Branching algebras for the general linear Lie superalgebra

2024/03/18 by Soo Teck Lee, Lee, Soo Teck, R. B. Zhang +1
Mathematics · Physics and Astronomy · #05E10 #15A75 #20G05 #22E46 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2403.11393

openalex publication_date 2024/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop an algebraic approach to the branching of representations of the general linear Lie superalgebra \mathfrakgl_p|q(\mathbb C), by constructing certain super commutative algebras whose structure encodes the branching rules. Using this approach, we derive the branching rules for restricting any irreducible polynomial representation V of \mathfrakgl_p|q(\mathbb C) to a regular subalgebra isomorphic to \mathfrakgl_r|s(\mathbb C)⊕ \mathfrakgl_r'|s'(\mathbb C), \mathfrakgl_r|s(\mathbb C)⊕\mathfrakgl₁(\mathbb C)^r'+s' or \mathfrakgl_r|s(\mathbb C), with r+r'=p and s+s'=q. In the case of \mathfrakgl_r|s(\mathbb C)⊕\mathfrakgl₁(\mathbb C)^r'+s' with s=0 or s=1 but general r, we also construct a basis for the space of \mathfrakgl_r|s(\mathbb C) highest weight vectors in V; when r=s=0, the branching rule leads to explicit expressions for the weight multiplicities of V in terms of Kostka numbers.

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