日本筑波大学数学系佐垣大辅 (Sagaki Daisuke) 教授将在2014年12月22日至 26日在同济大学数学系致远楼107教室作关于《路径模型引论 (Introduction to path models)》的系列讲座, 每次二小时左右, 热烈欢迎有兴趣的同行参加, 具体日程安排如下:

12月22，23，25，26日 (星期一，二，四，五) 下午14:00-16:30

12月24日(星期三) 上午9:30-12:00

题目: Introduction to path models

摘要: First, I'd like to explain Littelmann's path models. Let be a symmetrizable Kac-Moody algebra (such as, etc.). In the papers "A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras (1994)" and "Paths and root operators in representation theory (1995)", Littelmann introduced Lakshmibai-Seshadri (LS) paths of shape λ (where λ is an integral weight for), and gave the set of them a crystal structure in terms of his root operators; Kashiwara and Joseph proved independently that if λ is dominant, then the crystal of LS paths of shape λ is isomorphic to the crystal basis of the integrable highest weight module of highest weight λ. Using Littelmann's path model (consisting of LS paths), we can describe

1) the decomposition rule of the tensor products of integrable highest weight modules;

2) the branching rule of integrable highest weight modules, regarded as a module over a Levi subalgebra of by restriction;

3) crystal bases of Demazure submodules (Demazure crystals) in terms of “initial direction” of LS paths.

Next, I’ll explain my joint works with Satoshi Naito: From now on, assume that is an affine Lie algebra, and let(resp.) be the quantum affine algebra associated to, with (resp., without) the degree operator. Naito and I showed that if λ is a “level-zero” integral weight (note that such an integral weight is neither dominant nor antidominant), then the crystal of LS paths of shape λ is related to

a) an extremal weight module over(which is neither highest nor lowest);

b) a tensor product of Kirillov-Reshetikhin (KR) modules over(which is finite- dimensional and irreducible).

Finally, if I have time, I'll also explain quantum LS paths and semi-infinite LS paths, which were introduced in my recent joint work with Lenart, Naito, Schilling, Shimozono, and joint work with Ishii and Naito, respectively. Quantum (resp., semi-infinite) LS paths are defined in terms of the quantum (resp., semi-infinite) Bruhat graph, and the crystal of them is isomorphic to the crystal basis of a tensor product of KR modules (resp., an extremal weight module).

同济大学数学系

2014年12月15日