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For a finite field $\mathbf{k}$ with $|\mathbf{k}|=q$, and an object $M$ of a $\mathbf{k}$-linear category, suppose we have finite basis for $\mathrm{End}(M)$. How can we compute the size of $\mathrm{Aut}(M)$? What do we need to know about those basis elements to compute this?

Mike Pierce
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    With a result along these lines (or more precisely, along the lines of this paper for instance), we could produce lower bounds for the size of $\operatorname{Aut}(M)$ using only the dimension of $\operatorname{End}(M)$, assuming that the homs are maps over finite dimensional spaces. – Ben Grossmann Jul 20 '20 at 17:51
  • Otherwise, you are essentially counting solutions to $\det(x_1 f_1 + \cdots + x_n f_n) = 0$, assuming once again that $M$ is a finite dimensional space. – Ben Grossmann Jul 20 '20 at 17:55

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