A new algorithm for decomposing modular tensor products

Abstract

Let $p$ be a prime and let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field $F$ of characteristic $p$. For positive integers $r$ and $s$ with $r \leq s$, the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has the form $J_{\lambda_1} \oplus J_{\lambda_2} \oplus \dots \oplus J_{\lambda_{r}}$ where $\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_{r}>0$. This decomposition determines a partition $\lambda(r,s,p)=(\lambda_1,\lambda_2,\dots, \lambda_{r})$ of $r s$ but the values of the parts depend on $r$, $s$, and $p$. Shifting focus to the multiplicities of the parts of $\lambda(r,s,p)$, write \[(\lambda_1,\lambda_2,\dots, \lambda_{r})=(\overbrace{\mu_1,\dots,\mu_1}^{n_1},\overbrace{\mu_2,\dots,\mu_2}^{n_2},\dots, \overbrace{\mu_k,\dots,\mu_k}^{n_k}) =(n_1 \cdot \mu_1, \dots,n_k \cdot \mu_k),\] where $\mu_1>\mu_2>\dots>\mu_k>0$. Then $c(r,s,p)=(n_1,\dots,n_k)$ is a composition of $r$, from which $\lambda(r,s,p)$ can be computed easily. We present a new bottom-up algorithm for computing $c(r,s,p)$ directly from the base-$p$ expansions for $r$ and $s$.