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

Title & Authors
YOUNG TABLEAUX, CANONICAL BASES, AND THE GINDIKIN-KARPELEVICH FORMULA
Lee, Kyu-Hwan; Salisbury, Ben;

Abstract
A combinatorial description of the crystal $\small{\mathcal{B}({\infty})}$ 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;
Language
English
Cited by
1.
CRYSTAL B(λ) IN B(∞) FOR G2 TYPE LIE ALGEBRA,;;

대한수학회지, 2014. vol.51. 2, pp.427-442
1.
The flush statistic on semistandard Young tableaux, Comptes Rendus Mathematique, 2014, 352, 5, 367
2.
Connecting Marginally Large Tableaux and Rigged Configurations via Crystals, Algebras and Representation Theory, 2016, 19, 3, 523
3.
Crystal ℬ ( λ ) $\mathcal {B}(\lambda )$ as a Subset of the Tableau Description of ℬ ( ∞ ) $\mathcal {B}(\infty )$ for the Classical Lie Algebra Types, Algebras and Representation Theory, 2015, 18, 1, 137
4.
Description of B ( ∞ ) $\mathsf {B}(\infty )$ through Kashiwara Embedding for Lie Algebra Types E 6 and E 7, Algebras and Representation Theory, 2017, 20, 4, 871
References
1.
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.

2.
A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases, and totally positive varieties, Invent. Math. 143 (2001), no. 1, 77-128.

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.

8.
J. Kamnitzer, Mirkovic-Vilonen cycles and polytopes, Ann. of Math. (2) 171 (2010), no. 1, 245-294.

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.

10.
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.

11.
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.

12.
M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-516.

13.
M. Kashiwara, The crystal base and Littelmann's refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839-858.

14.
M. Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994), 155-197, CMS Conf. Proc., 16, Amer. Math. Soc., Providence, RI, 1995.

15.
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.

16.
M. Kashiwara and Y. Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), no. 1, 9-36.

17.
H. H. Kim and K.-H. Lee, Representation theory of p-adic groups and canonical bases, Adv. Math. 227 (2011), no. 2, 945-961.

18.
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.

19.
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).

20.
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.

21.
P. Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), no. 3, 499-525.

22.
P. Littelmann, Cones, crystals, and patterns, Transform. Groups 3 (1998), no. 2, 145-179.

23.
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.

24.
G. Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhauser Boston Inc., Boston, MA, 1993.

25.
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.

26.
P. J. McNamara, Metaplectic Whittaker functions and crystal bases, Duke Math. J. 156 (2011), no. 1, 1-31.

27.
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).

28.
The Sage-Combinat community, Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, 2008; http://combinat.sagemath.org.

29.
A. Savage, Geometric and combinatorial realizations of crystal graphs, Algebr. Represent. Theory 9 (2006), no. 2, 161-199.

30.
W. A. Stein et al., Sage Mathematics Software (Version 5.11), The Sage Development Team, 2013; http://www.sagemath.org.