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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The symmetric group possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of itself, coincide with the irreducible representations of the algebra generated by indeterminates for , subject to the relations

Recently M. Nazarov realized irreducible representations of and Young symmetrizers by means of the Howe duality between the Lie superalgebra and the Hecke algebra , the semidirect product of with the Clifford algebra on indeterminates.

Here I construct one more analog of Young symmetrizers in as well as the analogs of Specht modules for and .

**[BL]**Bernstein J. and Leites D., The superalgebra , 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 -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 -modules (Russian), Funktsional. Anal. i Prilozhen. 20, no. 1, 1986, 37-45. MR**87j:17033****[P]**Pragacz P., Algebro-geometric applications of Schur - and - 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 and , 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 , 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:
mleites@matematik.su.se (subject: for Sergeev)

DOI:
https://doi.org/10.1090/S1088-4165-99-00085-0

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