Coincidence of dimension-its representation theoretical aspect, ring theoretical aspect and combinatorial aspect.

2009 Fiscal Year Final Research Report

• Project/Area Number: 19540155
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Muroran Institute of Technology (2008-2009); Oyama National College of Technology (2007)
• Principal Investigator: HIDEAKI Morita Muroran Institute of Technology, 大学院・工学研究科, 准教授 (90435412)
• Project Period (FY): 2007 – 2009
• Keywords: 代数学 / 組合せ論

Research Abstract

A representation theoretical interpretation is obtained for the coincidence of dimension of the singly graded submodules of Garsia-Haiman modules corresponding to hooks. It is also obtained the factorization formula and the plethystic formula for the corresponding Macdonald polynomials at roots of unity.

Research Products

• [Journal Article] Zero-dimensional Gorenstein algebras with the action of the symmetric group (2009)
  • Author(s): H. Morita, A. Wachi, J. Watanabe
  • Journal Title: Rend. Semin. Mat. Univ. Padova 121 Pages: 45-71
  • Peer Reviewed
• [Journal Article] Factorization formulas for Macdonald polynomials (2008)
  • Author(s): F. Descouens, H. Morita
  • Journal Title: Euro. J. Combin 29 Pages: 395-410
  • Peer Reviewed
• [Journal Article] Garsia-Haiman modules for hook partitions and Green polynomials with two variables (2008)
  • Author(s): H. Morita
  • Journal Title: J. Alg 319 Pages: 235-247
  • Peer Reviewed
• [Journal Article] Green polynomials at roots of unity and Springer modules for the symmetric groups (2007)
  • Author(s): H. Morita
  • Journal Title: Adv. Math 212 Pages: 277-292
  • Peer Reviewed
• [Journal Article] Decomposition of Green polynomials of type A and Springer modules for hooks and rectangles (2007)
  • Author(s): H. Morita
  • Journal Title: Adv Math 210 Pages: 479-497
  • Peer Reviewed
• [Presentation] Garsia-Haiman modules for hooks and its graded characters at roots of unity (2008)
  • Author(s): H. Morita
  • Organizer: Combinatorics and Representation Theory
  • Place of Presentation: Nagoya University
  • Year and Date: 2008-09-04
• [Presentation] Garsia-Haiman modules and modified Macdonald polynomials at roots of unity (2008)
  • Author(s): 森田英章
  • Organizer: 第11回代数群と量子群の表現論研究集会
  • Place of Presentation: 岡山大学
  • Year and Date: 2008-05-28
• [Presentation] The differential module of the polynomial ring with the action of the symmetric group (2007)
  • Author(s): 森田英章-和地輝仁-渡辺純三
  • Organizer: 第29回可換環論シンポジウム
  • Place of Presentation: 愛知県厚生年金会館ウェルシティなごや
  • Year and Date: 20071119-20071122
• [Presentation] A combinatorial property of the Garsia-Haiman modules and the Green polynomials with two variables (2007)
  • Author(s): H. Morita
  • Organizer: Northern Workshop in representation theory of Lie groups and Lie algebras
  • Place of Presentation: Hokkaido University
  • Year and Date: 2007-03-09