Decomposing modular tensor products: `Jordan partitions', their parts and p-parts

Abstract

Determining the Jordan canonical form of the tensor product of Jordan blocks has many applications including to the representation theory of algebraic groups, and to tilting modules. Although there are several algorithms for computing this decomposition in literature, it is difficult to predict the output of these algorithms. We call a decomposition of the form $J_r\otimes J_s=J_{{\lambda }_1}\oplus \cdots \oplus J_{{\lambda }_b}$ a `Jordan partition'. We prove several deep results concerning the $p$-parts of the ${\lambda }_i$ where $p$ is the characteristic of the underlying field. Our main results include the proof of two conjectures made by McFall in 1980, and the proof that $\mathrm{lcm}(r,s)$ and $\gcd ({\lambda }_1,\dots ,{\lambda }_b)$ have equal $p$-parts. Finally, we establish some explicit formulas for Jordan partitions when $p=2$.

arXiv commentOld Theorems 14 and 16 in v1 have been shortened and combined using comments by M.J.J. Barry. A reference of Gow and Laffey added, and an additional ARC grant acknowledgement

References in corpus (2)

• On the Jordan decomposition of tensored matrices of Jordan canonical forms
• Decomposing modular tensor products, and periodicity of `Jordan partitions'

Cited by in corpus (1)

• Decomposing modular tensor products, and periodicity of `Jordan partitions'