The centers of spin symmetric group algebras and Catalan numbers
Jill Tysse
and Weiqiang Wang
University of Virginia Department of Mathematics Charlottesville VA 22904 USA
DOI: 10.1007/s10801-008-0128-1
Abstract
Generalizing the work of Farahat-Higman on symmetric groups, we describe the structures of the even centers Z n \mathcal{Z}_{n} of integral spin symmetric group superalgebras, which lead to universal algebras termed as the spin FH-algebras. A connection between the odd Jucys-Murphy elements and the Catalan numbers is developed and then used to determine the algebra generators of the spin FH-algebras and of the even centers Z n \mathcal{Z}_{n} .
Pages: 175–193
Keywords: keywords spin symmetric groups; jucys-murphy elements; Catalan numbers
Full Text: PDF
References
1. Brundan, J., Kleshchev, A.: Representation theory of symmetric groups and their double covers. In: Groups, Combinatorics & Geometry, Durham, 2001, pp. 31-53. World Scientific, Singapore (2003)
2. Farahat, H., Higman, G.: The centres of symmetric group rings. Proc. Roy. Soc. (A) 250, 212-221 (1959)
3. Goulden, I., Jackson, D.: Combinatorial Enumeration. Series in Discrete Math. Wiley-Interscience, New York (1983)
4. Józefiak, T.: Characters of projective representations of symmetric groups. Expo. Math. 7, 193-247 (1989)
5. Józefiak, T.: Semisimple superalgebras. In: Algebra-Some Current Trends, Varna,
1986. Lect. Notes in Math., vol. 1352, pp. 96-113. Springer, Berlin (1988)
6. Jucys, A.: Symmetric polynomials and the center of the symmetric group rings. Rep. Math. Phys. 5, 107-112 (1974)
7. Kleshchev, A.: Linear and Projective Representations of Symmetric Groups. Cambridge Tracts in Mathematics, vol.
163. Cambridge University Press, Cambridge (2005)
8. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Clarendon Press, Oxford (1995)
9. Murphy, G.: A new construction of Young's seminormal representation of the symmetric group. J. Al- gebra 69, 287-291 (1981)
10. Murray, J.: Generators for the centre of the group algebra of a symmetric group. J. Algebra 271, 725-748 (2004)
11. Nazarov, M.: Young's symmetrizers for projective representations of the symmetric group. Adv. Math. 127, 190-257 (1997)
12. Schur, I.: Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. J. Reine Angew. Math. 139, 155-250 (1911)
13. Sergeev, A.: The Howe duality and the projective representations of symmetric groups. Represent. Theory 3, 416-434 (1999)
14. Wang, W.: The Farahat-Higman ring of wreath products and Hilbert schemes. Adv. Math. 187, 417- 446 (2004)
15. Wang, W.: Universal rings arising in geometry and group theory. In: Cutkosky, S.D., Edidin, D., Qin, Z., Zhang, Q. (eds.) Vector Bundles and Representation Theory. Contemp. Math., vol. 322, pp. 125- 140 (2003)
16. Wang, W.: Double affine Hecke algebras for the spin symmetric group. Preprint (2006).
2. Farahat, H., Higman, G.: The centres of symmetric group rings. Proc. Roy. Soc. (A) 250, 212-221 (1959)
3. Goulden, I., Jackson, D.: Combinatorial Enumeration. Series in Discrete Math. Wiley-Interscience, New York (1983)
4. Józefiak, T.: Characters of projective representations of symmetric groups. Expo. Math. 7, 193-247 (1989)
5. Józefiak, T.: Semisimple superalgebras. In: Algebra-Some Current Trends, Varna,
1986. Lect. Notes in Math., vol. 1352, pp. 96-113. Springer, Berlin (1988)
6. Jucys, A.: Symmetric polynomials and the center of the symmetric group rings. Rep. Math. Phys. 5, 107-112 (1974)
7. Kleshchev, A.: Linear and Projective Representations of Symmetric Groups. Cambridge Tracts in Mathematics, vol.
163. Cambridge University Press, Cambridge (2005)
8. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Clarendon Press, Oxford (1995)
9. Murphy, G.: A new construction of Young's seminormal representation of the symmetric group. J. Al- gebra 69, 287-291 (1981)
10. Murray, J.: Generators for the centre of the group algebra of a symmetric group. J. Algebra 271, 725-748 (2004)
11. Nazarov, M.: Young's symmetrizers for projective representations of the symmetric group. Adv. Math. 127, 190-257 (1997)
12. Schur, I.: Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. J. Reine Angew. Math. 139, 155-250 (1911)
13. Sergeev, A.: The Howe duality and the projective representations of symmetric groups. Represent. Theory 3, 416-434 (1999)
14. Wang, W.: The Farahat-Higman ring of wreath products and Hilbert schemes. Adv. Math. 187, 417- 446 (2004)
15. Wang, W.: Universal rings arising in geometry and group theory. In: Cutkosky, S.D., Edidin, D., Qin, Z., Zhang, Q. (eds.) Vector Bundles and Representation Theory. Contemp. Math., vol. 322, pp. 125- 140 (2003)
16. Wang, W.: Double affine Hecke algebras for the spin symmetric group. Preprint (2006).