Combinatorics of Casselman-Shalika formula in type $A$
HTML articles powered by AMS MathViewer
- by Kyu-Hwan Lee, Philip Lombardo and Ben Salisbury PDF
- Proc. Amer. Math. Soc. 142 (2014), 2291-2301 Request permission
Abstract:
In the recent works of Brubaker-Bump-Friedberg, Bump-Nakasuji, and others, the product in the Casselman-Shalika formula is written as a sum over a crystal. The coefficient of each crystal element is defined using the data coming from the whole crystal graph structure. In this paper, we adopt the tableau model for the crystal and obtain the same coefficients using data from each individual tableau; i.e., we do not need to look at the graph structure. We also show how to combine our results with tensor products of crystals to obtain the sum of coefficients for a given weight. The sum is a $q$-polynomial which exhibits many interesting properties. We use examples to illustrate these properties.References
- Jennifer Beineke, Ben Brubaker, and Sharon Frechette, A crystal definition for symplectic multiple Dirichlet series, Multiple Dirichlet series, L-functions and automorphic forms, Progr. Math., vol. 300, Birkhäuser/Springer, New York, 2012, pp. 37–63. MR 2952571, DOI 10.1007/978-0-8176-8334-4_{2}
- Arkady Berenstein and Andrei Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), no. 1, 77–128. MR 1802793, DOI 10.1007/s002220000102
- Benjamin Brubaker, Daniel Bump, Gautam Chinta, Solomon Friedberg, and Jeffrey Hoffstein, Weyl group multiple Dirichlet series. I, Multiple Dirichlet series, automorphic forms, and analytic number theory, Proc. Sympos. Pure Math., vol. 75, Amer. Math. Soc., Providence, RI, 2006, pp. 91–114. MR 2279932, DOI 10.1090/pspum/075/2279932
- Ben Brubaker, Daniel Bump, and Solomon Friedberg, Weyl group multiple Dirichlet series, Eisenstein series and crystal bases, Ann. of Math. (2) 173 (2011), no. 2, 1081–1120. MR 2776371, DOI 10.4007/annals.2011.173.2.13
- Ben Brubaker, Daniel Bump, and Solomon Friedberg, Weyl group multiple Dirichlet series: type A combinatorial theory, Annals of Mathematics Studies, vol. 175, Princeton University Press, Princeton, NJ, 2011. MR 2791904, DOI 10.1515/9781400838998
- Daniel Bump and Maki Nakasuji, Integration on $p$-adic groups and crystal bases, Proc. Amer. Math. Soc. 138 (2010), no. 5, 1595–1605. MR 2587444, DOI 10.1090/S0002-9939-09-10206-X
- Gautam Chinta and Paul E. Gunnells, Constructing Weyl group multiple Dirichlet series, J. Amer. Math. Soc. 23 (2010), no. 1, 189–215. MR 2552251, DOI 10.1090/S0894-0347-09-00641-9
- Gautam Chinta and Paul E. Gunnells, Littelmann patterns and Weyl group multiple Dirichlet series of type $D$, Multiple Dirichlet series, L-functions and automorphic forms, Progr. Math., vol. 300, Birkhäuser/Springer, New York, 2012, pp. 119–130. MR 2952574, DOI 10.1007/978-0-8176-8334-4_{5}
- Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971, DOI 10.1090/gsm/042
- Joel Kamnitzer, The crystal structure on the set of Mirković-Vilonen polytopes, Adv. Math. 215 (2007), no. 1, 66–93. MR 2354986, DOI 10.1016/j.aim.2007.03.012
- Masaki Kashiwara and Toshiki Nakashima, Crystal graphs for representations of the $q$-analogue of classical Lie algebras, J. Algebra 165 (1994), no. 2, 295–345. MR 1273277, DOI 10.1006/jabr.1994.1114
- Henry H. Kim and Kyu-Hwan Lee, Representation theory of $p$-adic groups and canonical bases, Adv. Math. 227 (2011), no. 2, 945–961. MR 2793028, DOI 10.1016/j.aim.2011.02.017
- Henry H. Kim and Kyu-Hwan Lee, Quantum affine algebras, canonical bases, and $q$-deformation of arithmetical functions, Pacific J. Math. 255 (2012), no. 2, 393–415. MR 2928558, DOI 10.2140/pjm.2012.255.393
- Kyu-Hwan Lee and Ben Salisbury, A combinatorial description of the Gindikin-Karpelevich formula in type $A$, J. Combin. Theory Ser. A 119 (2012), no. 5, 1081–1094. MR 2891384, DOI 10.1016/j.jcta.2012.01.011
- Peter J. McNamara, Metaplectic Whittaker functions and crystal bases, Duke Math. J. 156 (2011), no. 1, 1–31. MR 2746386, DOI 10.1215/00127094-2010-064
- The Sage-Combinat community, Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, 2008, http://combinat.sagemath.org.
- W. A. Stein et al., Sage Mathematics Software (Version 4.7.1), The Sage Development Team, 2011, http://www.sagemath.org.
Additional Information
- Kyu-Hwan Lee
- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009 – and – Korea Institute for Advanced Study, Seoul, Korea
- MR Author ID: 650497
- Email: khlee@math.uconn.edu
- Philip Lombardo
- Affiliation: Department of Mathematics and Computer Science, St. Joseph’s College, Patchogue, New York 11772
- Email: plombardo@sjcny.edu
- Ben Salisbury
- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
- Address at time of publication: Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan 48859
- Email: ben.salisbury@cmich.edu
- Received by editor(s): November 3, 2011
- Received by editor(s) in revised form: July 24, 2012
- Published electronically: March 11, 2014
- Communicated by: Kailash C. Misra
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 2291-2301
- MSC (2010): Primary 17B37; Secondary 05E10
- DOI: https://doi.org/10.1090/S0002-9939-2014-11961-7
- MathSciNet review: 3195754