Math 6310 • Algebra

Set \[ J= \begin {pmatrix} 0&1&0&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0 \end {pmatrix} \in M_{10}(\mathbb {C}); \] here it is understood that unless otherwise noted, the entries of \(J\) are 0. Let \[ S=\{X\in M_{10}(\mathbb {C})\mid X^2=gJg^{-1}\text { for some }g\in \operatorname {GL}(10,\mathbb {C})\}. \] Observe that \(\operatorname {GL}(10,\mathbb {C})\) acts on \(S\) by conjugation. Compute the number of orbits for this action.