Hilbert series of typical representations for Lie superalgebras

TL;DR

This paper derives explicit formulas for the Hilbert series of typical representations of basic classical Lie superalgebras, revealing their structure through elementary symmetric and Eulerian polynomials.

Contribution

It provides a novel closed-form expression for the Hilbert series of typical representations, using elementary symmetric polynomials, Eulerian polynomials, and differential operators.

Findings

Explicit Hilbert series formulas for typical representations

Connection to elementary symmetric and Eulerian polynomials

Simplified closed-form expressions using differential operators

Abstract

Let g be a basic classical Lie superalgebra over C. In the case of a typical weight whose every nonnegative integer multiple is also typical, we compute a closed form for the Hilbert series whose coefficients encode the dimensions of finite-dimensional irreducible typical g-representations. We give a formula for this Hilbert series in terms of elementary symmetric polynomials and Eulerian polynomials. Additionally, we show a simple closed form in terms of differential operators.