Crystal bases for Kirillov-Reshetikhin modules and their combinatorial realization
| Project/Area Number |
15K04803
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | University of Tsukuba |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 量子アフィン代数 / 結晶基底 / パス模型 / 有限次元既約表現 / Kirilov-Reshetikhin 加群 / Lakshmibai-Seshadri パス / Macdonald 多項式 / 標準単項式理論 / Kirillov-Reshetikhin 加群 / Chevalley 型の公式 / Monk 型の公式 / van der Kallen 加群 / Lakshmiba-Seshadri パス |
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| Outline of Final Research Achievements |
(1) I constructed a quotient module of a Demazure-type submodule in an extremal weight module whose graded character is identical to a specialization of a nonsymmetric Macdonald polynomial.
(2) I extended the (classical) standard monomial theory (due to Peter Littelmann) to the case of semi-infinite Lakshmibai-Seshadri paths, and gave the semi-infinite standard monomial theory. As applications, I gave Chevalley-type formulas for graded characters of Demazure-type submodules in extremal weight modules for dominant and antidominant integral weights.
(3) I constructed a level-zero van der Kallen module as a quotient of an extremal weight module, and proved that its graded character is identical to a specialization of a nonsymmetric Macdonald polynomial.
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| Academic Significance and Societal Importance of the Research Achievements |
当研究課題の主な研究対象である Kirilov-Reshetikhin 加群のうち，レベルが１のものについては「レベル・ゼロ基本表現」と呼ばれる（結晶基底を持つ）有限次元表現と一致していることが知られている．また，レベル・ゼロ基本表現のいくつかのテンソル積は量子 Weyl 加群と呼ばれており，エクストリーマル・ウェイト加群の (Demazure 型の部分加群の) 商加群として得られることが知られている．今回の結果は，非対称 Macdonald 多項式の特殊化と，レベル１のKirilov-Reshetikhin 加群を結びつける重要な研究成果である．
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Report
(6 results)
Research Products
(16 results)
