2011年12月10日  在华东师范大学数学系举行
9:30 - 10:30  芮和兵  华东师范大学  Birman-Murakami-Wenzl 代数的奇异参数
10:40 - 11:40  筱田健一  上智大学  Gelfand-Graev 表示的自同态代数的特征标
13:30 - 14:30  付  强  同济大学  改良量子仿射gln的Z形式
14:40 - 15:40  坂内英一  上海交通大学  关于各种情形下的t-设计与紧t-设计

2011年12月11日  在同济大学数学系举行
9:00 - 10:00  苏育才  同济大学  某些Z-阶化李代数的拟有限表示
10:10 - 11:10  和田坚太郎  信州大学  待定
11:20 - 12:20  玛﹒娜丽莎 阿巴拉  菲律宾大学  Gauss 和与对称矩阵方程的解的个数
14:00 - 15:00  张红莲  上海大学  经过有限群和Mckay对应的双参数量子顶点表示
15:10 - 16:10  万金奎  北京理工大学  Frobenius 特征标公式和Hecke-Clifford 代数的旋一般次数
16:20 - 17:20  宮地兵衛  名古屋大学  与Hecke代数相伴的拟遗传覆盖的基

Hebing Rui                                         Jiachen Ye
Department of Mathematics             Department of Mathematics
East China Normal University      Nagoya University                   Tongji University
hebingrui@gmai.com             shoji@math.nagoya-u.ac.jp            jcye@tongji.edu.cn

Hebing Rui (East China Normal Univ.)  Singular parameters for Birman-Murakami-Wenzl algebra

Koenig and Xi defined the notion of singular parameters for Brauer algebras. They proved that Brauer algebra is Morita equivalent to the direct sum of certain group algebras of symmetric groups if the corresponding parameter is not singular. However, there is no criterion to determine whether a parameter is singular or not. In this talk, I will classify the singular parameters for Birman- Murakami-Wenzl algebra. As the Brauer algebra is the classical limit of Birman -Murakami-Wenzl algebra, we also give the classification of singular parameters for Brauer algebras. This is a joint work with Mei Si.

Ken-ichi Shinoda (Sophia Univ.)  Characters of Endomorphism Algebras of Gelfand-Graev Representations

I will review the characters of endomorphism algebras of Gelfand-Garaev representations of finite reductive groups, focusing on the relation with Gauss sums over finite groups.

Qiang Fu (Tongji Univ.)  The $\mathcal{Z}$-form for the modified quantum affine $\frak{gl}_n$

Let ${\boldsymbol{\mathfrak{D}}_{\!\vartriangle\!}}(n)$ be the double Ringel-Hall algebra of the cyclic quiver $\triangle(n)$ and let $\dot {\boldsymbol{\mathfrak{D}}_{\!\vartriangle\!}}(n)$ be the modified quantum affine algebra of ${\boldsymbol{\mathfrak{D}}_{\!\vartriangle\!}}(n)$. We will study the $\mathcal{Z}$-form for $\dot{\boldsymbol{\mathfrak{D}}_ {\!\vartriangle\!}}(n)$, where $\mathcal{Z}=3D\mathbb{Z}[v,v^{-1}]$.

Eiichi Bannai (Shanghai Jiaotong Univ.)  On t-designs and tight t-designs in various situations

The concept of $t$-design is a way to measure how well a subset approximates the whole set. This applies for combinatorial $t$-design (i.e., $t-(v,k,\lambda)$ design in classical design theory in combinatorics) as well as for spherical $t$-designs. Tight $t$-designs are those $t$-designs with the size attaining a natural (representation theoretical) lower bound. The first purpose of this talk is, aiming at non-specialists with various mathematical background, to explain $t$-designs and the classification problem of tight $t$-designs in various situations, including in $Q$-polynomial association schemes, in spheres and compact symmetric spaces of rank one, real Euclidean spaces, real hyperbolic spaces, etc.. The second purpose of this talk is to discuss the following two remarks. (i)  "The relation between spherical $t$-designs and Euclidean $t$-designs" has close resemblance with "the relation between combinatorial $t$-designs and combinatorial $t$-designs with different block sizes allowed". (Cf. Delsarte-Seidel, 1989.)
(ii) Generalizing above remark (i) which can be regarded as the case of binary Hamming association schemes $H(n,2)$, we see that the "relation between spherical $t$-designs and Euclidean $t$-designs" has close resemblance with the "relation between $t$-designs in a $Q$-polynomial association schemes and relative $t$-designs (in the sense of Delsarte 1977) in the $Q$-polynomial association scheme". We discuss some new results as well as some conjectures which may have some representation theoretical flavor.

Toshiaki Shoji (Nagoya Univ.) Symmetric space over a finite field and Kato's exotic nilcone

It is known by Bannai-Kawanaka-Song that the representation theory of the Hecke algebra $H(GL_{2n}(F_q), Sp_{2n}(F_q))$ associated to a finite symmetric space $(GL_{2n}/Sp_{2n}) (F_q)$ is closely related to the representation theory of $GL_n(F_q)$  due to Green.  Green's work was reconstructed geometrically by Lusztig in terms of his theory of character sheaves, and the geometric approach for $GL_{2n}/Sp_{2n}$ is also studied by Henderson. On the other hand, the theory of character sheves on $GL_n$ can be extended to the theory on $GL_n \times V$, where $V$ is a natural $GL_n$-module.  In this talk , we discuss the theory of character sheaves on $(GL_{2n}/Sp_{2n}) \times V$, where $V$ is a natural $Sp_{2n}$-module. In this setting, the role of unipotent variety for $GL_n$ is played by Kato's exotic nilcone. We prove the exotic Springer correspondence (originally proved by Kato by a different method)  based on the theory of character sheaves.

Yucai Su (Tongji Univ.)  Quasi-finite representations of some $\mathbb{Z}$-graded Lie algebras

In this talk, we present some results on quasi-finite representations of some $\mathbb{Z}$-graded Lie algebras (in particular those Lie algebras which contain the Virasoro algebra as a subalgebra). These Lie algebras include: the generalized Virasoro algebras; the generalized Block type Lie algebras; the W-infinity algebras (or Lie algebras of differential operators on the circle); and the Lie algebra which is the core of the extended affine Lie algebras of type $A_1$ with coordinates in rank 2 quantum torus. This talk is based on several joint works with coauthors.

Ma. Nerissa M. Abara (Univ. of the Philippines )  Gauss sums and number of solutions to a symmetric matrix equation

Let $F_q$ denote a finite field with $q = p^d$ elements, where $p$ is a prime different from 2 and let $\Lambda_2$ be the set of all 2-by-2 symmetric matrices with entries from $F_q$. We consider the number of equation \begin{equation*} X_1^2 + \cdots + X_m^2 = B, \end{equation*} where $X_i, B \in \Lambda_2$.  We classify the different quadratic forms Tr$(AX^2)$ under isometry and consider the sum $\displaystyle\sum_{X \in \Lambda_2}e\text{Tr}(AX^2)$, where $e(\alpha) = e^{\frac{2\pi i}{p}\text{tr}_{F_q/F_p}(\alpha)}$. The sum $e(\text{Tr}(-AB))$ will be investigated over all possible Jordan canonical forms of $B$ and Gauss sums on $F_q$ and lifted Gauss sums on $F_{q^2}$ will be used to obtain these sums.

Honglian Zhang (Shanghai Univ.)  Two-parameter quantum vertex representations via finite groups and the Mckay correspondence

In this talk, We provide a group theoretic realization of two-parameter quantum toroidal algebras using finite subgroups of $SL_2(\mathbb C)$ via McKay correspondence. In particular our construction contains the vertex representation of the two-parameter quantum affine algebras of $ADE$ types as special subalgebras. This is a joint works with Prof. N. Jing

Jinkui Wan (Beijing Institute of Technology)  Frobenius character formula and spin generic degrees for Hecke-Clifford algebra

The spin analogues of several classical concepts and results for Hecke algebras are established. A Frobenius type formula is obtained for irreducible characters of the Hecke-Clifford algebra. A precise characterization of the trace functions allows us to define the character table for the algebra. The algebra is endowed with a canonical symmetrizing trace form, with respect to which the spin generic degrees are formulated and shown to coincide with the spin fake degrees. We further provide a characterization of the trace functions and the symmetrizing trace form on the spin Hecke algebra which is Morita super-equivalent to the Hecke-Clifford algebra. This is a joint work with Weiqiang Wang.

Hyohe Miyachi (Nagoya Univ.)  Bases for quasihereditary covers associated with Hecke algebras

In my talk in Shanghai, I discussed about a construction of quasihereditary covers in finite dimensional contexts for Iwahori-Hecke algebras of type $A_n, B_3, B_4, D_4, D_5, F_4$ and $E_6$  and equivalences between them and the category $\mathcal{O}$'s of rational Cherednik algebras. At that time, the construction, especially, to show the quasihereditary, was a hand made one by looking at the composition series. Following G. Lusztig, R. Curtis, J. Du and M. Geck (historical order), I will try to have nicer bases to get the desired equivalences.

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