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The Howe duality and the projective representations of symmetric groups. (English) Zbl 0999.17014
Summary: 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 \[ \begin{gathered}\tau_{i, j}=-\tau_{j, i}, \quad \tau_{i, j}^{2}=1, \quad \tau_{i, j}\tau_{m, l}=-\tau_{m, l}\tau_{i, j}\text{ if }\{i, j\}\cap\{m, l\}=\emptyset;\\ \tau_{i, j}\tau_{j, m}\tau_{i, j}=\tau_{j, m}\tau_{i, j}\tau_{j, m}=-\tau_{i, m} \text{ for any } i, j, l, m. \end{gathered} \] Recently M. L. Nazarov [Funkts. Anal. Prilozh. 22, No. 1, 77-78 (1988; Zbl 0658.20010), Adv. Math. 127, 190-257 (1997; Zbl 0930.20011)] 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 the author constructs one more analog of Young symmetrizers in \(H_{k}\) as well as the analogs of Specht modules for \({\mathfrak{A}}_{k}\) and \(H_{k}\).

MSC:
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
20C25 Projective representations and multipliers
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