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

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–Poincaré 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–Räıs–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. School of Mathematics and Statistics University of Sydney, NSW 2006, Australia [email protected] Department of Mathematical Sciences Indiana University – Purdue University Indianapolis 402 North Blackford St, Indianapolis, IN 46202-3216, USA [email protected] 1 This is the author's manuscript of the article published in final edited form as: Molev, A. I., & Mukhin, E. E. (2015). Invariants of the vacuum module associated with the Lie superalgebra gl(1|1). Journal of Physics A: Mathematical and Theoretical, 48(31), 314001. http://doi.org/10.1088/1751-8113/48/31/314001 Dedicated to Rodney Baxter on his 75th birthday