Plethysm and fast matrix multiplication

Abstract

Abstract. Motivated by the symmetric version of matrix multiplication we study the plethysm S k (sln) of the adjoint representation sln of the Lie group SLn. In particular, we describe the decomposition of this representation into irreducible components for k = 3, and find highest-weight vectors for all irreducible components. Relations to fast matrix multiplication, in particular the Coppersmith-Winograd tensor, are presented.

Algebraic Geometry and Representation theory in the study of matrix multiplication complexity and other problems in theoretical computer science

“…Remark 10.3. In [97] T. Seyannaeve decomposed S 3 gl n and noticed that several of the highest weight vectors that appeared were Coppersmith-Winograd tensors. This gave rise to the idea that one might look among the highest weight vectors in S 3 gl n to find ones useful for the laser method.…”

Bounds on complexity of matrix multiplication away from CW tensors

“…In [16], the fourth author studied the highest weight vectors of the sl n -representation S 3 (gl n ). A basis of gl n is given by {E i,j } 1≤i,j≤n , where E i,j is the matrix with a 1 at position (i, j) and 0 elsewhere.…”

“…One of the highest weight vectors in S 3 (gl n ) is the symmetrized matrix multiplication tensor i,j,k E i,j E j,k E k,i [7]. The other highest weight vectors are listed in [16,Table 2]. An interesting observation is that many of these highest weight vectors are, up to a change of variables, equal to T cw,m or T CW,m for some value of m. * The border rank cannot be lower than n + 1, as T is a concise tensor.…”