Primary matrix centralizer formula
= Primary matrix centralizer formula
For $M=\bigoplus_j(F_q[t]/(f^j))^{m_j}$ with $f$ irreducible of degree d and partition $\lambda=(j^{m_j})$, its automorphism group has order $q^{d\sum_i(\lambda'_i)^2}\prod_j\prod_{r=1}^{m_j}(1-q^{-dr})$. The <endomorphism> algebra has dimension $d\sum_{a,b}\min(a,b)m_am_b=d\sum_i(\lambda'_i)^2$; its semisimple quotient is $\prod_jM_{m_j}(F_{q^d})$, and units are precisely lifts of units in that quotient.