The Ariki–Koike algebras and Rogers–Ramanujan type partitions

S Chern; Z Li; D Stanton; T Xue; AJ Yee

Journal article

The Ariki–Koike algebras and Rogers–Ramanujan type partitions

S Chern, Z Li, D Stanton, T Xue, AJ Yee

Journal of Algebraic Combinatorics | SPRINGER | Published : 2024

DOI: 10.1007/s10801-024-01340-z

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Abstract

Ariki and Mathas (Math Z 233(3):601–623, 2000) showed that the simple modules of the Ariki–Koike algebras HC,v;Q1,…,Qm(G(m,1,n)) (when the parameters are roots of unity and v≠1) are labeled by the so-called Kleshchev multipartitions. This together with Ariki’s categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl–Kac character formula. In this paper, we revisit their generating function relation for the v=-1 case. In particular, this v=-1 scenario is of special interest as the corresponding Kleshchev multipartitions are closely tied with generalized Rogers–Ramanujan type partitions when Q1=⋯=Q..

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University of Melbourne Researchers

Ting Xue Author

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Geometric methods in representation theory and the Langlands program

This research project aims to study questions in representation theory of groups using geometric methods. A central role is played by Langla..

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Awarded by Killam Trusts

Funding Acknowledgements

We would like to express our gratitude to the referees for suggestions that have improved the exposition of our paper to a great extent. S. Chern was supported by a Killam Postdoctoral Fellowship from the Killam Trusts. T. Xue was supported in part by the ARC Grant DP150103525. T. Xue thanks Thomas Gerber, Andrew Mathas, Emily Norton and Peng Shan for helpful discussions. A. J. Yee was supported in part by a Grant ( # 633963 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\#633963$\end{document} ) from the Simons Foundation.