The Howe duality and the projective representations of symmetric groups

The symmetric group $\mathfrak S_{k}$ possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of $\mathfrak S_{k}$ itself, coincide with the irreducible representations of the algebra $\mathfrak A_{k}$ generated by indeterminates $\tau _{i, j}$ for $i\neq j$, $1\leq i, j\leq n$ subject to the relations

Recently M. Nazarov realized irreducible representations of ${\mathfrak{A}}_{k}$ and Young symmetrizers by means of the Howe duality between the Lie superalgebra ${\mathfrak{q}}(n)$ and the Hecke algebra $H_{k}={\mathfrak{S}}_{k}\circ Cl_{k}$, the semidirect product of ${\mathfrak{S}}_{k}$ with the Clifford algebra $Cl_{k}$ on $k$ indeterminates.

Here I construct one more analog of Young symmetrizers in $H_{k}$ as well as the analogs of Specht modules for ${\mathfrak{A}}_{k}$ and $H_{k}$.