayama for computing the irreducible characters of the symmetric group. Along the way one finds connections, apparently unexplored, between the irreducible characters of the Hecke algebra and Hall-Littlewood symmetric functions and Kronecker products of symmetric groups.

Work partially supported by an NSF grant at the University of California, San Diego