A Determinantal Formula for Supersymmetric Schur Polynomials
E.M. Moens
and J. Van der Jeugt
University of Ghent Department of Applied Mathematics and Computer Science Krijgslaan 281-S9 B-9000 Gent Belgium
DOI: 10.1023/A:1025048821756
Abstract
We derive a new formula for the supersymmetric Schur polynomial s 
( x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl( m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression for s ( x/y). This new expression gives rise to a determinantal formula for s ( x/y). In particular, the denominator identity for gl( m/n) corresponds to a determinantal identity combining Cauchy's double alternant with Vandermonde's determinant. We provide a second and independent proof of the new determinantal formula by showing that it satisfies the four characteristic properties of supersymmetric Schur polynomials. A third and more direct proof ties up our formula with that of Sergeev-Pragacz.
( x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl( m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression for s ( x/y). This new expression gives rise to a determinantal formula for s ( x/y). In particular, the denominator identity for gl( m/n) corresponds to a determinantal identity combining Cauchy's double alternant with Vandermonde's determinant. We provide a second and independent proof of the new determinantal formula by showing that it satisfies the four characteristic properties of supersymmetric Schur polynomials. A third and more direct proof ties up our formula with that of Sergeev-Pragacz.Pages: 283–307
Keywords: supersymmetric Schur polynomials; Lie superalgebra $gl( m/n)$; characters; covariant tensor representations; determinantal identities
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References
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13. P. Pragacz and A. Thorup, “On a Jacobi-Trudi identity for supersymmetric polynomials,” Advances in Mathematics 95 (1992), 8-17.
14. M. Scheunert, The Theory of Lie Superalgebras, Springer, Berlin, 1979.
15. A.N. Sergeev, “Tensor algebra of the identity representation as a module over the Lie superalgebras Gl(n, m) and Q(n) (Russian),” Matematicheski\?ı Sbornik 123 (1984), 422-430.
16. G.E. Shilov, Linear Algebra, Dover Publications, New York, 1977.
17. J. Stembridge, “A characterization of supersymmetric polynomials,” Journal of Algebra 95 (1985), 439-444.
18. J. Van der Jeugt and V. Fack, “The Pragacz identity and a new algorithm for Littlewood-Richardson coefficients,” Computers and Mathematics with Applications 21 (1991), 39-47.
19. J. Van der Jeugt, J.W.B. Hughes, R.C. King, and J. Thierry-Mieg, “Character formulas for irreducible modules of the Lie superalgebra sl(m/n),” Journal of Mathematical Physics 31 (1990), 2278-2304.
20. J. Van der Jeugt, J.W.B. Hughes, R.C. King, and J. Thierry-Mieg, “A character formula for singly atypical modules of the Lie superalgebra sl(m/n),” Communications in Algebra 18 (1990), 3453-3480.
21. J. Van der Jeugt and R.B. Zhang, “Characters and composition factor multiplicities for the Lie superalgebra gl(m/n),” Letters in Mathematical Physics 47 (1999), 49-61.
22. Private communication: This was outlined to us by Alain Lascoux as an advisor of FPSAC2002, where the results of this paper were presented as a talk.
2. N. Bergeron and A. Garsia, “Sergeev's formula and the Littlewood-Richardson rule,” Linear and Multilinear Algebra 27 (1990), 79-100.
3. A. Berele and A. Regev, “Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras,” Advances in Mathematics 64 (1987), 118-175.
4. V.G. Kac, “Lie superalgebras,” Advances in Mathematics 26 (1977), 8-96.
5. V.G. Kac, “Representations of classical Lie superalgebras,” Lecture Notes in Mathematics, Springer, Berlin, 1978, Vol. 676, pp. 597-626.
6. V.G. Kac and M. Wakimoto, “Integrable highest weight modules over affine superalgebras and number theory,” Progress in Mathematics 123 (1994), 415-456.
7. R.C. King, “Supersymmetric functions and the Lie supergroup U (m/n),” Ars Combinatoria 16A (1983), 269-287.
8. C. Krattenthaler, “Advanced determinant calculus,” Séminaire Lotharingien Combinatoire 42 (1999), Article B42q, 67 pp.
9. A. Lascoux and P. Pragacz, “Ribbon Schur functions,” European Journal of Combinatorics 9 (1988), 561-574.
10. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 2nd edition, 1995.
11. I. Penkov and V. Serganova, “On irreducible representations of classical Lie superalgebras,” Indagationes Mathematica 3 (1992), 419-466.
12. P. Pragacz, “Algebro-geometric applications of Schur S- and Q-polynomials, Topics in invariant theory (Paris, 1989/1990),” Lecture Notes in Mathematics, Springer, Berlin, 1991, Vol. 1478, pp. 130-191.
13. P. Pragacz and A. Thorup, “On a Jacobi-Trudi identity for supersymmetric polynomials,” Advances in Mathematics 95 (1992), 8-17.
14. M. Scheunert, The Theory of Lie Superalgebras, Springer, Berlin, 1979.
15. A.N. Sergeev, “Tensor algebra of the identity representation as a module over the Lie superalgebras Gl(n, m) and Q(n) (Russian),” Matematicheski\?ı Sbornik 123 (1984), 422-430.
16. G.E. Shilov, Linear Algebra, Dover Publications, New York, 1977.
17. J. Stembridge, “A characterization of supersymmetric polynomials,” Journal of Algebra 95 (1985), 439-444.
18. J. Van der Jeugt and V. Fack, “The Pragacz identity and a new algorithm for Littlewood-Richardson coefficients,” Computers and Mathematics with Applications 21 (1991), 39-47.
19. J. Van der Jeugt, J.W.B. Hughes, R.C. King, and J. Thierry-Mieg, “Character formulas for irreducible modules of the Lie superalgebra sl(m/n),” Journal of Mathematical Physics 31 (1990), 2278-2304.
20. J. Van der Jeugt, J.W.B. Hughes, R.C. King, and J. Thierry-Mieg, “A character formula for singly atypical modules of the Lie superalgebra sl(m/n),” Communications in Algebra 18 (1990), 3453-3480.
21. J. Van der Jeugt and R.B. Zhang, “Characters and composition factor multiplicities for the Lie superalgebra gl(m/n),” Letters in Mathematical Physics 47 (1999), 49-61.
22. Private communication: This was outlined to us by Alain Lascoux as an advisor of FPSAC2002, where the results of this paper were presented as a talk.