A combinatorial identity for the Jacobian of $t$-shifted invariants

TL;DR

This paper introduces a modified formula for the Jacobian of invariants associated with simple Lie algebras, extending classical results to Takiff Lie algebras, which could impact representation theory and invariant theory.

Contribution

It provides a new combinatorial identity for the Jacobian of $t$-shifted invariants in the context of Takiff Lie algebras, expanding classical invariant formulas.

Findings

New combinatorial identity for Jacobians of $t$-shifted invariants

Extension of classical formulas to Takiff Lie algebras

Potential applications in representation theory

Abstract

Let $\mathfrak g$ be a simple Lie algebra. There are classical formulas for the Jacobians of the generating invariants of the Weyl group of $\mathfrak g$ and of the images under the Harich-Chandra projection of the generators of $ZU(\mathfrak g)$. We present a modification of these formulas related to Takiff Lie algebras.