Commutative Subalgebras in Martrices
by
Karavayev Aleksey
Moscow State University

Let F be a field. By MCS_n(F) we will denote the set of all maximal commutative subalgebras with respect to inclusion of the matrix algebra M_{n ×n}(F). In 1900 I.Schur provided an example of algebra A in MCS_n(F) wich has F-linear space dimention dim_F A = 1 + [n²/4]. Schur Theorem states this value as sharp upper bound for dimention of commutative subalgebras of M_{n ×n}. It was conjectured in 1950 by M.Gerstenhaber that dim_F A >= n for all A in MCS_n(F). However in 1965 R.Courter proposed an algebra C in MCS₁₄(F) with dim_F(C)=13. Though the class of algebras A in MCS_n (F) having dimention strictly less than n is far from being understood.

In 1993-94 W.Brown introduced two constructions of such algebras. Later he noticed that the second one could be generalised. We give a direct proof of the last statement. Also we provide a new invariant of this construction.

Date received: February 13, 2002