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Here we consider two algebras, a free unital associative complex algebra(denoted by ${\mathcal{B}}$) equiped with a multiparametric\textbf{\emph{q}}-differential structure and a twisted group algebra (denotedby ${\mathcal{A}(S_{n})}$), with the motivation to represent the algebra${\mathcal{A}(S_{n})}$ on the (generic) weight subpaces of the algebra${\mathcal{B}}$. One of the fundamental problems in ${\mathcal{B}}$ is todescribe the space of all constants (the elements which are annihilated by allmultiparametric partial derivatives). To solve this problem, one needs somespecial matrices and their factorizations in terms of simpler matrices. Asimpler approach is to study first certain canonical elements in the twistedgroup algebra ${\mathcal{A}(S_{n})}$. Then one can use certain naturalrepresentation of ${\mathcal{A}(S_{n})}$ on the weight subspaces of${\mathcal{B}}$, which are the subject of this paper.

A representation of twisted group algebra of symmetric groups on weight subspaces of free associative complex algebra