The Howe Duality and the Projective Representations of Symmetric Groups

The symmetric group Sk possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of Sk itself, coincide with the irreducible representations of the algebra Ak generated by indeterminates τi,j for i 6= j, 1 ≤ i, j ≤ n subject to the relations τi,j = −τj,i, τ i,j = 1, τi,jτm,l = −τm,lτi,j if {i, j} ∩ {m, l} = ∅; τi,jτj,mτi,j = τj,mτi,jτj,m = −τi,m for any i, j, l, m. Recently M. Nazarov realized irreducible representations of Ak and Young symmetrizers by means of the Howe duality between the Lie superalgebra q(n) and the Hecke algebra Hk = Sk ◦Clk, the semidirect product of Sk with the Clifford algebra Clk on k indeterminates. Here I construct one more analog of Young symmetrizers in Hk as well as the analogs of Specht modules for Ak and Hk. 1. Summary Lately, we have witnessed an increase of interest in the study of representations of symmetric groups. In particular, in their projective representations. Recall that the symmetric group Sk has a nontrivial central extension whose irreducible representations do not reduce to those of Sk but coincide (may be identified) with the irreducible representations of the algebra Ak determined by generators τi,j for i 6= j, 1 ≤ i, j ≤ n subject to the relations τi,j = −τj,i, τ i,j = 1, τi,jτm,l = −τm,lτi,j if {i, j} ∩ {m, l} = ∅; τi,jτj,mτi,j = τj,mτi,jτj,m = −τi,m for any i, j, l, m. (1.1) In [N1] Nazarov realized irreducible representations of Ak by means of an orthogonal basis constructed in each of the spaces of the representations and indicating the action of the generators τi,i+1 on them (an analog of the Young orthogonal form). In [N2], with the help of an “odd” analog of the degenerate affine Hecke algebra Nazarov constructed elements of the algebra Hk = Sk ◦Clk, the semidirect product of Sk with the Clifford algebra Clk on k indeterminates. Certain elements of Hk serve as analogs of Young symmetrizers. Received by the editors September 4, 1998 and, in revised form, September 8, 1999. 1991 Mathematics Subject Classification. Primary 20C30, 20C25, 17A70.