Co-operative Research in Representation Theory

Toshiyuki Tanisaki和Masaaki Kashiwara成功地证明了Kac-Moody李代数的不可约最高权表示特征的Kazhdan-Lusztig猜想。这就完成了所谓的Lusztig程序，该程序旨在同时证明Kac-Moody李代数、半单代数群和参数为单位根的量子群的Kazhdan-Lusztig猜想。(这三个猜想的等价性已经被其他人证明了。)Eiichi Bannai和Etsuko Bannai在代数水平上证明了由Hamming结合方案构造的融合代数的模不变性。Bannai和Michio Ozeki还证明了通过将Jacobi theta函数化为某些有限酉映射群的不变多项式，可以得到模形式。班奈和他的合作者的这些研究开辟了代数组合学的一个新的研究领域。竹内光弘引入了量子群的Q-表示的概念，并研究了量子特殊线性群的Q-表示。武口还刻画了量子一般线性群的所有自同构，以及量子特殊线性群的所有自同构和自同态。吉田智之研究了有限群之间的内胚射的数目。他的一个结果说，从有限交换群A到有限群G的自同态的个数可以被A和G阶的最大公约数整除。许多其他显著的结果发表在1995年7月在东京大学举行的第12届代数组合学研讨会论文集上。

Toshiyuki Tanisaki, with Masaaki Kashiwara, succeeded in proving the Kazhdan-Lusztig type conjecture for the characters of irreducible highest weight representations of Kac-Moody Lie algebras. This completed the so-called Lusztig program, which intended to prove Kazhdan-Lusztig type conjectures for Kac-Moody Lie algebras, semisimple algebraic groups, and quantum groups whose parameters are roots of unity, simultaneously. (The equivalence of these three conjectures has already been proved by other people.) Eiichi Bannai, with Etsuko Bannai, proved the modular invariance property of the fusion algebras at algebraic level constructed from Hamming association scheme. Bannai, with Michio Ozeki, also showed that one can get modular forms by putting Jacobi theta functions into invariant polynomials of certain finite unitary reflection groups. These investigations of Bannai and his collaborators are opening a new reserch area of algebraic combinatorics. Mitsuhiro Takeuchi introduced the notion of q-representations of quantum groups, and studied q-representations of quantum special linear groups. Takeuchi also described all the automorphisms of the quantum general linear groups, and all the automorphisms and endomorphisms of quantum special linear groups. Tomoyuki Yoshida studied the number of endomorhisms between finite groups. One of his results says that the number of endomorphisms from a finite abelian group A to a finite group G is divisible by the greatest common divisors of the orders of A and G.Many of other remarkable results are reported in the Proceedings of the 12th Symposium in Algebraic Combinatorics held at Tokyo University in July, 1995.