A NEW ALGORITHM FOR DECOMPOSING MODULAR TENSOR PRODUCTS

Abstract

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 $J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _{r}}$ . This decomposition determines a partition $\lambda (r,s,p)=(\lambda _1,\lambda _2,\ldots , \lambda _{r})$ of $r s$ . Let $n_1, \ldots , n_k$ be the multiplicities of the distinct parts of the partition and set $c(r,s,p)=(n_1,\ldots ,n_k)$ . Then $c(r,s,p)$ is a composition of r. We present a new bottom-up algorithm for computing $c(r,s,p)$ and $\lambda (r,s,p)$ directly from the base-p expansions for r and s.