Complexity of multiplication in some group algebras
Valeriy B. Alekseev, Alexey Pospelov · Discrete Mathematics and Applications · 2005
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 ( S 3 ) 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 ( S 3 ), in particular, we show that the algebra C ( S 3 ) is decomposed into a direct product of the algebra of 2 × 2 matrices and two one-dimensional algebras.