Remote Access Representation Theory
Green Open Access

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
Full-text PDF Free Access

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?)

  • 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
  • G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
  • 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, DOI
  • Jones A., The structure of the Young’s symmetrizers for spin representations of the symmetric group. II., J. Algebra, 213, 1999, 381–404.
  • 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.
  • 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, DOI
  • 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
  • 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
  • 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
  • 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, DOI
  • 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, DOI
  • Maxim Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), no. 2, 190–257. MR 1448714, DOI
  • 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, DOI
  • I. B. Penkov, Characters of typical irreducible finite-dimensional ${\mathfrak q}(n)$-modules, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 37–45, 96 (Russian). MR 831047
  • 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, DOI
  • Arun Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), no. 1, 99–133. MR 1444315, DOI
  • Schepochkina I., Maximal subalgebras of matrix Lie superalgebras, hep-th/9702122.
  • A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras ${\rm Gl}(n,\,m)$ and $Q(n)$, Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422–430 (Russian). MR 735715
  • A. N. Sergeev, The centre of enveloping algebra for Lie superalgebra $Q(n,\,{\bf C})$, Lett. Math. Phys. 7 (1983), no. 3, 177–179. MR 706205, DOI
  • Sergeev A., Irreducible representations of solvable Lie superalgebras, math.RT/9810109.
  • Weyl H., Classical groups, their invariants and representations, Princeton Univ. Press, Princeton, 1939.
  • Yamaguchi M., A duality of the twisted group algebra of the symmetric group and a Lie superalgebra, math.RT/9811090.
  • Yamaguchi M., A duality of the twisted group algebra of the hyperoctaedral group and the queer Lie superalgebra, math.RT/9903159.

Similar Articles

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: (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