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A proof of the second Rogers-Ramanujan identity via Kleshchev multipartitions. (English) Zbl 1528.11111

J. Lepowsky and S. C. Milne [Adv. Math. 29, 15–59 (1978; Zbl 0384.10008)] observed a similarity between some Lie algebraic identities and the infinite products of the Rogers-Ramanujan identities \[ \sum_{n=0}^{\infty}\frac {q^{n^2}}{(1-q)(1-q^2)\cdots (1-q^n)}= \prod_{n=0}^{\infty}\frac{1}{(1-q^{5n+1})(1-q^{5n+4})} \;\;\;\;\; (|q|<1) \] and \[ \sum_{n=0}^{\infty}\frac {q^{n^2+n}}{(1-q)(1-q^2)\cdots (1-q^n)}= \prod_{n=0}^{\infty}\frac{1}{(1-q^{5n+2})(1-q^{5n+3})} \;\;\;\;\; (|q|<1). \] J. Lepowsky and R. L. Wilson [Invent. Math. 77, 199–290 (1984; Zbl 0577.17009)] gave a Lie theoretic interpretation of the infinite sums in the Rogers-Ramanujan identities. S. Corteel [Proc. Am. Math. Soc. 145, 2011–2022 (2017; Zbl 1357.05007)] gave a simple combinatorial proof of the second Rogers-Ramanujan identity by using cylindric plane partitions and the Robinson-Schensted-Knuth algorithm. Several relationships between Rogers-Ramanujan type identities and Kashiwara crystals are known, see [J. Dousse and J. Lovejoy, Proc. Am. Math. Soc. 146, 55–67 (2018; Zbl 1375.05018)].
In the paper under review the author proves the second Rogers-Ramanujan identity using certain \(k\)-tuples of unequal partitions, the Kleshchev multipartitions.

MSC:

11P84 Partition identities; identities of Rogers-Ramanujan type
05A15 Exact enumeration problems, generating functions
17B37 Quantum groups (quantized enveloping algebras) and related deformations

Software:

qMultiSum
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Full Text: DOI arXiv

References:

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