The Howe duality and the projective representations of symmetric groups

Abstract

The symmetric group S k \mathfrak {S}_{k} possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of S k \mathfrak {S}_{k} itself, coincide with the irreducible representations of the algebra A k \mathfrak {A}_{k} generated by indeterminates τ i , j \tau _{i, j} for i ≠ j i\neq j , 1 ≤ i , j ≤ n 1\leq i, j\leq n subject to the relations τ i , j = − τ j , i , τ i , j 2 = 1 , τ i , j τ m , l = − τ m , l τ i , j if { i , j } ∩ { m , l } = ∅ ; τ i , j τ j , m τ i , j = τ j , m τ i , j τ j , m = − τ i , m for any i , j , l , m . \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 A k \mathfrak {A}_{k} and Young symmetrizers by means of the Howe duality between the Lie superalgebra q ( n ) \mathfrak {q}(n) and the Hecke algebra H k = S k ∘ C l k H_{k}=\mathfrak {S}_{k}\circ Cl_{k} , the semidirect product of S k \mathfrak {S}_{k} with the Clifford algebra C l k Cl_{k} on k k indeterminates. Here I construct one more analog of Young symmetrizers in H k H_{k} as well as the analogs of Specht modules for A k \mathfrak {A}_{k} and H k H_{k} .