Bounds for Bilinear Complexity of Noncommutative Group Algebras

TL;DR

This paper investigates the bilinear complexity of multiplication in noncommutative group algebras, establishing bounds and conditions related to matrix multiplication complexity and algebra structure.

Contribution

It characterizes minimal bilinear complexity in semisimple group algebras and provides lower bounds for other group algebras, connecting to matrix multiplication complexity.

Findings

Characterization of minimal bilinear complexity in semisimple group algebras

Nontrivial lower bounds for general group algebras

Subquadratic upper bounds under certain matrix multiplication conditions

Abstract

We study the complexity of multiplication in noncommutative group algebras which is closely related to the complexity of matrix multiplication. We characterize such semisimple group algebras of the minimal bilinear complexity and show nontrivial lower bounds for the rest of the group algebras. These lower bounds are built on the top of Bl\"aser's results for semisimple algebras and algebras with large radical and the lower bound for arbitrary associative algebras due to Alder and Strassen. We also show subquadratic upper bounds for all group algebras turning into "almost linear" provided the exponent of matrix multiplication equals 2.