A basic relation between invariants of matrices under the action of the special orthogonal group.
Let \(k\) be a field of characteristic zero, let \(n\) be a positive integer, and let \((k)^n_{2n}\) be the space of \(n\)-tuples of \(2n\times 2n\) matrices with entries in \(k\). Consider on \((k)^n_{2n}\) the polynomial function \(Q\) defined by the equality \[ Q(A_1,\dots,A_n)=\text{Pf}_L\bigl((1/2)(A_1-A_1^t),\dots,(1/2)(A_n-A_n^t)\bigl), \] where \(\text{Pf}_L\) is the complete polarization of the Pfaffian of an antisymmetric \(2n\times 2n\) matrix. Then the product \(Q(X_1,\dots,X_n)Q(Y_1,\dots,Y_n)\) can be written as a polynomial in the basic invariants \(\text{tr}(U_{i_1}U_{i_2}\cdots U_{i_d})\), where the factors \(U_i\) are components of the tuple \((X_1,\dots,X_n,Y_1,\dots,Y_n)\) or their transposes. The main result of this paper gives an explicit form of this polynomial.
fundamental theorems of invariant theory
scientific article; zbMATH DE number 5177386