Highest weight vectors and transmutation

TL;DR

This paper constructs finite spanning sets for modules of highest weight vectors in polynomial functions on Lie algebras and matrix tuples, extending results to positive characteristic using transmutation and Howe duality.

Contribution

It provides new finite spanning sets for highest weight modules in characteristic zero and positive characteristic, utilizing transmutation techniques and extending to matrix tuples under diagonal conjugation.

Findings

Finite homogeneous spanning sets for modules of highest weight vectors in characteristic zero.

Extension of results to positive characteristic and matrix tuples.

Introduction of transmutation technique based on Howe duality for simplifying the problem.

Abstract

Let $G={\rm GL}_n$ be the general linear group over an algebraically closed field $k$, let $\mathfrak g=\mathfrak gl_n$ be its Lie algebra and let $U$ be the subgroup of $G$ which consists of the upper uni-triangular matrices. Let $k[\mathfrak g]$ be the algebra of polynomial functions on $\mathfrak g$ and let $k[\mathfrak g]^G$ be the algebra of invariants under the conjugation action of $G$. In characteristic zero, we give for all dominant weights $\chi\in\mathbb Z^n$ finite homogeneous spanning sets for the $k[\mathfrak g]^G$-modules $k[\mathfrak g]_\chi^U$ of highest weight vectors. This result (with some mistakes) was already given without proof by J.~F.~Donin. Then we do the same for tuples of $n\times n$-matrices under the diagonal conjugation action. Furthermore we extend our earlier results in positive characteristic and give a general result which reduces the problem to giving spanning sets of the highest weight vectors for the action of ${\rm GL}_r\times{\rm GL}_s$ on tuples of $r\times s$ matrices. This requires the technique called "transmutation" by R.~Brylinsky which is based on an instance of Howe duality. In the cases that $\chi_{{}_n}\ge -1$ or $\chi_{{}_1}\le 1$ this leads to new spanning sets for the modules $k[\mathfrak g]_\chi^U$.