Matrix Multiplication via Matrix Groups

Authors Jonah Blasiak , Henry Cohn , Joshua A. Grochow , Kevin Pratt , Chris Umans

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Author Details

Jonah Blasiak
  • Department of Mathematics, Drexel University, Philadelphia, PA, USA
Henry Cohn
  • Microsoft Research New England, One Memorial Drive, Cambridge, MA, USA
Joshua A. Grochow
  • Departments of Computer Science and Mathematics, University of Colorado Boulder, CO, USA
Kevin Pratt
  • School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
Chris Umans
  • Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA

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Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, and Chris Umans. Matrix Multiplication via Matrix Groups. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


In 2003, Cohn and Umans proposed a group-theoretic approach to bounding the exponent of matrix multiplication. Previous work within this approach ruled out certain families of groups as a route to obtaining ω = 2, while other families of groups remain potentially viable. In this paper we turn our attention to matrix groups, whose usefulness within this framework was relatively unexplored. We first show that groups of Lie type cannot prove ω = 2 within the group-theoretic approach. This is based on a representation-theoretic argument that identifies the second-smallest dimension of an irreducible representation of a group as a key parameter that determines its viability in this framework. Our proof builds on Gowers' result concerning product-free sets in quasirandom groups. We then give another barrier that rules out certain natural matrix group constructions that make use of subgroups that are far from being self-normalizing. Our barrier results leave open several natural paths to obtain ω = 2 via matrix groups. To explore these routes we propose working in the continuous setting of Lie groups, in which we develop an analogous theory. Obtaining the analogue of ω = 2 in this potentially easier setting is a key challenge that represents an intermediate goal short of actually proving ω = 2. We give two constructions in the continuous setting, each of which evades one of our two barriers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Fast matrix multiplication
  • representation theory
  • matrix groups


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