Commutant algebra of superderivations on a Grassmann algebra (Q1917598)

The authors consider the representation of the Cartan-type Lie superalgebra \(W(n)\) on the \(m\)-fold tensor product of the Grassman algebra \(\wedge (n)\). In the case \(m \leq n\) the commutant of the algebra of representation operators in \(\text{End} (\otimes^m \wedge (n))\) is proved to be equal to the following algebra: Consider the semigroup of mappings of \(\{1,2,\dots,m\}\) into itself, and its natural action on the tensor product \(\otimes^m \wedge (n)\). The image of the semigroup ring under this representation is just the commutant algebra in question. The authors conjecture, that the theorem holds true also in the general case. There is a proof for arbitrary \(m\) if \(n = 1\), and in the special case \(m = 3\) and \(n = 2\).