Combinatorics in representation theory of classical groups and quantum groups, and applications

经典群和量子群表示论中的组合学及其应用

• 批准号: 10440004
• 负责人: OKADA Soichi
• 金额: $ 8.26万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10440004/
• 关键词: representations; classical groups; quantum groups; Young diagrams; Schur functions; RS correspondence; alternating sign matrix; KL conjecture; D加群; 小行列式の和公式; crystal base; representations; classical groups; quantum groups; Young diagrams; Schur functions; RS correspondence; alternating sign matrix; KL conjecture; D加群; 小行列式の和公式; crystal base

In this research project, we obtained the following results.1. Okada obtained explicit branching rules for the tensor products and restrictions of the irreducible representations of the classical groups corresponding to nearly-rectangular shaped Young diagrams. And he proved that the partition functions of the square ice model related to the alternating sign matrices with symmetry can be expressed in terms of the irreducible characters of the classical groups.2. Kashiwara described the crystal bases for the quantum group U_q (gl(m, n))(with G.Benkart and S.Kang). Also, in the study of D-modules on the flag varieties, he proved the Kazhdan-Lusztig conjecture for the affine Lie algebras at a non-critical level (with T.Tanisaki), and showed that the duality for D-modules on the flag variety corresponds to that of Harish-Chandra modules (with D.Bartlet).3. Koike described, in terms of generalized Brauer diagrams, the structure of the centralizer algebra of the spin groups on the tensor product of the basic spin representation and the tensor powers of the natural representation. Also he gave a realization of irreducible representations of the spin groups in the above tensor products.4. Terada gave an geometric interpretation to the Robinson-Schensted correpondence between Brauer diagrams and up-down tableaux. And he constructed an Robinson-Schensted-type bijection for the Weil representation of sp (2n)(with T.Roby).5. Yamada described weight vectors in the basic representations of some affine Lie algebras in terms of symmetric functions (with T.Nakajima), and found an interesting facts on the spin decomposition matrices of the symmetric groups. And He found the Littlewood's multiple formula for the spin irreducible characters of the symmetric groups (with H.Mizukawa).

在该研究项目中，我们获得了以下结果1。冈田获得了张量产品的明确分支规则，并对与几乎矩形的年轻图相对应的经典组的不可约定表示的限制。他证明了与与对称的交替符号矩阵有关的平方冰模型的分区函数可以用经典组的不可约特征来表示。2。 Kashiwara描述了量子组U_Q（GL（M，N））的晶体碱基（带G.Benkart和S.Kang）。此外，在对国旗品种的D模型的研究中，他证明了仿生的Kazhdan-Lusztig猜想是非关键水平（带有T.Tanisaki）的代数，并证明了旗帜上D模型的双重性与Harish-Chandra Modules（与D.Bartlet）相对应。 Koike在广义的Brauer图中描述了自旋组在基本自旋表示的张量和自然代表的张量量的张量产物上的中央式代数的结构。他还实现了上述张量产品中自旋组的不可还原表示。4。 Terada对Brauer图和上向下的Tableaux之间的Robinson-Schensted Correponcence进行了几何解释。他为SP（2n）（带有T.Roby）的Weil表示构建了Robinson-Schensted-type boovion。5。 Yamada用对称函数（使用T.Nakajima）描述了一些仿射代数的基本表示中的重量向量，并在对称组的自旋分解矩阵上找到了一个有趣的事实。他发现了利特伍德（Littlewood）的多重公式，用于对称群体的旋转不可减少字符（与H.Mizukawa）。