# Some properties of the Specht polynomials

Speaker:

Junzo Watanabe, Tokai University

Date and Time:

Thursday, May 18, 2023 - 11:45am to 12:30pm

Location:

Fields Institute, Room 230

Abstract:

The graded Artinian algebra $A$ $=$ $K[x_{1},\cdot\cdot\cdot,x_{n}]/(x_{1}^{2},\cdot\cdot\cdot,x_{n}^{2})$ can have the set of square-free monomials $E$ $=$ $\{$$x_{1}^{i_{1}}$$\cdot\cdot\cdot$$x_{n}^{i_{n}}$ $\mid$ $0$ $\leq$ $i_{j}$ $<$ $2$$\}$ as a basis. $E$ is a graded finite dimensional vector subspace of the polynomial ring $R$.

Let $D$ $=$ $\frac{\partial}{{\partial}x_{1}}$ $+$ $\cdot\cdot\cdot$ $+$ $\frac{\partial}{{\partial}x_{n}}$. Specht polynomials can be used to obtain the Jordan basis of $E$ to represent the map $D$ $:$ $E$ $\rightarrow$ $E$ as a Jordan canonical form and it can be used to to prove the basic theorem which says $A$ has the SLP. In this talk I would like to discuss to what extent this fact can be generalized.