Kostka-Foulkes Polynomials Cyclage Graphs and Charge Statistic for the Root System C n
Cédric Lecouvey
DOI: 10.1007/s10801-005-6909-x
Abstract
We establish a Morris type recurrence formula for the root system C n. Next we introduce cyclage graphs for the corresponding Kashiwara-Nakashima 
s tableaux and use them to define a charge statistic. Finally we conjecture that this charge may be used to compute the Kostka-Foulkes polynomials for type C n.
s tableaux and use them to define a charge statistic. Finally we conjecture that this charge may be used to compute the Kostka-Foulkes polynomials for type C n.Pages: 203–240
Keywords: keywords crystal graphs; cyclage graphs; Kostka-foulkes polynomials
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References
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2. P. Caldero, “On harmonic elements for semi-simple Lie algebras,” Advances in Mathematics 166, (2002), 73-99.
3. V. Chari and A. Presley, A Guide to Quantum Groups, Cambridge University Press, 1994.
4. J. Hong and S.J. Kang, Introduction to Quantum Groups and Crystals Bases, A.M.S 2002, GSM/12.
5. J.C. Jantzen, “Lectures on quantum groups,” Graduate Studies in Math. 6 (A.M.S 1995).
6. A. Joseph, G. Letzer, and S. Zelikson, “On the Brylinski-Kostant filtration,” J. Amer. Math. Soc. 13(4) (2000), 945-970. LECOUVEY
7. M. Kashiwara, “Crystallizing the q-analogue of universal enveloping algebra,” Commun. Math. Phys. 133 (1990), 249-260.
8. M. Kashiwara, “On crystal bases of the q-analogue of universal enveloping algebras,” Duke Math. J., 63 (1991), 465-516.
9. M. Kashiwara, “Crystallization of quantized universal enveloping algebras,” Sugaku Expositiones 7 (1994), 99-115
10. M. Kashiwara, “On crystal bases,” Canadian Mathematical Society, Conference Proceedings 16 (1995) 155- 197.
11. M. Kashiwara and T. Nakashima, “Crystal graphs for representations of the q-analogue of classical Lie algebras,” Journal of Algebra 165 (1994), 295-345.
12. A. Lascoux, B. Leclerc, and J-Y. Thibon, “Crystal graphs and q-analogue of weight multiplicities for the root system An,” Letters in Mathematical Physics 35 (1995), 359-374.
13. A. Lascoux and M.-P. Sch\ddot utzenberger, “Le mono\ddot ide plaxique,” in A. de Luca (Ed.), Non Commutative Structures in Algebra and Geometric Combinatorics, Quaderni della Ricerca Scientifica del C.N.R., Roma, 1981.
14. A. Lascoux and M.-P. Sch\ddot utzenberger, “Sur une conjecture de H.O Foulkes,” CR Acad Sci Paris 288 (1979), 95-98.
15. C. Lecouvey, “Schensted-type correspondence, Plactic Monoid and Jeu de Taquin for type Cn,” J. Algebra 247 (2002), 295-331.
16. M. Lothaire, “Encyclopedia of mathematics and its applications,” Algebraic Combinatorics of Words, 90, 164-196.
17. G. Lusztig, “Singularities, character formulas, and a q-analog of weight multiplicities, Analyse et topologie sur les espaces singuliers (II-III),” Asterisque 101/102 (1983), 208-227.
18. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Mathematical Monograph, Oxford University Press, New York, 1995.
19. K. Nelsen and A. Ram, Kostka-Foulkes Polynomials and Macdonald Spherical Functions in Surveys in Combinatorics, 2003 (Bangor), Lect. Notes London Math. Soc. 307, Cambridge Univ. Press, Cambridge, pp. 325-370.
20. M-P. Sch\ddot utzenberger, “Propriétés nouvelles des tableaux de Young, Séminaire Delange-Pisot-Poitou,” 19`eme année 26 (1977/78).