Browse > Article
http://dx.doi.org/10.4134/JKMS.2014.51.2.289

YOUNG TABLEAUX, CANONICAL BASES, AND THE GINDIKIN-KARPELEVICH FORMULA  

Lee, Kyu-Hwan (Department of Mathematics University of Connecticut)
Salisbury, Ben (Department of Mathematics The City College of New York, The Institute for Computational and Experimental Research in Mathematics Brown University)
Publication Information
Journal of the Korean Mathematical Society / v.51, no.2, 2014 , pp. 289-309 More about this Journal
Abstract
A combinatorial description of the crystal $\mathcal{B}({\infty})$ 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 Gindikin-Karpelevich formula as a sum over the set of Young tableaux.
Keywords
Gindikin-Karpelevich; Kostant partition; Young tableaux; canonical basis; Lusztig parametrization; crystal;
Citations & Related Records
연도 인용수 순위
  • Reference
1 A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases, and totally positive varieties, Invent. Math. 143 (2001), no. 1, 77-128.   DOI
2 A. Berenstein and A. Zelevinsky, Canonical bases for the quantum group of type $A_r$ and piecewise-linear combinatorics, Duke Math. J. 82 (1996), no. 3, 473-502.   DOI
3 N. Bourbaki, Lie groups and Lie algebras. Chapters 4-6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002.
4 D. Bump and M. Nakasuji, Integration on p-adic groups and crystal bases, Proc. Amer. Math. Soc. 138 (2010), no. 5, 1595-1605.
5 S. G. Gindikin and F. I. Karpelevic, Plancherel measure for symmetric Riemannian spaces of non-positive curvature, Dokl. Akad. Nauk SSSR 145 (1962), 252-255.
6 J. Hong and S.-J. Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002.
7 J. Hong and H. Lee, Young tableaux and crystal B(${\infty}$) for finite simple Lie algebras, J. Algebra 320 (2008), no. 10, 3680-3693.   DOI   ScienceOn
8 J. Kamnitzer, Mirkovic-Vilonen cycles and polytopes, Ann. of Math. (2) 171 (2010), no. 1, 245-294.   DOI
9 S.-J. Kang, Crystal bases for quantum affine algebras and combinatorics of Young walls, Proc. London Math. Soc. (3) 86 (2003), no. 1, 29-69.   DOI
10 M. Kashiwara, The crystal base and Littelmann's refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839-858.   DOI
11 S.-J. Kang, K.-H. Lee, H. Ryu, and B. Salisbury, A combinatorial description of the Gindikin-Karpelevich formula in type $A_n^{(1)}$, arXiv:1203.1640.   DOI   ScienceOn
12 S.-J. Kang and K. C. Misra, Crystal bases and tensor product decompositions of $U_q(G_2)$-modules, J. Algebra 163 (1994), no. 3, 675-691.   DOI   ScienceOn
13 H. H. Kim and K.-H. Lee, Representation theory of p-adic groups and canonical bases, Adv. Math. 227 (2011), no. 2, 945-961.   DOI   ScienceOn
14 M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-516.   DOI
15 M. Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994), 155-197, CMS Conf. Proc., 16, Amer. Math. Soc., Providence, RI, 1995.
16 M. Kashiwara and Y. Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), no. 1, 9-36.   DOI
17 R. Langlands, Euler products, A James K. Whittemore Lecture in Mathematics given at Yale University, 1967.Yale Mathematical Monographs, 1. Yale University Press, New Haven, Conn.-London, 1971.
18 K.-H. Lee, P. Lombardo, and B. Salisbury, Combinatorics of the Casselman-Shalika formula in type A, to appear in Proc. Amer. Math. Soc. (arXiv:1111.1134).
19 K.-H. Lee and B. Salisbury, A combinatorial description of the Gindikin-Karpelevich formula in type A, J. Combin. Theory Ser. A 119 (2012), no. 5, 1081-1094.   DOI   ScienceOn
20 G. Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhauser Boston Inc., Boston, MA, 1993.
21 P. Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), no. 3, 499-525.   DOI
22 P. Littelmann, Cones, crystals, and patterns, Transform. Groups 3 (1998), no. 2, 145-179.   DOI   ScienceOn
23 The Sage-Combinat community, Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, 2008; http://combinat.sagemath.org.
24 G. Lusztig, Singularities, character formulas, and a q-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981), 208-229, Asterisque, 101-102, Soc. Math. France, Paris, 1983.
25 P. J. McNamara, Metaplectic Whittaker functions and crystal bases, Duke Math. J. 156 (2011), no. 1, 1-31.   DOI
26 S. Morier-Genoud, Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties, Trans. Amer. Math. Soc. 360 (2008), no. 1, 215-235 (electronic).   DOI   ScienceOn
27 A. Savage, Geometric and combinatorial realizations of crystal graphs, Algebr. Represent. Theory 9 (2006), no. 2, 161-199.   DOI
28 W. A. Stein et al., Sage Mathematics Software (Version 5.11), The Sage Development Team, 2013; http://www.sagemath.org.
29 M. Kashiwara and T. Nakashima, Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra 165 (1994), no. 2, 295-345.   DOI   ScienceOn
30 I. G. Macdonald, Spherical functions on a group of p-adic type, Publications of the Ramanujan Institute, No. 2. Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras, Madras, 1971.