  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.
