Let $\alpha, \alpha’, \beta, \beta’$ be distinct non-identity elements of $S_n$. Suppose there exists $\tau \in S_n$ such that $\alpha’ = \tau \alpha \tau^{-1}$ and $\beta’ = \tau \beta \tau^{-1}$.
Given only $\alpha$, $\alpha’$, $\beta$, and $\beta’$, how could I find some $\tau’$ with the above property? One method is of course to just generate all witnesses for $\alpha \sim \alpha’$ and $\beta \sim \beta’$ then check for common elements up to inverses but I would like to find a better way.
If there isn’t a simple-ish answer, where can I go to read more about this? I tried reading up in my textbook but there isn’t much information in there.
You could consider the system of linear equations $\tau \alpha - \alpha' \tau$ and $\tau \beta - \beta' \tau$ as well as $(1,\ldots,1) \tau = (1,\ldots,1)$ and $\tau \pmatrix{1\cr \ldots\cr 1} = \pmatrix{1\cr \ldots \cr 1}$ on $\mathbb R^{n \times n}$, and the inequalities $\tau \ge 0$, and solve with linear programming. Basic feasible solutions should correspond to permutation matrices.