A superdimension formula for gl(m|n) modules
Michael Chmutov, Rachel Karpman, Shifra Reif
TL;DR
This paper presents a formula for calculating the superdimension of finite-dimensional simple gl(m|n)-modules, providing a new proof of a conjecture relating superdimension and atypicality degree, with implications for representation theory.
Contribution
It introduces a superdimension formula based on the Su-Zhang character formula and offers a simplified proof of the Kac-Wakimoto conjecture for gl(m|n).
Findings
Superdimension formula for gl(m|n) modules derived.
Proof that nonzero superdimension corresponds to maximal atypicality.
Simplified algebraic proof of Kac-Wakimoto conjecture.
Abstract
We give a formula for the superdimension of a finite-dimensional simple gl(m|n)-module using the Su-Zhang character formula. As a corollary, we obtain a simple algebraic proof of a conjecture of Kac-Wakimoto for gl(m|n), namely, a simple module has nonzero superdimension if and only if it has maximal degree of atypicality. This conjecture was proven originally by Serganova using the Duflo-Serganova associated variety.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Algebra and Geometry · Algebraic Geometry and Number Theory