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.
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Work partially supported by an NSF grant at the University of California, San Diego
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Ram, A. A Frobenius formula for the characters of the Hecke algebras. Invent Math 106, 461–488 (1991). https://doi.org/10.1007/BF01243921
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DOI: https://doi.org/10.1007/BF01243921