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Parabolic Kostka polynomials, quiver varieties, crystal bases and tropicalcombinatorics

抛物线 Kostka 多项式、箭袋变量、晶体基和热带组合

基本信息

批准号：
15340006
负责人：
KIRILLOV Anatol N.
金额：
$ 6.53万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340006/
关键词：
Parabolic Kostka polynomials Tropical RSK Amoebas Torus knots Virasoro algebra Nichols-Woronowicz algebras Schubert Calculus Flat connections Parabolic Kostka polynomials Tropical RSK Discrete integrable systems Nichols algebras We

项目摘要

During 2003-2006 years in the course of the Project "Parabolic Kostka polynomials, quiver varieties, crystal bases and tropical combinatorics", the members of the Project have published about 25 papers in the leading mathematical journals, have organized and attended several International and Domestic Conferences and Workshops, participated regularly in the joint discussions and collaboration.The main event of 2003 year term of the Project "Parabolic Kostka polynomials, quiver varieties, crystal bases and tropical combinatorics" was the "International Workshop on Quantum Cohomology" to be held during June 16-19, 2003 at RIMS, Kyoto University, and organized by A.N.Kirillov and M.Guest (Tokyo Metropolitan University). The Workshop collected several leading specialists in the field such as Profs. H.Nakajima (Kyodai), K.Saito (RIMS), B.Kim (S.Korea), A.-L.Mare (Canada), A.Buch (Sweden), as well as many (about 50) domestic participants.The main event of 2004 year term of the Project "Parab … More olic Kostka polynomials, quiver varieties, crystal bases and tropical combinatorics" was the International Workshop "Tropical algebraic geometry and tropical combinatorics" carried out August 8-22, RIMS, Kyoto University, and organized by A.N.Kirillov and M.Noumi. The Workshop collected several leading specialist in "Tropical Mathematics" such as Profs. A.Knutson (UC Berkeley, USA), E.Miller (Univ. of Minnesota, USA), G.Mikhalkin (Toronto Univ., Canada), D.Speyer (UC Berkeley, USA), O.Viro (Uppsala Univ., Sweden), M.Kashiwara (RIMS), M.Okado (Osaka), Y.Yamada (Kobe), as well as many (near 60) domestic participants.The both Workshops were recognized to be successful and gave rise to notable interest to Tropical Mathematics and Quantum Cohomology in Japan.In the body of the Project, A.N.Kirillov attended as invited speaker the International Workshop "Combinatorics, Special Functions and Physics, 2004" held August 2-4, Nankai Univ., China, and several domestic Conferences.Part of basic results about parabolic Kostka polynomials obtained by A.N.Kirillov in the body of project, has been published in the Publications of RIMS, vol. 40. In particular, this paper contains a proof of the so-called Generalized Saturation Conjecture, as well as proofs of several new interesting properties of parabolic Kostka polynomials, Schur functions and so on.Several important results about connections between Schubert Calculus and non-commutative differential calculus have been obtained and published by A.N.Kirillov and T.Maeno. In particular, we described the algebra generated by flat connections for some noncommutative algebraic varieties, and prove Monk formula for noncommutative B_n Schubert polynomials. Less

在2003-2006年期间，在“抛物型Kostka多项式、多变量、晶体基和热带组合数学”项目的过程中，项目成员在主要数学期刊上发表了约25篇论文，组织和参加了若干国际和国内会议和讲习班，定期参加联合讨论和合作。2003年项目“抛物型Kostka多项式，多变量，晶体基和热带组合学”是2003年6月16-19日在京都大学RIMS举行的“量子上同调国际研讨会”，由A.N.Kirillov和M.Guest（东京首都大学）组织。研讨会汇集了该领域的几位主要专家，如教授。H.Nakajima（Kyodai）、K.Saito（RIMS）、B.Kim（韩国）、A.- L.Mare（加拿大），A.Buch（瑞典），以及许多（约50）国内参与者。 ...更多信息 “热带代数几何和热带组合数学”国际研讨会于8月8日至22日在京都大学RIMS举行，由A.N.Kirillov和M. Noumi组织。该讲习班汇集了几个领先的专家在“热带数学”，如教授。A.Knutson（美国加州大学伯克利分校），E.米勒（美国明尼苏达大学），G.Mikhalkin（多伦多大学，加拿大），D.Speyer（美国加州大学伯克利分校），O.Viro（乌普萨拉大学，瑞典），M.Kashiwara（RIMS），M.Okado（Osaka），Y.Yamada（科比），以及许多（近60）国内参与者。这两个讲习班被认为是成功的，并引起了日本对热带数学和量子上同调的显著兴趣。在项目的主体中，A.N.Kirillov作为特邀演讲者参加了国际讲习班“组合学，特殊函数和物理，2004年8月2日至4日在南开大学举行，A. N. Kirillov在项目主体中得到的关于抛物型Kostka多项式的部分基本结果已发表在RIMS出版物第40卷上。特别地，本文证明了所谓的广义饱和猜想，证明了抛物型Kostka多项式、Schur函数等几个新的有趣性质，A. N. Kirillov和T. Maeno得到并发表了关于Schubert微积分与非交换微分学之间联系的几个重要结果。特别地，我们描述了某些非交换代数簇的平坦联络生成的代数，并证明了非交换B_n舒伯特多项式的Monk公式.少