Combinatorial aspects of representations of groups and algebras
Project/Area Number  09640001 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Algebra

Research Institution  Hokkaido University 
Principal Investigator 
YAMADA Hirofumi Grad.School of Science, Hokkaido Univ., Asso.Prof., 大学院・理学研究科, 助教授 (40192794)

CoInvestigator(Kenkyūbuntansha) 
NAKAJIMA Tatsuhiro Faculty of Economics, Meikai Univ., Lec., 経済学部, 講師 (00286006)
SGIBUKAWA Youichi Grad.School of Science, Hokkaido Univ., Inst., 大学院・理学研究科, 助手 (90241299)
SAITO Mutsumi Grad.School of Science, Hokkaido Univ., Asso.Prof., 大学院・理学研究科, 助教授 (70215565)
YAMASHITA Hiroshi Grad.School of Science, Hokkaido Univ., Asso.Prof., 大学院・理学研究科, 助教授 (30192793)
TERAO Hiroaki Grad.School of Science, Tokyo Metropolitan Univ., Prof., 大学院・理学研究科, 教授 (90119058)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,400,000 (Direct Cost : ¥2,400,000)
Fiscal Year 1998 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1997 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  affine Lie algebras / Schur functions / decomposition matrices / complex reflection groups / アフィンリー環 / シューア函数 / 分解行列 / 複素鏡映群 / シューア函教 / ウエイトベクトル 
Research Abstract 
I focused on a relationship of Schur's Qfunctions and affine Lie algebras. First I found that the Qfunctions, expressed as polynomials of power sum symmetric functions, form a weight basis for the basic representation of certain affine Lie algebras, realized on a polynomial ring. Qfunctions are parametrized by the strict partitions. Using some combinatorics of Young diagrams, I determined the weight of the given Qfunction. This procedure was applied to the simplest affine lie algebra $A^{(1)}_1$ to find an identity satisfied by Schur functions and Qfunctions indexed by some specific partitions. At first this identity seemed funny : However this was proved to be true by making use of decomposition matrices of the spin representations of the symmetric group. By virtue of this fact, I turned to a study of the decomposition matrices themselves. As a first result I proved that the determinant of the decomposition matrix of the spin representations is equal to a power of two when the characteristic equals two. Another feature of my research is the so called "higher Specht polynomials" for the complex reflection group G(r, p, n). The group G(r, p, n) acts on the polynomial ring of n variables. The "coinvariant ring" is the quotient by the ideal which is generated by invariants over the group. It is known that the action of G(r, p, n) on this coinvariant ring is isomorphic to the regular representation. The higher Specht polynomials appear naturally as basis vectors of each irreducible component.

Report
(4results)
Research Output
(8results)