Project/Area Number  08454001 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
Algebra

Research Institution  HOKKAIDO UNIVERSITY 
Principal Investigator 
YOSHIDA Tomoyuki Hokkaido Univ., Fac.Research Science, Prof., 大学院・理学研究科, 教授 (30002265)

CoInvestigator(Kenkyūbuntansha) 
BANNAI Eiichi Kyushu Univ., Fac.Research Math., Prof., 大学院・数理学研究科, 教授 (10011652)
TUJISITA Toru Fac.Research Science, Hokkaido Univ., Prof., 大学院・理学研究科, 教授 (10107063)
YAMAMADA Hirofumi Fac.Research Science, Hokkaido Univ., Assoc.Prof., 大学院・理学研究科, 助教授 (40192794)
NAKAMURA Iku Fac.Research Science, Hokkaido Univ., Prof., 大学院・理学研究科, 教授 (50022687)
SAITO Mutsumi Fac.Research Science, Hokkaido Univ., Assoc.Prof., 大学院・理学研究科, 助教授 (70215565)

Project Fiscal Year 
1996 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥5,200,000 (Direct Cost : ¥5,200,000)
Fiscal Year 1996 : ¥5,200,000 (Direct Cost : ¥5,200,000)

Keywords  finite group / Burnside ring / crossed Gset / Mackey functor / Frobenius theorem / TQFT / monoidal category / modular representaion / Burnsidering / Macher functor / Frohemins theoren / modnlar representation / modular representation 
Research Abstract 
The purpose of this reserch was the study of classical problems for discrete groups and its applications. In this research, the investigators obtained the following results. These results will be arranged and published in order. 1. On the crossed Burnside ring of a finite group, (a) we discovered its relation with the quantum double of the group algebra ; (b) we proved the fundamental theorem (an embedding into the products of some group alegabras) ; (c) we obtained an idempotent formula and applied it to the classical problems. We have arranged them as a preprint (Crossed Gsets and crossed Burnside rings) and gave lectures on them in some conferences (Seattle, Yamagata, Kusatsu). 2. On a relationship between our classical problems and Topological Quantum Field Theory (TQFT), we checked that DijkgraafWitten invariants are, in some cases, almost algebraic integers. For example, the invariant for a 3torus is surely a rational integer. Furthermore, we have a weak result for cyclic gauge group case ; however, in this case, the original conjecture had to be revised. These statements will be found in the proceeding of Symposium on Algebra held in Yamagata. 3. We obtained many important results on Schur functions, especially a deep connection with affine Lie algebras. These results was expressed in a conference on combinatorics held in Mineapolis. 4. Investigators have a lot of results in some other area which related with our project : ring theory, real algebraic geometry, theory of monoidal categories (Kumamoto), a relationship between dynamical system and intuitional logic (Sapporo). 5. Using the funds for equipment, we purchased a workstation and a personal computer, which were used to run some formula manipulation programs (GAP,Mathematica).
