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Polynomial super representations of \(U_{q}^{\mathrm{res}}(\mathfrak {gl}_{m|n})\) at roots of unity

  • Jie DuEmail author
  • Yanan Lin
  • Zhongguo Zhou
Article
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Abstract

As a homomorphic image of the hyperalgebra \(U_{q,R}(m|n)\) associated with the quantum linear supergroup \(U_{\varvec{\upsilon }}(\mathfrak {gl}_{m|n})\), we first give a presentation for the q-Schur superalgebra \(S_{q,R}(m|n,r)\) over a commutative ring R. We then develop a criterion for polynomial supermodules of \(U_{q,F}(m|n)\) over a field F and use this to determine a classification of polynomial irreducible supermodules at roots of unity. This also gives classifications of irreducible \(S_{q,F}(m|n,r)\)-supermodules for all r. As an application when \(m=n\ge r\) and motivated by the beautiful work (Brundan and Kujawa in J Algebraic Combin 18:13–39, 2003) in the classical (non-quantum) case, we provide a new proof for the Mullineux conjecture related to the irreducible modules over the Hecke algebra \(H_{q^2,F}({{\mathfrak {S}}}_r)\); see Brundan (Proc Lond Math Soc 77:551–581, 1998) for a proof without using the super theory.

Keywords

Quantum linear supergroup Quantum hyperalgebra Polynomial representation q-Schur superalgebra Hecke algebra Mullineux conjecture 

Mathematics Subject Classification

17B35 17B37 17B70 20C08 20G43 

Notes

Acknowledgements

The first author would like to thank Jonathan Brundan for his comments made at the 2016 Charlottesville and 2017 Sydney conferences. The authors also thank Weiqiang Wang and Hebing Rui for several discussions and thank the referee for some helpful comments.

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.School of Mathematical SciencesXiamen UniversityXiamenChina
  3. 3.College of ScienceHohai UniversityNanjingChina

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