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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 $H_{k}$.

References [Enhancements On Off] (What's this?)

  • [BL] Bernstein J. and Leites D., The superalgebra $Q(n)$, the odd trace and the odd determinant. C.R. Acad. Bulgare Sci. v. 35, no.3, 1982, 285-286. MR 84c:17003
  • [Ja] James G., The representation theory of the symmetric groups, Lect. Notes Math. 682, 1978. MR 80g:20019
  • [Jo1] Jones A., The structure of the Young's symmetrizers for spin representations of the symmetric group. I., J. Algebra, 205, 1998, 626-660. MR 99j:20017
  • [Jo2] Jones A., The structure of the Young's symmetrizers for spin representations of the symmetric group. II., J. Algebra, 213, 1999, 381-404. CMP 99:10
  • [JN] Jones A. and Nazarov M., Affine Sergeev algebra and $q$-analogs of the Young's symmetrizers for projective representations of the symmetric group, Proc. London Math. Soc., 78, 1999, 481-512. CMP 99:09
  • [Ju1] Jucys A., Symmetric polynomials and the center of the symmetric group ring, Report Math. Phys. 5, 1974, 107-112. MR 54:7597
  • [Ju2] Jucys A., Factorization of Young's projection operators for symmetric groups, Litovsk. Fiz. Sb. 5, 1971, 1-10. MR 44:7851
  • [L] Leites D., Lie superalgebras. In: Modern Problems of Mathematics. Recent developments, v. 25, VINITI, Moscow, 1984, 3-49 (in Russian; the English translation: JOSMAR, v. 30 (6), 1985, 2481-2512). MR 86f:17019
  • [Ma] Macdonald I., Symmetric functions and Hall polynomials, Oxford Univ. Press, 1995. MR 96h:05207
  • [Mu] Murphy G., A new construction of the Young seminormal representations of the symmetric group, J. Algebra 69, 1981, 287-291. MR 82h:20014
  • [N1] Nazarov M., An orthogonal basis in the irreducible projective representations of the symmetric group, Funct. Anal. Appl., 22, no. 2, 1988, 66-68. MR 89f:20019
  • [N2] Nazarov M., Young's symmetrizers for projective representations of the symmetric group, Adv. Math., 127, no. 2, 1997, 190-257. MR 98m:20019
  • [OV] Okunkov A. and Vershik A., A new approach to representation theory of symmetric groups. Selecta Math., new series, 2, no. 4, 1996, 581-605. MR 99g:20024
  • [Pe] Penkov I., Characters of typical irreducible representations of finite dimensional ${\mathfrak{q}}(n)$-modules (Russian), Funktsional. Anal. i Prilozhen. 20, no. 1, 1986, 37-45. MR 87j:17033
  • [P] Pragacz P., Algebro-geometric applications of Schur $S$- and $Q$- polynomials, Lect. Notes Math. 1478, 1991, 130-191. MR 93h:05170
  • [R] Ram A., Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3), 75, 1997, 99-133. MR 98d:20007
  • [Sch] Schepochkina I., Maximal subalgebras of matrix Lie superalgebras, hep-th/9702122.
  • [S1] Sergeev A., The tensor algebra of the standard representation as a module over the Lie superalgebras $GL(n, m)$ and $Q(n)$, Math. Sbornik, 123 (165), no.3, 1984, 422-430. MR 85h:17010
  • [S2] Sergeev A., The centre of the enveloping algebra for the Lie superalgebra $Q(n,\mathbf C)$, Lett. Math. Phys., 7, 1983, 177-179. MR 85i:17004
  • [S3] Sergeev A., Irreducible representations of solvable Lie superalgebras, math.RT/9810109.
  • [W] Weyl H., Classical groups, their invariants and representations, Princeton Univ. Press, Princeton, 1939. MR 1:42c
  • [Ya1] Yamaguchi M., A duality of the twisted group algebra of the symmetric group and a Lie superalgebra, math.RT/9811090.
  • [Ya2] Yamaguchi M., A duality of the twisted group algebra of the hyperoctaedral group and the queer Lie superalgebra, math.RT/9903159.

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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

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