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

In this paper, we study the ring of invariants under the action of SL(m, K) x SL(n, K) and SL(m, K) x SL(n, K) x SL(2, K) on the 3-dimensional tensor of indeterminates of form m x n x 2, where K is an infinite field. We show that if m = n >= 2, then the ring of SL(n, K) x 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) x SL(n, K)-invariants and SL(m, K) x SL(n, K) x SL(2, K)-invariants completely in the case where m not equal n. (C) 2016 Elsevier Inc. All rights reserved.