Complexity of multiplication in some group algebras

Abstract

In this paper, we present some results on algebraic complexity theory, which is a relatively new region of complexity theory. We consider the complexity of multiplication (bilinear complexity) in group algebras. In the 6-dimensional group algebra C(S3) over the field of complex numbers with permutations of the third order as the basis, we find a bilinear algorithm for multiplication with multiplicative complexity equal to 9 (instead of trivial 36) and prove that this bound is unimprovable. We prove a series of assertions on the structure of the group algebra C(S3), in particular, we show that the algebra C(S3) is decomposed into a direct product of the algebra of 2 × 2 matrices and two one-dimensional algebras.