The Lascoux, Leclerc and Thibon algorithm and Soergel's tilting algorithm
Steen Ryom-Hansen
Univerdad de Talca Instituto de Matemática y Física Chile Chile
DOI: 10.1007/s10801-006-6026-5
Abstract
We generalize Soergel's tilting algorithm to singular weights and deduce from this the validity of the Lascoux-Leclerc-Thibon conjecture on the connection between the canonical basis of the basic submodule of the Fock module and the representation theory of the Hecke-algebras at root of unity.
Pages: 5–20
Keywords: keywords tilting modules; crystal basis; Fock module; kashdan-Lusztig polynomials
Full Text: PDF
References
1. H.H. Andersen, “Tensor products of quantized tilting modules,” Commun. Math. Phys. 149 (1992), 149- 159.
2. S. Ariki, “On the decomposition numbers of the Hecke algebra G(m, 1, n),” J. Math. Kyoto Univ. 36 (1996) 789-808.
3. H.H. Andersen, J.C. Jantzen and W. Soergel, “Representations of quantum groups at a p-th root of unity and of semisimple Groups in characteristic p: Independence of p,” Astérisque 220 (1994).
4. S. Donkin, “On Schur algebras and related algebras. IV. The blocks of the Schur algebras,” J. Algebra 168(2), (1994) 400-429.
5. K. Erdmann, “Symmetric groups and quasi-hereditary algebras. Finite dimensional algebras and related topics,” in V. Dlab and L.L. Scott, (eds.), Kluwer, 1994, pp. 123-161.
6. F. Goodman and H. Wenzl, “Crystal basis of quantum affine algebras and affine Kazhdan-Lusztig polynomials,” Internat. Math. Res. Notices 5 (1999), 251-275 MR1675980 (2000b:17017)
7. F. Goodman and H. Wenzl, “A path algorithm for affine Kazhdan-Lusztig polynomial,” Math. Z. 237(2) (2001) 235-249. MR1838309 (2002i:20006)
8. G. James, “The decomposition matrices of Gln(q) for n \leq 10,” Proc. London Math. Soc., 60 (1990) 225-265.
9. A. Lascoux, B. Leclerc, and J.-Y. Thibon, “Hecke algebras at roots of unity and crystal bases of quantum affine algebras,” Commun. Math. Phys. 181 (1996) 205-263.
10. B. Leclerc and J.-Y. Thibon, “Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials, Combinatorial Methods in Representation Theory,” Advanced Studies in Pure Mathematics 28 (2000) 155-220.
11. B. Leclerc, J.-Y. Thibon, “Canonical bases of q-deformed Fock spaces,” Int. Math. Res. Notices, 9 (1996) 447-456.
12. W. Soergel and Kazhdan-Lusztig-Polynome und eine Kombinatorik f\ddot ur Kipp-Moduln, Represent. Theory 1 (1997) 37-68.
13. W. Soergel, “Characterformeln f\ddot ur Kipp-Moduln \ddot uber Kac-Moody Algebren, Representation Theory,” 1 (1997) 115-132.
14. M. Varagnolo and E. Vasserot, On the decomposition numbers of the quantized Schur algebra, Duke Math.
2. S. Ariki, “On the decomposition numbers of the Hecke algebra G(m, 1, n),” J. Math. Kyoto Univ. 36 (1996) 789-808.
3. H.H. Andersen, J.C. Jantzen and W. Soergel, “Representations of quantum groups at a p-th root of unity and of semisimple Groups in characteristic p: Independence of p,” Astérisque 220 (1994).
4. S. Donkin, “On Schur algebras and related algebras. IV. The blocks of the Schur algebras,” J. Algebra 168(2), (1994) 400-429.
5. K. Erdmann, “Symmetric groups and quasi-hereditary algebras. Finite dimensional algebras and related topics,” in V. Dlab and L.L. Scott, (eds.), Kluwer, 1994, pp. 123-161.
6. F. Goodman and H. Wenzl, “Crystal basis of quantum affine algebras and affine Kazhdan-Lusztig polynomials,” Internat. Math. Res. Notices 5 (1999), 251-275 MR1675980 (2000b:17017)
7. F. Goodman and H. Wenzl, “A path algorithm for affine Kazhdan-Lusztig polynomial,” Math. Z. 237(2) (2001) 235-249. MR1838309 (2002i:20006)
8. G. James, “The decomposition matrices of Gln(q) for n \leq 10,” Proc. London Math. Soc., 60 (1990) 225-265.
9. A. Lascoux, B. Leclerc, and J.-Y. Thibon, “Hecke algebras at roots of unity and crystal bases of quantum affine algebras,” Commun. Math. Phys. 181 (1996) 205-263.
10. B. Leclerc and J.-Y. Thibon, “Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials, Combinatorial Methods in Representation Theory,” Advanced Studies in Pure Mathematics 28 (2000) 155-220.
11. B. Leclerc, J.-Y. Thibon, “Canonical bases of q-deformed Fock spaces,” Int. Math. Res. Notices, 9 (1996) 447-456.
12. W. Soergel and Kazhdan-Lusztig-Polynome und eine Kombinatorik f\ddot ur Kipp-Moduln, Represent. Theory 1 (1997) 37-68.
13. W. Soergel, “Characterformeln f\ddot ur Kipp-Moduln \ddot uber Kac-Moody Algebren, Representation Theory,” 1 (1997) 115-132.
14. M. Varagnolo and E. Vasserot, On the decomposition numbers of the quantized Schur algebra, Duke Math.