Geometric realization of the crystal bases of standard modules over quantum affine algebras

2016 Fiscal Year Final Research Report

• Project/Area Number: 24540010
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: Naito Satoshi 東京工業大学, 理工学研究科, 教授 (60252160)
• Co-Investigator(Renkei-kenkyūsha): SAITO Yoshihisa 東京大学, 大学院数理科学研究科, 准教授 (20294522) ; KATO Syu 京都大学, 大学院理学研究科, 准教授 (40456760) ; SAGAKI Daisuke 筑波大学, 数理物質系, 准教授 (40344866)
• Research Collaborator: Lenart Cristian State University of New York at Albany, Department of Mathematics and Statistics, 教授 ; Schilling Anne University of California, Department of Mathematics, 教授 ; Shimozono Mark Virginia Tech, Department of Mathematics, 教授
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 表現論 / アフィン量子群の表現論 / レベル・ゼロ表現 / extremal ウエイト加群 / Demazure 部分加群 / マクドナルド多項式 / Lakshmibai-Seshadri パス / 半無限旗多様体

Outline of Final Research Achievements

First, we got an explicit description, in terms of the quantum Bruhat graph, of the graded character of an arbitrary Demazure submodule of a level-zero extremal weight module over a quantum affine algebra. Also, we got an explicit description, in terms of the quantum Bruhat graph, of the specializations at t = 0 and t = infinity of an arbitrary nonsymmetric Macdonald polynomial. By combining these results, we proved that the graded character of the Demazure submodule corresponding to the identity element (resp., the longest element) of a finite Weyl group is identical to the product of a certain factor (which is an explicit rational function in q) and the specialization at t = 0 (resp., at t = infinity) of the symmetric (resp., nonsymmetric) Macdonald polynomial associated to a dominant integral weight (resp., anti-dominant integral weight).
Moreover, we studied the connection of level-zero Demazure submodules above with Schubert subvarieties of a semi-infinite flag manifold.

Free Research Field

アフィン量子群の表現論で、特に可積分表現が持つ結晶基底を研究している。