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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
##### Citations:
Zbl 0658.20010; Zbl 0930.20011
Full Text:
##### References:
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