The Howe duality and the projective representations of symmetric groups

Abstract:

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 \begin{gather*} \tau _{i, j}=-\tau _{j, i}, \quad \tau _{i, j}^{2}=1, \quad \tau _{i, j}\tau _{m, l}=-\tau _{m, l}\tau _{i, j}\text { if }\{i, j\}\cap \{m, l\}=\emptyset ;\ \tau _{i, j}\tau _{j, m}\tau _{i, j}=\tau _{j, m}\tau _{i, j}\tau _{j, m}=-\tau _{i, m}\; \; \text { for any } i, j, l, m. \end{gather*} 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 $H_{k}$.