The number of nonconstant invariant polynomials of matrices with several prescribed blocks (Q1399235)

Let \(C=M_n(F)\) be a square matrix partitioned into \(k\times k\) blocks \([C_{i,j}]\) such that the diagonal blocks \(C_{1,1},\dots,C_{k,k}\) are square. The main theorem of this paper claims that: Let \(F\) be an algebraically closed field, \(\sigma\in S_k\) be a permutation, and \(A_{j,\sigma(j)}\) (\(1\leq j\leq k\)) be given matrices. Then the interval of the number \(i(C)\) of nonconstant invariant polynomials is given where the blocks \(C_{j,\sigma(j)}=A_{j,\sigma(j)}\) of \(C\) are fixed for every \(1\leq j\leq k\).