Primary matrix centralizer formula

For $M={⨁}_j(F_q[t]/(f^j){)}^{m_j}$ with $f$ irreducible of degree d and partition $\lambda =(j^{m_j})$, its automorphism group has order $q^{d{\sum }_i({\lambda }_i^{\prime }{)}^2}{\prod }_j{\prod }_{r=1}^{m_j}(1-q^{-dr})$. The endomorphism algebra has dimension $d{\sum }_{a,b}\min (a,b)m_a{m}_b=d{\sum }_i({\lambda }_i^{\prime }{)}^2$; its semisimple quotient is ${\prod }_j{M}_{m_j}(F_{q^d})$, and units are precisely lifts of units in that quotient.

Fix an irreducible polynomial f of degree d with nonzero constant term. In $G{L}_{dm}(q)$, the number of matrices whose minimal polynomial is a power of f is $|G{L}_{dm}(q)|q^{d(m^2-m)}/|G{L}_m(q^d)|$. The primary matrix centralizer formula is identical to the unipotent centralizer formula over $F_{q^d}$, and summing reciprocal centralizer orders reduces to counting nilpotent matrices by the Fitting decomposition.