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 Free Access

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]**J. N. Bernstein and D. A. Leĭtes,*The superalgebra 𝑄(𝑛), the odd trace and the odd determinant*, C. R. Acad. Bulgare Sci.**35**(1982), no. 3, 285–286. MR**677839****[Ja]**G. D. James,*The representation theory of the symmetric groups*, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR**513828****[Jo1]**Andrew R. Jones,*The structure of the Young symmetrizers for spin representations of the symmetric group. I*, J. Algebra**205**(1998), no. 2, 626–660. MR**1632785**, https://doi.org/10.1006/jabr.1997.7400**[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]**A.-A. A. Jucys,*Symmetric polynomials and the center of the symmetric group ring*, Rep. Mathematical Phys.**5**(1974), no. 1, 107–112. MR**419576**, https://doi.org/10.1016/0034-4877(74)90019-6**[Ju2]**A. Jucis,*Factorization of Young’s projection operators for symmetric groups*, Litovsk. Fiz. Sb.**11**(1971), 1–10 (Russian, with English and Lithuanian summaries). MR**290671****[L]**D. A. Leĭtes,*Lie superalgebras*, Current problems in mathematics, Vol. 25, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 3–49 (Russian). MR**770940****[Ma]**I. G. Macdonald,*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144****[Mu]**G. E. Murphy,*A new construction of Young’s seminormal representation of the symmetric groups*, J. Algebra**69**(1981), no. 2, 287–297. MR**617079**, https://doi.org/10.1016/0021-8693(81)90205-2**[N1]**M. L. Nazarov,*An orthogonal basis in irreducible projective representations of the symmetric group*, Funktsional. Anal. i Prilozhen.**22**(1988), no. 1, 77–78 (Russian); English transl., Funct. Anal. Appl.**22**(1988), no. 1, 66–68. MR**936708**, https://doi.org/10.1007/BF01077731**[N2]**Maxim Nazarov,*Young’s symmetrizers for projective representations of the symmetric group*, Adv. Math.**127**(1997), no. 2, 190–257. MR**1448714**, https://doi.org/10.1006/aima.1997.1621**[OV]**Andrei Okounkov and Anatoly Vershik,*A new approach to representation theory of symmetric groups*, Selecta Math. (N.S.)**2**(1996), no. 4, 581–605. MR**1443185**, https://doi.org/10.1007/PL00001384**[Pe]**I. B. Penkov,*Characters of typical irreducible finite-dimensional 𝔮(𝔫)-modules*, Funktsional. Anal. i Prilozhen.**20**(1986), no. 1, 37–45, 96 (Russian). MR**831047****[P]**Piotr Pragacz,*Algebro-geometric applications of Schur 𝑆- and 𝑄-polynomials*, Topics in invariant theory (Paris, 1989/1990) Lecture Notes in Math., vol. 1478, Springer, Berlin, 1991, pp. 130–191. MR**1180989**, https://doi.org/10.1007/BFb0083503**[R]**Arun Ram,*Seminormal representations of Weyl groups and Iwahori-Hecke algebras*, Proc. London Math. Soc. (3)**75**(1997), no. 1, 99–133. MR**1444315**, https://doi.org/10.1112/S0024611597000282**[Sch]**Schepochkina I., Maximal subalgebras of matrix Lie superalgebras, hep-th/9702122.**[S1]**A. N. Sergeev,*Tensor algebra of the identity representation as a module over the Lie superalgebras 𝐺𝑙(𝑛,𝑚) and 𝑄(𝑛)*, Mat. Sb. (N.S.)**123(165)**(1984), no. 3, 422–430 (Russian). MR**735715****[S2]**A. N. Sergeev,*The centre of enveloping algebra for Lie superalgebra 𝑄(𝑛,𝐶)*, Lett. Math. Phys.**7**(1983), no. 3, 177–179. MR**706205**, https://doi.org/10.1007/BF00400431**[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.

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (1991):
20C30,
20C25,
17A70

Retrieve articles in all journals with MSC (1991): 20C30, 20C25, 17A70

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