Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.3792/pjaa.101.007

Web of Science

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1112/jlms.70152

Web of Science

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1007/s11139-024-00895-6

Web of Science

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1353/ajm.2024.a923238

Scopus

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Publishing type：Research paper (scientific journal)  

DOI： 10.3792/PJAA.99.005

Scopus

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In a sequence of two papers, S. Capparelli, A. Meurman, A. Primc, M. Primc
(CMPP) and then M. Primc put forth three remarkable sets of conjectures,
stating that the generating functions of coloured integer partition in which
the parts satisfy restrictions on the multiplicities admit simple infinite
product forms. While CMPP related one set of conjectures to the principally
specialised characters of standard modules for the affine Lie algebra
$\mathrm{C}_n^{(1)}$, finding a Lie-algebraic interpretation for the remaining
two sets remained an open problem. In this paper, we use the work of Griffin,
Ono and the fourth author on Rogers-Ramanujan identities for affine Lie
algebras to solve this problem, relating the remaining two sets of conjectures
to non-standard specialisations of standard modules for $\mathrm{A}_{2n}^{(2)}$
and $\mathrm{D}_{n+1}^{(2)}$. We also use their work to formulate conjectures
for the bivariate generating function of one-parameter families of CMPP
partitions in terms of Hall-Littlewood symmetric functions. We make a detailed
study of several further aspects of CMPP partitions, obtaining (i) functional
equations for bivariate generating functions which generalise the well-known
Rogers-Selberg equations, (ii) a partial level-rank duality in the
$\mathrm{A}_{2n}^{(2)}$ case, and (iii) (conjectural) identities of the
Rogers-Ramanujan type for $\mathrm{D}_3^{(2)}$.

arXiv

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Other Link： http://arxiv.org/pdf/2404.03851v1

Publishing type：Internal/External technical report, pre-print, etc.  

We propose a generalization of Schur regular partitions for each odd integer<br />
$p\geq 3$. Applying Kashiwara crystal theory, we prove that the number of<br />
partitions of $n$ with this condition is equinumerous to the number of strict<br />
$p$-class regular partitions of $n$. At $p=3$, it is Schur&#039;s 1926 partition<br />
theorem found as a mod 6 analog of the Rogers-Ramanujan partition theorem<br />
(RRPT). The statement for $p=5$ was conjectured by Andrews in 1970s in a course<br />
of his 3 parameter generalization of RRPT and proved in 1994 by<br />
Andrews-Bessenrodt-Olsson with an aid of computer.

arXiv

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