基于广义切比雪夫多项式的正则分量簇乘

Graded Lie algebras and regular prehomogeneous vector spaces with one-dimensional scalar multiplication

The aim of this paper is to study relations between regular reductive PVs with one-dimensional scalar multiplication and the structure of graded Lie algebras. We will show that the regularity of such PVs is described by an $\mathfrak{sl}_2$-triplet of a graded Lie algebra.

Multiplication formulas and semisimplicity for q-Schur superalgebras

We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for q-Schur superalgebras. This gives a combinatorialisation of the relative norm approach developed by the first two authors. We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for q-Schur superalgebras. We also construct infinitesimal and little q-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.

温和代数中簇特征子的乘法公式

A multiplication formula for cluster characters in gentle algebras

本文证明了温和代数中由生成扩张诱导的簇特征子满足乘法公式，推广了已有结果，并为带主系数的簇代数交换关系提供了表示论解释，同时应用于正多边形特殊三角剖分产生的代数，联系了B型与A型簇变量。

The regular representation of $U_v(\mathfrak{gl}_{m|n})$

Using quantum differential operators, we construct a super representation of $U_v(\mathfrak{gl}_{m|n})$ on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a $U_v(\mathfrak{gl}_{m|n})$-submodule isomorphic to the regular representation of $U_v(\mathfrak{gl}_{m|n})$. In this way, we obtain a presentation of $U_v(\mathfrak{gl}_{m|n})$ by a basis together with explicit multiplication formulas of the basis elements by generators.

queer q-Schur 超代数中的一些乘法公式

Some multiplication formulas in queer $q$-Schur superalgebras

本文推导了 queer q-Schur 超代数标准基元素与偶、奇生成元的乘法公式，并利用 Hecke--Clifford 代数中的交换公式解决技术难点，为构造量子 queer 超群新实现及单位根处的 Schur--Olshanski 对偶应用奠定基础。

由簇管嵌入诱导的广义簇代数同态

Generalized Cluster Algebra Homomorphisms Induced by Cluster Tube Embeddings

该研究关联仿射簇代数正则模乘法公式与虚交换关系，引入参数z定义簇管加权特征，证明非齐次管嵌入诱导仿射簇代数单同态，为广义交换参数提供表示论解释。