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Representation Theory

ISSN 1088-4165



The Howe duality
and the projective representations
of symmetric groups

Author: Alexander Sergeev
Journal: Represent. Theory 3 (1999), 416-434
MSC (1991): Primary 20C30, 20C25, 17A70
Published electronically: November 9, 1999
MathSciNet review: 1722115
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Abstract | References | Similar Articles | Additional Information

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 $H_{k}$.

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Additional Information

Alexander Sergeev
Affiliation: On leave of absence from Balakovo Institute of Technique of Technology and Control, Branch of Saratov State Technical University, Russia; Department of Mathematics, University of Stockholm, Roslagsv. 101, Kräftriket hus 6, S-106 91, Stockholm, Sweden
Email: (subject: for Sergeev)

Keywords: Projective representations, symmetric group, Howe duality
Received by editor(s): September 4, 1998
Received by editor(s) in revised form: September 8, 1999
Published electronically: November 9, 1999
Additional Notes: I am thankful to D. Leites for support; to him and the referee for help
Article copyright: © Copyright 1999 American Mathematical Society