Study on 1-dimensional sums and special polynomials in mathematical physics in terms of path models

• Project/Area Number: 19740004
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: University of Tsukuba
• Principal Investigator: SAGAKI Daisuke University of Tsukuba, 大学院・数理物質科学研究科, 講師 (40344866)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥4,020,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥720,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2007: ¥900,000 (Direct Cost: ¥900,000)
• Keywords: 量子アフィン代数 / 結晶基底 / パス模型 / Kirillov-Reshetikhin加群 / Mirkovic-Vilonen多面体 / 代数一般 / 表現論 / Kirillov-Reshetikhin 加群 / Mirkovic-Vilonen 多面体

Research Abstract

In the theory of Mirkovic-Vilonen (MV for short) polytopes, we obtain the following three results.
(1) Let P and Q be MV polytopes. We have proved that the MV polytope corresponding to the tensor product of P and Q is contained in the Minkowski sum of P and Q.
(2) We have described the crystal bases of Demazure modules and opposite Demazure modules in terms of MV polytopes.
(3) We have extended the theory of MV polytopes to the case of the quantum group associated to the nontwisted affine Lie algebra of type A.

Research Products

• [Journal Article] Polytopal estimate of Mirkovic-Vilonen polytopes lying in a Demazure crystal (2011)
  • Author(s): Syu Kato, Satoshi Naito, Daisuke Sagaki
  • Journal Title: Advances in Mathematics Volume 226 Pages: 2587-2617
  • Peer Reviewed
• [Journal Article] Polytopal estimate of Mirkovic-Vilonen polytopes lying in a Demazure crystal (2011)
  • Author(s): Syu Kato, Satoshi Naito, Daisuke Sagaki
  • Journal Title: Advances in Mathematics Volume: 226 Pages: 2587-2617
  • Peer Reviewed
• [Journal Article] Crystal base elements of an extremal weight module fixed by a diagram automorphism II : case of affine Lie algebras (2010)
  • Author(s): Satoshi Naito, Daisuke Sagaki
  • Journal Title: Progress in Mathematics Volume 284 Pages: 225-255
  • Peer Reviewed
• [Journal Article] 量子アファイン展開環上のextremalウェイト加群の結晶基底とLittelmannのパス模型 (2010)
  • Author(s): 内藤聡・佐垣大輔
  • Journal Title: 雑誌「数学」 62巻 Pages: 57-84
  • NAID: 130004558900
  • Peer Reviewed
• [Journal Article] Simplicity of a vertex operator algebra whose Griess algebra is the Jordan algebra of symmetric matrices (2010)
  • Author(s): Hidekazu Niibori, Daisuke Sagaki
  • Journal Title: Communications in Algebra Volume 38 Pages: 848-875
  • Peer Reviewed
• [Journal Article] Crystal base elements of an extremal weight module fixed by a diagram automorphism II : case of affine Lie algebras (2010)
  • Author(s): Satoshi Naito, Daisuke Sagaki
  • Journal Title: Progress in Mathematics Volume: 284 Pages: 225-255
  • Peer Reviewed
• [Journal Article] 量子アファイン展開環上のextremalウェイト加群の結晶基底とLittelmannのパス模型 (2010)
  • Author(s): 内藤聡・佐垣大輔
  • Journal Title: 数学 Volume: 62 Pages: 57-84
  • NAID: 130004558900
  • Peer Reviewed
• [Journal Article] Simplicity of a vertex operator algebra whose Griess algebra is the Jordan algebra of symmetric matrices (2010)
  • Author(s): 新堀秀和・佐垣大輔
  • Journal Title: Communications in Algebras 38 Pages: 848-875
  • Peer Reviewed
• [Journal Article] Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal (2009)
  • Author(s): Satoshi Naito, Daisuke Sagaki
  • Journal Title: Advances in Mathematics Volume 221 Pages: 1804-1842
  • Peer Reviewed
• [Journal Article] Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal (2009)
  • Author(s): 内藤聡・佐垣大輔
  • Journal Title: Advances in Mathematics 221 Pages: 1804-1842
  • Peer Reviewed
• [Journal Article] Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal (2009)
  • Author(s): 内藤聡・佐垣大輔
  • Journal Title: Advances in Mathematics (To appear)
  • Peer Reviewed
• [Journal Article] Simplicity of a vertex operator algebra whose Griess algebra is the Jordan algebra of symmetric matrices (2009)
  • Author(s): 新堀秀和・佐垣大輔
  • Journal Title: Communications in Algebras (To appear)
  • Peer Reviewed
• [Journal Article] Lakshmibai-Seshadri paths of level-zero shape and one-dimensional sums associated to level-zero fundamental representations (2008)
  • Author(s): Satoshi Naito, Daisuke Sagaki
  • Journal Title: Compositio Mathematica Volume 144 Pages: 1525-1556
  • Peer Reviewed
• [Journal Article] A modification of the Anderson-Mirkovic conjecture for Mirkovic-Vilonen polytopes in types B and C (2008)
  • Author(s): Satoshi Naito, Daisuke Sagaki
  • Journal Title: Journal of Algebra Volume 320 Pages: 387-416
  • Peer Reviewed
• [Journal Article] Crystal structure on the set of Lakshmibai-Seshadri paths of an arbitrary level-zero weight shape (2008)
  • Author(s): Satoshi Naito, Daisuke Sagaki
  • Journal Title: Proceeding of London Mathematical Society Volume 96 Pages: 582-622
  • Peer Reviewed
• [Journal Article] Lakshmibai-Seshadri paths of level-zero shape and one-dimensional sums associated to level-zero fundamental representations (2008)
  • Author(s): Satoshi Naito, Daisuke Sagaki
  • Journal Title: Compositio Mathematica Volume144 Pages: 1525-1556
  • Peer Reviewed
• [Journal Article] Lakshmibai-Seshadri paths of level-zero shape and one-dimensional sums associated to level-zero fundamental representations (2008)
  • Author(s): 内藤聡・佐垣大輔
  • Journal Title: Compositio Mathematica 144 Pages: 1525-1556
  • Peer Reviewed
• [Journal Article] Crystal structure on the set of Lakshmibai-Seshadri paths of an arbitrary level-zero weight shape (2008)
  • Author(s): 内藤, 聡・佐垣, 大輔
  • Journal Title: Proceeding of London Mathematical Society To appear
  • Peer Reviewed
• [Journal Article] A modification of the Anderson-Mirkovic conjecture for Mirkovic-Vilonen polytopes in types BandC (2008)
  • Author(s): 内藤, 聡・佐垣, 大輔
  • Journal Title: Journal of Algebra To appear
  • Peer Reviewed
• [Journal Article] Tensor products and Minkowski sums of Mirkovic-Vilonen polytopes
  • Author(s): Syu Kato, Satoshi Naito, Daisuke Sagaki
  • Journal Title: Transformation Groups に掲載予定
  • Peer Reviewed
• [Journal Article] Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of type B∞, C∞, and D∞
  • Author(s): Satoshi Naito, Daisuke Sagaki
  • Journal Title: Transactions of the American Mathematical Society に掲載予定
  • Peer Reviewed
• [Presentation] Tensor products and Minkowski sums of Mirkovic-Vilonen polytopes (2011)
  • Author(s): Daisuke Sagaki
  • Organizer: Trimester Programs "On the Interaction of Representation Theory with Geometry and Combinatorics"
  • Place of Presentation: Hausdorff Institute for Mathematics (ドイツ・ボン大学).
  • Year and Date: 2011-02-28
• [Presentation] Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of type B∞, C∞, and D∞ (2010)
  • Author(s): Daisuke Sagaki
  • Organizer: 国際研究集会「Representation Theory of Algebraic Groups and Quantum Groups 2010」
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-08-06
• [Presentation] Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of type B_∞, C_∞, and, D_∞ (2010)
  • Author(s): Daisuke Sagaki
  • Organizer: Representation Theory of Algebraic Groups and Quantum Groups 2010
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-08-06
• [Presentation] Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of type B∞, C∞, and D∞ (2010)
  • Author(s): 佐垣大輔
  • Organizer: 第13回「代数群と量子群の表現論」
  • Place of Presentation: すいとぴあ江南(愛知県江南市)
  • Year and Date: 2010-06-10
• [Presentation] Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of type B_∞, C_∞, and, D_∞ (2010)
  • Author(s): 佐垣大輔
  • Organizer: 第13回「代数群と量子群の表現論」
  • Place of Presentation: すいとぴあ江南(愛知県江南市)
  • Year and Date: 2010-06-10
• [Presentation] 量子アファイン環のレベル・ゼロ基本表現とそのテンソル積のパス模型 (2009)
  • Author(s): 佐垣大輔
  • Organizer: 2009年度日本数学会秋季総合分科会 特別講演
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-27
• [Presentation] 量子アファイン環のレベル・ゼロ基本表現とそのテンソル積のパス模型 (2009)
  • Author(s): 佐垣大輔
  • Organizer: 2009年度日本数学会秋季総合分科会特別講演
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-27
• [Presentation] 量子アファィン環のレベル・ゼロ基本表現とそのテンソル積のパス模型 (2009)
  • Author(s): 佐垣大輔
  • Organizer: 2009日本数学会秋季総合分科会 特別講演
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-27
• [Presentation] Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal (2009)
  • Author(s): Daisuke Sagaki
  • Organizer: 国際研究集会「International workshop on combinatorial and geometric approach to representation theory」
  • Place of Presentation: Seoul National University (韓国)
  • Year and Date: 2009-09-21
• [Presentation] Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal (2009)
  • Author(s): 佐垣大輔
  • Organizer: International workshop on combinatorial and geometric approach to representation theory
  • Place of Presentation: Seoul National University(韓国)
  • Year and Date: 2009-09-21
• [Presentation] On tensor products of Mirkovic-Vilonen polytopes in type A (2009)
  • Author(s): 斉藤義久, 佐垣大輔 ,内藤聡
  • Organizer: 研究集会「表現論と組合せ論」
  • Place of Presentation: 北海道大学
  • Year and Date: 2009-08-26
• [Presentation] Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal (2008)
  • Author(s): 内藤聡, 佐垣大輔
  • Organizer: 研究集会「組合せ論的表現論の拡がり」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-10-07
• [Presentation] Berenstein-Zelevinsky datum for the affine Lie algebra of type Al^<(1)> (2008)
  • Author(s): Daisuke Sagaki, Satoshi Naito
  • Organizer: 国際研究集会「Crystals and Tropical Combinatorics」
  • Place of Presentation: 関西セミナーハウス(京都)
  • Year and Date: 2008-08-30
• [Presentation] Berenstein-Zelevinsky datum for the affine Lie algebra of type A_{\ell}^{(1)} (2008)
  • Author(s): Daisuke Sagaki, Satoshi Naito
  • Organizer: 国際研究集会「Crystals and Tropical Combinatorics」
  • Place of Presentation: 関西セミナーハウス(京都)
  • Year and Date: 2008-08-30
• [Presentation] Berenstein-Zelevinsky datum for the affine Lie algebra of type A_{\ell}^{(1)} (2008)
  • Author(s): 佐垣大輔(内藤聡氏との連名)
  • Organizer: 研究集会「Crystals and Tropical Combinatoric
  • Place of Presentation: 関西セミナーハウス(京都)
  • Year and Date: 2008-08-30
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  • URL: http://www.math.tsukuba.ac.jp/~sagaki/
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