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)×SL(n,K) and SL(m,K)×SL(n,K)×SL(2,K) on the 3-dimensional tensor of indeterminates of form m×n×2, where K is an infinite field. We show that if m=n≥2, then the ring of SL(n,K)×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)×SL(n,K)-invariants and SL(m,K)×SL(n,K)×SL(2,K)-invariants completely in the case where m≠n.