A geometric approach to the Kronecker problem II : rectangular shapes, invariants of n × n matrices, and a generalization of the Artin-Procesi theorem, (under preparation

Abstract. In this paper we describe the ring of invariants of the space ofm-tuples of n×nmatrices, under the action of SL(n)×SL(n) given by (A,B)·(X1, X2, · · · , Xm) 7→ (AX1Bt, AX2Bt, · · · , AXmBt). Determining the ring of invariants is the first step in the geometric approach to finding multiplicities of representations of the symmetric group in the tensor product of rectangular shaped representations. We show that these invariants are given by multi-determinants and can also be described in terms of certain magic squares. We compute the null cone for this action. We also study a birational subring of invariants and an analysis thereof results into a different proof of the Artin-Procesi theorem for the ring of invariants for several matrices under the simultaneous conjugation action of SL(n)