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

• 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

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

2003-2006年，在“抛物线Kostka多项式、箭袋簇、晶体基和热带组合”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，组织并参加了多次国际国内会议会议和研讨会，定期参加联合讨论和合作。2003 年项目“抛物线 Kostka 多项式， “箭袋品种、晶体基和热带组合学”是“国际量子上同调研讨会”，将于 2003 年 6 月 16 日至 19 日在京都大学 RIMS 举行，由 A.N.Kirillov 和 M.Guest（东京都立大学）组织。聚集了该领域的几位领先专家，例如 H.Nakajima (Kyodai)、K.Saito (RIMS)、B.Kim 教授。 （韩国）、A.-L.Mare（加拿大）、A.Buch（瑞典）以及许多（约 50 名）国内参与者。 2004 年项目“Parab … More olic Kostka”的主要活动多项式、箭袋簇、晶体基和热带组合”是8月8日至22日举行的国际研讨会“热带代数几何和热带组合”， RIMS，京都大学，由 A.N.Kirilov 和 M.Noumi 组织，该研讨会聚集了几位“热带数学”领域的顶尖专家，例如 A.Knutson 教授（美国加州大学伯克利分校）、E.Miller 教授（美国明尼苏达大学）。 ）、G.Mikhalkin（加拿大多伦多大学）、D.Speyer（美国加州大学伯克利分校）、O.Viro（瑞典乌普萨拉大学）、 M.Kashiwara (RIMS)、M.Okado (Osaka)、Y.Yamada (Kobe) 以及许多（近 60 名）国内参与者。这两场研讨会都被认为是成功的，并引起了人们对热带数学和热带数学的浓厚兴趣。日本的量子上同调。在该项目的主体部分，A.N.Kirillov 作为受邀演讲者参加了“组合学、特殊函数和物理”国际研讨会， 2004”于8月2-4日在中国南开大学召开，并召开了多个国内会议。项目正文中A.N.Kirillov获得的关于抛物线Kostka多项式的部分基本结果已发表在RIMS出版物第40卷上。特别是，本文包含了所谓的广义饱和猜想的证明，以及抛物线 Kostka 多项式的几个新的有趣属性的证明， Schur 函数等。A.N.Kirillov 和 T.Maeno 获得并发表了关于舒伯特微积分和非交换微分微积分之间联系的几个重要结果。特别是，我们描述了一些非交换代数簇的平面连接生成的代数，并证明非交换 B_n Schubert 多项式的 Monk 公式。

项目成果

Wonderful Amoebas

奇妙的阿米巴原虫

• 发表时间: 2006
• 期刊: Sugaku 58
• 作者: Kirillov.A.N.;T.Maeno
• 通讯作者: T.Maeno

On some quadratic algebras, II

关于一些二次代数，II

• 期刊: Preprint (submitted)
• 作者: Kirillov.A.N.;van Diejen F.;Hiroyuki Ochiai;Kirillov.A.N.
• 通讯作者: Kirillov.A.N.

Elementary Askey-Wilson functions

基本 Askey-Wilson 函数

• 期刊: (submitted)
• 作者: Kirillov.A.N.;van Diejen F.
• 通讯作者: van Diejen F.

On some noncommutative algebras related with K-theory of flag varieties

与旗簇K理论相关的一些非交换代数

• 发表时间: 2005
• 期刊: International Research Mathematical Notices 60
• 作者: Kirillov.A.N.;T.Maeno
• 通讯作者: T.Maeno

Torus knot and minimal model

环面结和最小模型

• 发表时间: 2003
• 期刊: Physics Letters B 575
• 作者: Kirillov.A.N.;K.Hikami
• 通讯作者: K.Hikami