A Finiteness Property of Torus Invariants

TL;DR

This paper investigates the structure of invariant subrings under torus actions on multi-homogeneous polynomial rings, establishing a uniform bound on the degrees of generators of relations across all dimensions.

Contribution

It introduces a new approach using weight matrices to prove a finiteness property of relations in torus invariants, showing they are generated in bounded degree regardless of the number of tensor factors.

Findings

Existence of a uniform degree bound for relations in invariant subrings

Relations are generated in multi-homogeneous degree ≤ d_1 for all n

Provides a matrix-based method to analyze invariants under torus actions

Abstract

In this paper the invariant subring $R_n$ of an algebraic torus $T=(\mathbb{C}^\times)^r$ acting on the multi-homogenous polynomial ring $$S^{\boxtimes n}=\bigoplus_{d=0}^\infty (S^{(d)})^{\otimes n},$$ where $S^{(d)}$ is the $d$th graded piece of the polynomial ring $S=\mathbb{C}[x_1,\dots,x_k]$, is studied from the viewpoint of matrices whose entries sum to zero. Using these weight matrices we prove that there exists a $d_1$ such that for all positive integers $n$, the relations of the invariant subring $R_n$ are generated in multi-homogenous degree $\leq d_1$. Grant: 0943832