Polynomial super representations of the hyperalgebra of \mathfrakgl_m|n at roots of unity

2018/04/06 by Jie Du, Du, Jie, Yanan Lin +3
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1804.02126

openalex publication_date 2018/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

As a homomorphic image of the hyperalgebra U_q,R(m|n) associated with the quantum linear supergroup U_υ(\mathfrakgl_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 filed 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≥ r and motivated by the beautiful work \citebru 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_q²,F(\mathfrak S_r); see \citeBr for a proof without using the super theory.

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