YOUNG TABLEAUX, CANONICAL BASES, AND THE GINDIKIN-KARPELEVICH FORMULA
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
A combinatorial description of the crystal B(∞) for finite- dimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig’s parametrization, and obtain a combinatorial rule for expressing the Gindi- kin-Karpelevich formula as a sum over the set of Young tableaux.
Abstract
A combinatorial description of the crystal B(∞) for finite- dimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig’s parametrization, and obtain a combinatorial rule for expressing the Gindi- kin-Karpelevich formula as a sum over the set of Young tableaux.
참고문헌 (30)
- A. Berenstein, "Canonical bases for the quantum group of type Ar and piecewise-linear combinatorics", Duke Math. J, 82(3), pp.473~502 (1996)
- A. Berenstein, "Tensor product multiplicities, canonical bases, and totally positive varieties", Invent. Math, 143(1), pp.77~128 (2001)
- N. Bourbaki, "Lie groups and Lie algebras" (2002), Springer-Verlag
- D. Bump, "Integration on p-adic groups and crystal bases", Proc. Amer. Math. Soc, 138(5), pp.1595~1605 (2010)
- S. G. Gindikin, "Plancherel measure for symmetric Riemannian spaces of non-positive curvature", Dokl. Akad. Nauk SSSR, 145, pp.252~255 (1962)
- J. Hong, "Introduction to quantum groups and crystal bases" (2002), American Mathematical Society
- J. Hong, "Young tableaux and crystal B(∞) for finite simple Lie algebras", J. Algebra, 320(10), pp.3680~3693 (2008)
- J. Kamnitzer, "Mirkovic-Vilonen cycles and polytopes", Ann. of Math. (2), 171(1), pp.245~294 (2010)
- S.-J. Kang, "Crystal bases for quantum affine algebras and combinatorics of Young walls", Proc. London Math. Soc. (3), 86(1), pp.29~69 (2003)
- S.-J. Kang, "A combinatorial description of the Gindikin-Karpelevich formula in type An(1)", arXiv:1203.1640, ~
- S.-J. Kang, "Crystal bases and tensor product decompositions of Uq(G2)-modules", J. Algebra, 163(3), pp.675~691 (1994)
- M. Kashiwara, "On crystal bases of the q-analogue of universal enveloping algebras", Duke Math. J, 63(2), pp.465~516 (1991)
- M. Kashiwara, "The crystal base and Littelmann's refined Demazure character formula", Duke Math. J, 71(3), pp.839~858 (1993)
- M. Kashiwara, "On crystal bases, Representations of groups (Banff, AB, 1994)", CMS Conf. Proc., 16, pp.155~197 (1995), Amer. Math. Soc.
- M. Kashiwara, "Crystal graphs for representations of the q-analogue of classical Lie algebras", J. Algebra, 165(2), pp.295~345 (1994)
- M. Kashiwara, "Geometric construction of crystal bases", Duke Math. J, 89(1), pp.9~36 (1997)
- H. H. Kim, "Representation theory of p-adic groups and canonical bases", Adv. Math, 227(2), pp.945~961 (2011)
- R. Langlands, "Euler products, A James K. Whittemore Lecture in Mathematics given at Yale University" (1971), Yale University Press
- K.-H. Lee, "Combinatorics of the Casselman-Shalika formula in type A", to appear in Proc. Amer. Math. Soc. (arXiv:1111.1134), ~
- K.-H. Lee, "A combinatorial description of the Gindikin-Karpelevich formula in type A", J. Combin. Theory Ser. A, 119(5), pp.1081~1094 (2012)
- P. Littelmann, "Paths and root operators in representation theory", Ann. of Math. (2), 142(3), pp.499~525 (1995)
- P. Littelmann, "Cones, crystals, and patterns", Transform. Groups, 3(2), pp.145~179 (1998)
- G. Lusztig, "Singularities, character formulas, and a q-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981)", Asterisque, 101, pp.208~229 (1983), Soc. Math. France
- G. Lusztig, "Introduction to quantum groups" (1993), Birkhauser Boston Inc.
- I. G. Macdonald, "Spherical functions on a group of p-adic type" (1971), Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras
- P. J. McNamara, "Metaplectic Whittaker functions and crystal bases", Duke Math. J, 156(1), pp.1~31 (2011)
- S. Morier-Genoud, "Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties", Trans. Amer. Math. Soc, 360(1), pp.215~235 (2008)
- "The Sage-Combinat community, Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics" (2008)[URL]
- A. Savage, "Geometric and combinatorial realizations of crystal graphs", Algebr. Represent. Theory, 9(2), pp.161~199 (2006)
- "Sage Mathematics Software (Version 5.11)" (2013)[URL]
