Action of special linear groups to the tensor of indeterminates, classical invariants of binary forms and hyperdeterminant

TL;DR

This paper investigates the invariants of tensor actions by special linear groups, revealing their structure and generators, and connecting them to classical invariant theory of binary forms.

Contribution

It characterizes the invariant rings under specific group actions on tensors, including generators and relations, extending classical invariant theory results.

Findings

For m=n≥2, the invariant ring is generated by n+1 algebraically independent elements.

The action of SL(2,K) on the invariant ring aligns with classical binary form invariants.

Complete descriptions of invariant rings are provided for cases where m≠n.

Abstract

In this paper, we study the ring of invariants under the action of SL(m,K)\times SL(n,K) and SL(m,K)\times SL(n,K)\times SL(2,K) on the 3-dimensional array of indeterminates of form m\times n\times 2, where K is an infinite field. And we show that if m=n\geq 2, then the ring of SL(n,K)\times SL(n,K)-invariants is generated by n+1 algebraically independent elements over K and the action of SL(2,K) on that ring is identical with the one defined in the classical invariant theory of binary forms. We also reveal the ring of SL(m,K)\times SL(n,K)-invariants and SL(m,K)\times SL(n,K)\times SL(2,K)-invariants completely in the case where m\neq n.