When is the ring of $T$ invariants of the homogeneous coordinate ring of $G/B$ a polynomial algebra- connection with the Coxeter elements

TL;DR

This paper investigates conditions under which the ring of invariants of the homogeneous coordinate ring of a flag variety is polynomial, linking the algebraic structure to Coxeter elements and dominant characters.

Contribution

It establishes criteria involving Coxeter elements and dominant characters that determine when the invariant ring is a polynomial algebra.

Findings

Invariant ring is polynomial under certain Coxeter element conditions.

Dimension inequalities of global sections characterize the polynomial nature.

Results apply to indecomposable dominant characters of maximal tori.

Abstract

In this article, we prove that for any indecomposable dominant character of a maximal torus $T$ of a simple adjoint group $G$ such that there is a Coxeter element $w \in W$ for which $X(w)^{ss}_T(\mathcal L_\chi) \neq \emptyset$. If further, for any dominant character $\chi_1$ of $T$ such that $\chi_1\lneqq \chi$ with respect to the dominant ordering, $dim(H^0(G/B, \mathcal L_{\chi_1})^T) < dim (H^0(G/B, \mathcal L_\chi)^T)$, then the graded algebra $\oplus_{d \in \mathbb Z_{\geq 0}}H^0(G/B, \mathcal L_\chi^{\otimes d})^T$ is a polynomial ring in $r$ variables where $r\geq 2$.