A Finiteness Property of Torus Invariants

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 $$S^{⊠n}=\underset{d=0}{\overset{\infty }{⨁}}(S^{(d)}{)}^{\otimes n},$$ 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 $\le d_1$. Grant: 0943832

6 pages