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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 the affinization of the Lie superalgebra $\mathfrak{gl}(1|1)$. 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--Raïs--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.