The Rogers - Ramanujan Identities , the Finite General Linear Groups , and theHall
نویسنده
چکیده
The Rogers-Ramanujan identities have been studied from the viewpoints of combinatorics, number theory, a ne Lie algebras, statistical mechanics, and conformal eld theory. This note connects the Rogers-Ramanujan identities with the nite general linear groups and the HallLittlewood polynomials of symmetric function theory.
منابع مشابه
9 D ec 1 99 7 The Rogers - Ramanujan Identities , the Finite General Linear Groups , and the Hall - Littlewood Polynomials By Jason Fulman Dartmouth College
Gordon’s generalization of the Rogers-Ramanujan identities have been widely studied and appear in many places in mathematics and physics. Andrews [1] discusses combinatorial aspects of these identities. Berndt [4] describes some number theoretic connections. Feigen and Frenkel [8] interpret the Gordon identities as a character formula for the Virasoro algebra. Andrews, Baxter, and Forrester [2]...
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We connect Gordon’s generalization of the Rogers-Ramanujan identities with the Hall-Littlewood polynomials and with generating functions which arise in a probabilistic setting in the finite general linear groups. This yields a Rogers-Ramanujan type product formula for the n → ∞ probability that an element of GL(n, q) or Mat(n, q) is semisimple. 1. Background and notation The Rogers-Ramanujan id...
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The asymptotic probability theory of conjugacy classes of the finite general linear and unitary groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given and compared with work on the uniform measure. Elementary probabilistic proofs of the Rogers-Ramanujan identities follow. As a coro...
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Recently, starting from two infinite summation formulae for Hall-Littlewood polynomials, two of the present authors [7] have generalized a method due to Macdonald [9] to obtain new finite summation formulae for these polynomials. This approach permits them to extend Stembridge’s list of multiple qseries identities of Rogers-Ramanujan type [12]. Conversely these symmetric functions identities ca...
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