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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
DOI: https://doi.org/10.1090/S1088-4165-99-00085-0
Published electronically: November 9, 1999
MathSciNet review: 1722115
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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: mleites@matematik.su.se (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