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

Abstract

We describe the algebra of invariants of the vacuum module associated with an affinization of the Lie superalgebra gl(1|1). We give a formula for its Hilbert–Poincare´ 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–Raı¨s–Tauvel theorem which we prove by producing an explicit basis of invariants of the symmetric algebra of polynomial currents associated with gl(1|1). We identify the invariants with affine supersymmetric polynomials via a version of the Chevalley theorem.