Affine Type A Crystal Structure on Tensor Products of Rectangles, Demazure Characters, and Nilpotent Varieties
Mark Shimozono
DOI: 10.1023/A:1013894920862
Abstract
Answering a question of Kuniba, Misra, Okado, Takagi, and Uchiyama, it is shown that certain higher level Demazure characters of affine type A, coincide with the graded characters of coordinate rings of closures of conjugacy classes of nilpotent matrices.
Pages: 151–187
Keywords: crystal graph; tableau; Kostka polynomial; Littlewood-Richardson coefficient
Full Text: PDF
References
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7. S.-J. Kang, M. Kashiwara, K. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki, “Perfect crystals of quantum affine Lie algebras,” Duke Math. J. 68 (1992), 499-607.
8. M. Kashiwara, “Crystal base and Littelmann's refined Demazure character formula,” Duke Math. J. 71 (1993), 839-858.
9. M. Kashiwara, “Crystal bases of modified quantized enveloping algebra,” Duke Math. J. 73 (1994), 383-413.
10. M. Kashiwara and T. Nakashima, “Crystal graphs for representations of the q-analogue of classical Lie algebras,” J. Algebra 165 (1994), 295-345.
11. A.N. Kirillov and M. Shimozono, “A generalization of the Kostka-Foulkes polynomials,” preprint, math.QA/9803062.
12. J. Klimek, W. Kraśkiewicz, M. Shimozono, and J. Weyman, “On the Grothendieck group of modules supported in a nilpotent orbit in the Lie algebra gl(n),” J. Pure Appl. Algebra 153 (2000), 237-261.
13. D.E. Knuth, “Permutations, matrices, and generalized Young tableaux,” Pacific J. Math. 34 (1970), 709-727.
14. A. Kuniba, K. Misra, M. Okado, T. Takagi, and J. Uchiyama, “Paths, Demazure crystals, and symmetric functions,” J. Math. Phys. 41 (2000), 6477-6486.
15. A. Kuniba, K. Misra, M. Okado, and J. Uchiyama, “Demazure modules and perfect crystals,” Comm. Math. Phys. 192 (1998), 555-567.
16. A. Lascoux, “Cyclic permutations on words, tableaux and harmonic polynomials,” in Proc. of the Hyderabad Conference on Algebraic Groups, 1989, Manoj Prakashan, Madras, 1991, pp. 323-347.
17. A. Lascoux, B. Leclerc, and J.-Y. Thibon, “Crystal graphs and q-analogues of weight multiplicities for root systems of type An,” Lett. Math. Phys. 35 (1995), 359-374.
18. A. Lascoux and M.-P. Sch\ddot utzenberger, “Sur une conjecture de H.O. Foulkes,” C. R. Acad. Sc. Paris 286A (1978), 323-324.
19. A. Lascoux and M.P. Sch\ddot utzenberger, “Le mono\ddot ıde plaxique,” in Noncommutative structures in algebra and geometric combinatorics, A. de Luca (Ed.), Quaderni della Ricerca Scientifica del C. N. R., Roma, 1981, pp. 129-156.
20. D.E. Littlewood and A.R. Richardson, “Group characters and algebra,” Phil. Trans. Royal Soc. London Ser. A 233 (1934), 99-141.
21. G. Lusztig, “Green polynomials and singularities of unipotent classes,” Adv. Math. 42 (1981), 169-178.
22. I.G. MacDonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.
23. A.O. Morris, “The characters of the group G L(n, q),” Math. Zeitschr. 81 (1963), 112-123.
24. A. Nakayashiki and Y. Yamada, “Kostka polynomials and energy functions in solvable lattice models,” Selecta Math. (N.S.) 3 (1997), 547-599.
25. G. de B. Robinson, “On the representations of the symmetric group,” Amer. J. Math. 60 (1938), 745-760.
26. C. Schensted, “Longest increasing and decreasing sequences,” Canad. J. Math 13 (1961), 179-191.
27. A. Schilling and S. Warnaar, “Inhomogeneous lattice paths, generalized Kostka-Foulkes polynomials, and An - 1-supernomials,” Comm. Math. Phys. 202 (1999), 359-401.
28. M.-P. Sch\ddot utzenberger, “Promotion des morphisms d'ensembles ordonnes,” Disc. Math. 2 (1972), 73-94.
29. M.-P. Sch\ddot utzenberger, “La correspondance de Robinson,” in Combinatoire et Représentations du Groupe Symétrique, D. Foata (Ed.), Strasbourg, 1976, Springer Lecture Notes in Math.,Vol. 579, (1977), pp. 59-113.
30. M. Shimozono, “A cyclage poset structure for Littlewood-Richardson tableaux,” European J. Combin. 22 (2001), 365-393.
31. M. Shimozono, “Multi-atoms and a monotonicity property for generalized Kostka polynomials,” European J. Combin. 22 (2001), 395-414.
32. M. Shimozono, preprint math. QA/9804039.
33. M. Shimozono and J. Weyman, “Bases for coordinate rings of conjugacy classes of nilpotent matrices,” J. Algebra 220 (1999), 1-55.
34. M. Shimozono and J. Weyman, “Graded characters of modules supported in the closure of a nilpotent conjugacy class,” European J. Combin. 21 (2000), 257-288.
35. J. Stembridge, “Multiplicity-free products of Schur functions,” Annals of Combinatorics 5 (2001), 113-121.
36. J. Weyman, “The equations of conjugacy classes of nilpotent matrices,” Invent. Math. 98 (1989), 229-245.
37. D.E. White, “Some connections between the Littlewood-Richardson rule and the construction of Schensted,” J. Comb. Th. Ser. A 30 (1981), 237-247.
2. W. Fulton, Young tableaux, London Mathematical Society Student Texts, Vol. 35, Cambridge University Press, Cambridge, 1997.
3. M. Haiman, “Dual equivalence with applications, including a conjecture of Proctor,” Discrete Math. 99 (1992), 79-113.
4. G.-N. Han, “Croissance des polyn\hat omes de Kostka,” C. R. Acad. Sci. Paris Ser. I 311 (1990), 269-272.
5. V.G. Kac, Infinite Dimensional Lie Algebras, Cambridge University Press, Cambridge, 1990.
6. S.-J. Kang, M. Kashiwara, K. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki, “Affine crystals and vertex models,” Int. J. Modern Phys A. Suppl. 1A (1992), 449-484.
7. S.-J. Kang, M. Kashiwara, K. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki, “Perfect crystals of quantum affine Lie algebras,” Duke Math. J. 68 (1992), 499-607.
8. M. Kashiwara, “Crystal base and Littelmann's refined Demazure character formula,” Duke Math. J. 71 (1993), 839-858.
9. M. Kashiwara, “Crystal bases of modified quantized enveloping algebra,” Duke Math. J. 73 (1994), 383-413.
10. M. Kashiwara and T. Nakashima, “Crystal graphs for representations of the q-analogue of classical Lie algebras,” J. Algebra 165 (1994), 295-345.
11. A.N. Kirillov and M. Shimozono, “A generalization of the Kostka-Foulkes polynomials,” preprint, math.QA/9803062.
12. J. Klimek, W. Kraśkiewicz, M. Shimozono, and J. Weyman, “On the Grothendieck group of modules supported in a nilpotent orbit in the Lie algebra gl(n),” J. Pure Appl. Algebra 153 (2000), 237-261.
13. D.E. Knuth, “Permutations, matrices, and generalized Young tableaux,” Pacific J. Math. 34 (1970), 709-727.
14. A. Kuniba, K. Misra, M. Okado, T. Takagi, and J. Uchiyama, “Paths, Demazure crystals, and symmetric functions,” J. Math. Phys. 41 (2000), 6477-6486.
15. A. Kuniba, K. Misra, M. Okado, and J. Uchiyama, “Demazure modules and perfect crystals,” Comm. Math. Phys. 192 (1998), 555-567.
16. A. Lascoux, “Cyclic permutations on words, tableaux and harmonic polynomials,” in Proc. of the Hyderabad Conference on Algebraic Groups, 1989, Manoj Prakashan, Madras, 1991, pp. 323-347.
17. A. Lascoux, B. Leclerc, and J.-Y. Thibon, “Crystal graphs and q-analogues of weight multiplicities for root systems of type An,” Lett. Math. Phys. 35 (1995), 359-374.
18. A. Lascoux and M.-P. Sch\ddot utzenberger, “Sur une conjecture de H.O. Foulkes,” C. R. Acad. Sc. Paris 286A (1978), 323-324.
19. A. Lascoux and M.P. Sch\ddot utzenberger, “Le mono\ddot ıde plaxique,” in Noncommutative structures in algebra and geometric combinatorics, A. de Luca (Ed.), Quaderni della Ricerca Scientifica del C. N. R., Roma, 1981, pp. 129-156.
20. D.E. Littlewood and A.R. Richardson, “Group characters and algebra,” Phil. Trans. Royal Soc. London Ser. A 233 (1934), 99-141.
21. G. Lusztig, “Green polynomials and singularities of unipotent classes,” Adv. Math. 42 (1981), 169-178.
22. I.G. MacDonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.
23. A.O. Morris, “The characters of the group G L(n, q),” Math. Zeitschr. 81 (1963), 112-123.
24. A. Nakayashiki and Y. Yamada, “Kostka polynomials and energy functions in solvable lattice models,” Selecta Math. (N.S.) 3 (1997), 547-599.
25. G. de B. Robinson, “On the representations of the symmetric group,” Amer. J. Math. 60 (1938), 745-760.
26. C. Schensted, “Longest increasing and decreasing sequences,” Canad. J. Math 13 (1961), 179-191.
27. A. Schilling and S. Warnaar, “Inhomogeneous lattice paths, generalized Kostka-Foulkes polynomials, and An - 1-supernomials,” Comm. Math. Phys. 202 (1999), 359-401.
28. M.-P. Sch\ddot utzenberger, “Promotion des morphisms d'ensembles ordonnes,” Disc. Math. 2 (1972), 73-94.
29. M.-P. Sch\ddot utzenberger, “La correspondance de Robinson,” in Combinatoire et Représentations du Groupe Symétrique, D. Foata (Ed.), Strasbourg, 1976, Springer Lecture Notes in Math.,Vol. 579, (1977), pp. 59-113.
30. M. Shimozono, “A cyclage poset structure for Littlewood-Richardson tableaux,” European J. Combin. 22 (2001), 365-393.
31. M. Shimozono, “Multi-atoms and a monotonicity property for generalized Kostka polynomials,” European J. Combin. 22 (2001), 395-414.
32. M. Shimozono, preprint math. QA/9804039.
33. M. Shimozono and J. Weyman, “Bases for coordinate rings of conjugacy classes of nilpotent matrices,” J. Algebra 220 (1999), 1-55.
34. M. Shimozono and J. Weyman, “Graded characters of modules supported in the closure of a nilpotent conjugacy class,” European J. Combin. 21 (2000), 257-288.
35. J. Stembridge, “Multiplicity-free products of Schur functions,” Annals of Combinatorics 5 (2001), 113-121.
36. J. Weyman, “The equations of conjugacy classes of nilpotent matrices,” Invent. Math. 98 (1989), 229-245.
37. D.E. White, “Some connections between the Littlewood-Richardson rule and the construction of Schensted,” J. Comb. Th. Ser. A 30 (1981), 237-247.