| Project/Area Number |
17540008
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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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 |
NAITO Satoshi University of Tsukuba, Graduate School of Pure andApplied Sciences, Associate Professor (60252160)
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| Co-Investigator(Kenkyū-buntansha) |
MORITA Jun University of Tsukuba, Graduate School of Pure andApplied Sciences, Professor (20166416)
TAKEYAMA Yoshihiro University of Tsukuba, Graduate School of Pure and Applied Sciences, Lecturer (60375392)
DAISUKE Sagaki University of Tsukuba, Graduate School of Pure and Applied Sciences, Associate Professor (40344866)
MASATO Okado Osaka University, Graduate School of Engineering Sciences, Associate Professor (70221843)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,670,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥270,000)
Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2006: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2005: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | quantum affine algebra / level-zero representation / extremal weight module / crystal basis / Lakshmibai-Seshadri path / path model / energy function / Kostka polynomial / 量子群 / 既約最高ウェイト加群 / Mirkovic-Vilonen多面体 / tropical Plucker関係式 / Anderson-Mirkovic予想 / Demazure加群 / Demazureクリスタル / レベル・ゼロ表現 / エネルギー / degree関数 / extremalウエイト加群 / レベル・ゼロ基本表現 / テンソル積 / Lakshmiba-Seshadri path / path model |
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| Research Abstract |
Let g be an affine Lie algebra over the complex numbers, and let λ be a level-zero integral weight that is a sum of level-zero fundamental weights π_I, I=1, . N, with repetitions allowed. We consider the quantum Weyl module W_{q} (λ) over the quantum affine algebra U_{q} (g) associated to certain Drinfeld polynomials corresponding to λ. It is known that the classical limit (I. e., "q=1" limit) of W_{q} (λ) becomes the Weyl module W (λ) over the affine Lie algebra g. Also, it is known that the Weyl module W (λ) is isomorphic (as a module over the Current algebra corresponding to g) to a fusion product of the Weyl modules W (π_I), I=1, . N.
In our previous works, we showed that the crystal B (λ)_{cl} of Lakshmibai-Seshadri paths (modulo the null root δ of g) of shape λ is isomorphic as a crystal to the crystal basis of the quantum Weyl module W_{q} (λ). Moreover, we showed that the crystal B (λ)_{cl} is isomorphic as a crystal to a tensor product B of the crystals B (π_I), I=1, ., n
In our series of works from 2005 to 2007, we defined a certain (nonnegative) integer-valued function (which we call the degree function) on the crystal B (λ)_{cl} above, and proved that this degree function can be identified (through the isomorphism between B (λ)_{cl} and B) with the "energy function" on B, which arose from the study of solvable lattice models in statistical mechanics. In particular, by restricting ourselves to the case of affine Lie algebras of type A, we obtain a description of Kostka polynomials in terms of Lakshmibai-Seshadri paths.
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