On the problem of freeness of multiplicative matrix semigroups

Abstract

The following problem looking as a high-school exercise hides an unexpected difficulty. Do the matrices A = ( 2 0 0 3 ) and B = ( 3 5 0 5 ) satisfy any nontrivial equation with the multiplication symbol only? This problem was mentioned as open in Cassaigne et al. [J. Cassaigne, T. Harju, J. Karhumäki, On the undecidability of freeness of matrix semigroups, Internat. J. Algebra Comput. 9 (3–4) (1999) 295–305] and in a book by Blondel et al. [V. Blondel, J. Cassaigne, J. Karhumäki, Problem 10.3: Freeness of multiplicative matrix semigroups, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton University Press, 2004, pp. 309–314] as an intriguing instance of a natural computational problem of deciding whether a given finitely generated semigroup of 2 × 2 matrices is free. In this paper we present a new partial algorithm for the latter which, in particular, easily finds that the following equation A B 10 A 2 B A 2 B A 10 = B 2 A 6 B 2 A 2 B A B A B A 2 B 2 A 2 B A B 2 holds for the matrices above. 1 1 This equation has been obtained also by the mean of heavy computations by Cassaigne and Nicolas and reported earlier in the preprint [5] (see the remark in Section 1). Our algorithm turns out quite practical and allows us to settle also other related open questions posed in the mentioned article.