A formula of Lascoux-Leclerc-Thibon and representations of symmetric groups
Hideaki Morita1
and Tatsuhiro Nakajima2
1Tokai University School of Science Hiratsuka 259-1292 Japan
2Meikai University Faculty of Economics Urayasu 279-8550 Japan
2Meikai University Faculty of Economics Urayasu 279-8550 Japan
DOI: 10.1007/s10801-006-9101-z
Abstract
We consider Green polynomials at roots of unity, corresponding to partitions which we call l-partitions. We obtain a combinatorial formula for Green polynomials corresponding to l-partitions at primitive lth roots of unity. The formula is rephrased in terms of representation theory of the symmetric group.
Pages: 45–60
Full Text: PDF
References
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2005. Springer J Algebr Comb (2006) 24:45-60
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17. T. Shoji, “A variant of the induction theorem for Springer representations,” preprint 2005.
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2005. Springer J Algebr Comb (2006) 24:45-60
2. J. Désarménien, B. Leclerc, and J.-Y. Thibon, “Hall-Littlewood Functions and Kostka-Foulkes Polynomials in Representation Theory,” Seminaire Lotharingien de Combinatoire 32, 1994.
3. C. DeConcini and C. Procesi, “Symmetric functions, conjugacy classes, and the flag variety,” Inv. Math. 64 (1981), 203-230.
4. A. M. Garsia and C. Procesi, “On certain graded Sn-modules and the q-Kostka polynomials,” Adv. Math. 94 (1992), 82-138.
5. J. A. Green, “The character of the finite general linear groups,” Trans. Amer. Math. Soc., 80 (1955), 402-447.
6. J. E. Humphreys, “Reflection Groups and Coxeter Groups,” Cambridge studies in advances mathematics 29, Cambridge University Press, 1990.
7. W. Kraśkiewicz and J. Weyman, “Algebra of coinvariants and the action of Coxeter elements,” Bayreuth. Math. Schr. 63 (2001), 265-284.
8. A. Lascoux, B. Leclerc, and Y. -Y. Thibon, “Fonctions de Hall-Littlewood et polyn\hat omes de Kostka- Foulkes aux racines de l'unité,” C. R. Acad. Sci. Paris, 316, Ser. I (1993), 1-6.
9. A. Lascoux, B. Leclerc, and J. -Y. Thibon, “Green polynomials and Hall-Littlewood functions at roots of unity,” Euro. J. Comb. 15 (1994), 173-180.
10. G. Lusztig, “Green polynomials and singularities of unipotent classes,” Adv. Math. 42 (1981), 169-178.
11. I. G. Macdonald, “Symmetric Functions and Hall Polynomials,” 2nd ed., Oxford University Press, 1995.
12. H. Morita, “Decomposition of Green polynomial of type A and DeConcini-Procesi-Tanisaki algebras of certain types,” submitted.
13. H. Morita, “The Green polynomials at roots of unity and its application,” submitted.
14. H. Morita and T. Nakajima, “The coinvariant algebra of the symmetric group as a direct sum of induced modules,” Osaka J. Math. 42 (2005), 217-231.
15. A. O. Morris and N. Sultana, “Hall-Littlewood functions at roots of 1 and modular representations of the symmetric group,” Math. Proc. Camb. Phil. Soc., 110 (1991), 443-453.
16. V. Reiner, D. Stanton, and P. Webb, “Springer's regular elements over arbitrary fields,” preprint, 2004.
17. T. Shoji, “A variant of the induction theorem for Springer representations,” preprint 2005.
18. T. A. Springer, “Regular elements of finite reflection groups,” Invent. Math. 25 (1974), 159-198.
19. T. A. Springer, “Trigonometric sums, Green functions of finite groups and representations of Weyl groups,” Inv. math. 36, (1976), 173-207.
20. T. Tanisaki, “Defining ideals of the closures of conjugacy classes and representations of the Weyl groups,” Tohoku J. Math. 34 (1982), 575-585.