Invariants of the vacuum module associated with the Lie superalgebra gl(1|1)

TL;DR

This paper characterizes the invariants of the vacuum module for the affine Lie superalgebra gl(1|1), providing a formula for its Hilbert--Poincaré series and connecting it to plane partitions and supersymmetric polynomials.

Contribution

It introduces a super version of a classical theorem, explicitly constructs invariants, and links algebraic invariants to combinatorial objects and supersymmetric polynomials.

Findings

Hilbert--Poincaré series matches plane partition generating function

Explicit basis of invariants constructed

Invariants identified with affine supersymmetric polynomials

Abstract

We describe the algebra of invariants of the vacuum module associated with the affinization of the Lie superalgebra $\mathfrak{gl}(1|1)$. We give a formula for its Hilbert--Poincar\'{e} series in a fermionic (cancellation-free) form which turns out to coincide with the generating function of the plane partitions over the $(1,1)$-hook. Our arguments are based on a super version of the Beilinson--Drinfeld--Ra\"{i}s--Tauvel theorem which we prove by producing an explicit basis of invariants of the symmetric algebra of polynomial currents associated with $\mathfrak{gl}(1|1)$. We identify the invariants with affine supersymmetric polynomials via a version of the Chevalley theorem.