Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Specialization of nonsymmetric Macdonald polynomials at $ t=\infty$ and Demazure submodules of level-zero extremal weight modules


Authors: Satoshi Naito, Fumihiko Nomoto and Daisuke Sagaki
Journal: Trans. Amer. Math. Soc. 370 (2018), 2739-2783
MSC (2010): Primary 17B37; Secondary 33D52, 81R60, 81R50
DOI: https://doi.org/10.1090/tran/7114
Published electronically: November 16, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we give a representation-theoretic interpretation of the specialization $ E_{w_\circ \lambda } (q,\infty )$ of the nonsymmetric Macdonald polynomial $ E_{w_\circ \lambda }(q,t)$ at $ t=\infty $ in terms of the Demazure submodule $ V_{w_\circ }^- (\lambda )$ of the level-zero extremal weight module $ V(\lambda )$ over a quantum affine algebra of an arbitrary untwisted type. Here, $ \lambda $ is a dominant integral weight, and $ w_\circ $ denotes the longest element in the finite Weyl group $ W$. Also, for each $ x \in W$, we obtain a combinatorial formula for the specialization $ E_{x \lambda } (q, \infty )$ at $ t=\infty $ of the nonsymmetric Macdonald polynomial $ E_{x \lambda } (q,t)$ and also a combinatorical formula for the graded character $ \mathop {\rm gch}\nolimits V_{x}^- (\lambda )$ of the Demazure submodule $ V_{x}^- (\lambda )$ of $ V(\lambda )$. Both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape $ \lambda $.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 17B37, 33D52, 81R60, 81R50

Retrieve articles in all journals with MSC (2010): 17B37, 33D52, 81R60, 81R50


Additional Information

Satoshi Naito
Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8551, Japan
Email: naito@math.titech.ac.jp

Fumihiko Nomoto
Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8551, Japan
Email: nomoto.f.aa@m.titech.ac.jp

Daisuke Sagaki
Affiliation: Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
Email: sagaki@math.tsukuba.ac.jp

DOI: https://doi.org/10.1090/tran/7114
Received by editor(s): May 8, 2016
Received by editor(s) in revised form: October 21, 2016
Published electronically: November 16, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society