A New Plethysm Formula for Symmetric Functions
William F.IV Doran
California Institute of Technology Department of Mathematics Pasadena CA 91125
DOI: 10.1023/A:1008662732304
Abstract
This paper gives a new formula for the plethysm of power-sum symmetric functions and Schur symmetric functions with one part. The form of the main result is that for p m ( x) ^\circ s a ( x) = å T w maj m ( T) s sh( T) ( x) p_μ(\underline x) \circ s_a (\underline x) = \sum\limits_T {\underline ω^{maj_μ(T)} s_{sh(T)} (\underline x)}
Pages: 253–272
Keywords: symmetric function; plethysm; eigenvalue; representation of the symmetric group
Full Text: PDF
References
1. L. Butler, “Subgroup lattices and symmetric functions,” Memoirs of the Amer. Math. Soc. 539 (1994), Amer. Math. Soc., Providence, RI.
2. Y. Chen, A. Garsia, and J. Remmel, “Algorithms for plethysm,” Contemporary Mathematics 34 (1984), 109-153.
3. D. Gay, “Characters of the Weyl group of SU(n) on zero weight spaces and centralizers of permutation representations,” Rocky Moun. J. Math. 6 (1976), 449-455.
4. G. James, The Representation Theory of the Symmetric Group, Lecture Notes in Mathematics 682, Springer- Verlag, Berlin/Heidelberg/New York, 1978.
5. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics, Vol. 16, Addison-Wesley, Reading, MA, 1981.
6. A. Lascoux, B. Leclerc, and J.-Y. Thibon, “Green polynomials and Hall-Littlewood functions at roots of unity,” European J. Combinatorics 15 (1994), 173-180.
7. A. Lascoux, B. Leclerc, and J.-Y. Thibon, “Crystal graphs and q-analogues of weight multiplicities for the root system An,” Letters Math. Phys. 35 (1995), 359-374. P1: VBI Journal of Algebraic Combinatorics KL629-03-Doran August 20, 1998 14:9 272 DORAN
8. A. Lascoux and M. Sch\ddot utzenberger, “Sur une conjecture de H.O. Foulkes,” C.R. Acad. Sci. Paris 286A (1978), 323-324.
9. I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Clarendon Press, Oxford, 1995.
10. B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1991.
11. J. Stembridge, “On the eigenvalues of representations of reflection groups and wreath products,” Pac. J. Math. 140 (1989), 353-395.
2. Y. Chen, A. Garsia, and J. Remmel, “Algorithms for plethysm,” Contemporary Mathematics 34 (1984), 109-153.
3. D. Gay, “Characters of the Weyl group of SU(n) on zero weight spaces and centralizers of permutation representations,” Rocky Moun. J. Math. 6 (1976), 449-455.
4. G. James, The Representation Theory of the Symmetric Group, Lecture Notes in Mathematics 682, Springer- Verlag, Berlin/Heidelberg/New York, 1978.
5. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics, Vol. 16, Addison-Wesley, Reading, MA, 1981.
6. A. Lascoux, B. Leclerc, and J.-Y. Thibon, “Green polynomials and Hall-Littlewood functions at roots of unity,” European J. Combinatorics 15 (1994), 173-180.
7. A. Lascoux, B. Leclerc, and J.-Y. Thibon, “Crystal graphs and q-analogues of weight multiplicities for the root system An,” Letters Math. Phys. 35 (1995), 359-374. P1: VBI Journal of Algebraic Combinatorics KL629-03-Doran August 20, 1998 14:9 272 DORAN
8. A. Lascoux and M. Sch\ddot utzenberger, “Sur une conjecture de H.O. Foulkes,” C.R. Acad. Sci. Paris 286A (1978), 323-324.
9. I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Clarendon Press, Oxford, 1995.
10. B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1991.
11. J. Stembridge, “On the eigenvalues of representations of reflection groups and wreath products,” Pac. J. Math. 140 (1989), 353-395.