Schensted-Type Correspondences and Plactic Monoids for Types B n and D n
Cédric Lecouvey
DOI: 10.1023/A:1025154930381
Abstract
We use Kashiwara's theory of crystal bases to study plactic monoids for U q( so 2 n+1) and U q( so 2 n ). Simultaneously we describe a Schensted type correspondence in the crystal graphs of tensor powers of vector and spin representations and we derive a Jeu de Taquin for type B from the Sheats sliding algorithm.
Pages: 99–133
Keywords: combinatorics; quantum algebra; representation theory
Full Text: PDF
References
1. C. De Concini, “Symplectic standard tableaux,” Adv. in Math. 34 (1979), 1-27.
2. E. Date, M. Jimbo, and T. Miwa, “Representations of Uq (gl(n, C)) at q = 0 and the Robinson-Schensted correspondence,” in Physics and Mathematic of Strings, L. Brink, D. Friedman and A.M. Polyakov (Eds.), Word Scientific, Teaneck, NJ, 1990, pp. 185-211.
3. W. Fulton, Young Tableaux, London Mathematical Society, Student Text 35.
4. M. Kashiwara and T. Nakashima, “Crystal graphs for representations of the q-analogue of classical Lie algebras,” J. Algebra 165 (1994), 295-345.
5. M. Kashiwara, “On crystal bases,” in Canadian Mathematical Society, Conference Proceedings, 1995, Vol. 16.
6. M. Kashiwara, “Similarity of crystal bases,” AMS Contemporary Math. 194 (1996), 177-186.
7. R.C. King, “Weight multiplicities for the classical groups,” Lectures Notes in Physics 50 (New York; Springer, 1975), 490-499.
8. A. Lascoux, B. Leclerc, and J.Y. Thibon, “Crystal graph and q-analogues of weight multiplicities for the roots system A* n,” Lett. Math. Phys. 35 (1994), 359-374.
9. A. Lascoux and M.P. Sch\ddot utzenberger, “Le mono\ddot ıde plaxique,” in Non Commutative Structures in Algebra and Geometric Combinatorics, A. de Luca (Ed.), Quaderni della Ricerca Scientifica del C.N.R., Roma, 1981.
10. C. Lecouvey, “Schensted-type correspondence, plactic monoid and Jeu de Taquin for type Cn,” J. Algebra (to appear).
11. C. Lecouvey, “Algorithmique et combinatoire des alg`ebres enveloppantes quantiques de type classique,” Th`ese, Université de caen, 2001.
12. P. Littelmann, “A plactic algebra for semisimple Lie algebras,” Adv. in Math. 124 (1996), 312-331.
13. P. Littelmann, “Crystal graph and Young tableaux,” J. Algebra 175 (1995), 65-87.
14. P. Littelmann, “A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras,” Inv. Math. 116 (1994), 329-346.
15. T. Nakashima, “Crystal base and a generalization of the Littelwood-Richardson rule for the classical Lie algebras,” Comm. Math. Phys. 154 (1993), 215-243.
16. J.T. Sheats, “A symplectic jeu de taquin bijection between the tableaux of King and of De Concini,” Trans. A.M.S. 351(7) (1999), 3569-3607.
17. S. Sundaran, “Orthogonal tableaux and an insertion scheme for SO2n+1,” J. Combin. Theory, Ser. A 53 (1990), 239-256.
18. S. Sundaram, “Tableaux in the representation theory of the classical groups,” IMA. Volumes in Mathematics and its Applications 19 (Springer-Verlag, 1990).
2. E. Date, M. Jimbo, and T. Miwa, “Representations of Uq (gl(n, C)) at q = 0 and the Robinson-Schensted correspondence,” in Physics and Mathematic of Strings, L. Brink, D. Friedman and A.M. Polyakov (Eds.), Word Scientific, Teaneck, NJ, 1990, pp. 185-211.
3. W. Fulton, Young Tableaux, London Mathematical Society, Student Text 35.
4. M. Kashiwara and T. Nakashima, “Crystal graphs for representations of the q-analogue of classical Lie algebras,” J. Algebra 165 (1994), 295-345.
5. M. Kashiwara, “On crystal bases,” in Canadian Mathematical Society, Conference Proceedings, 1995, Vol. 16.
6. M. Kashiwara, “Similarity of crystal bases,” AMS Contemporary Math. 194 (1996), 177-186.
7. R.C. King, “Weight multiplicities for the classical groups,” Lectures Notes in Physics 50 (New York; Springer, 1975), 490-499.
8. A. Lascoux, B. Leclerc, and J.Y. Thibon, “Crystal graph and q-analogues of weight multiplicities for the roots system A* n,” Lett. Math. Phys. 35 (1994), 359-374.
9. A. Lascoux and M.P. Sch\ddot utzenberger, “Le mono\ddot ıde plaxique,” in Non Commutative Structures in Algebra and Geometric Combinatorics, A. de Luca (Ed.), Quaderni della Ricerca Scientifica del C.N.R., Roma, 1981.
10. C. Lecouvey, “Schensted-type correspondence, plactic monoid and Jeu de Taquin for type Cn,” J. Algebra (to appear).
11. C. Lecouvey, “Algorithmique et combinatoire des alg`ebres enveloppantes quantiques de type classique,” Th`ese, Université de caen, 2001.
12. P. Littelmann, “A plactic algebra for semisimple Lie algebras,” Adv. in Math. 124 (1996), 312-331.
13. P. Littelmann, “Crystal graph and Young tableaux,” J. Algebra 175 (1995), 65-87.
14. P. Littelmann, “A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras,” Inv. Math. 116 (1994), 329-346.
15. T. Nakashima, “Crystal base and a generalization of the Littelwood-Richardson rule for the classical Lie algebras,” Comm. Math. Phys. 154 (1993), 215-243.
16. J.T. Sheats, “A symplectic jeu de taquin bijection between the tableaux of King and of De Concini,” Trans. A.M.S. 351(7) (1999), 3569-3607.
17. S. Sundaran, “Orthogonal tableaux and an insertion scheme for SO2n+1,” J. Combin. Theory, Ser. A 53 (1990), 239-256.
18. S. Sundaram, “Tableaux in the representation theory of the classical groups,” IMA. Volumes in Mathematics and its Applications 19 (Springer-Verlag, 1990).