Maximal commutative algebras of linear transformations

Abstract

In 1905, Schur [4] proved that the maximum number of linearly independent n x n complex matrices is g(n) = [n*/4] + 1, where [ ] denotes the greatest integer function. He also determined the linearly independent sets of commuting matrices of cardinality g(n). Jacobson [3] extended these results to arbitrary fields, except in one exceptional situation. Other proofs of these theorems can be found in [5, 61. All of these proofs depend on rather explicit manipulation of matrices. We will prove these theorems by representation-theoretic methods. Let us suppose that S is a maximal linearly independent set of commuting linear transformations on a finite-dimensional vector space V over a field k. Then it is easy to see that the k-linear span of S is a commutative k-subalgebra R of Hom,(V, V) containing the identity transformation. Further, V is a faithful R-module. Since we are interested in the relationship between dim, V and 1 S / = dim,R, we give our attention to the k-dimensions of faithful modules over finite-dimensional commutative k-algebras. We will discuss this problem first for local algebras, and then obtain the general case as an easy corollary. Once we have proved Schur’s inequality dim,R 4, then Hom,(V, I’) contains many commutative subalgebras which are maximal with respect to inclusion, but of dimension smaller than g(d im, I’). For example, if R is a finite-dimensional commutative k-algebra, then its regular representation embeds it as a maximal commutative subalgebra of Hom,(R, R). Let Vn(k) denote the set of maximal commutative subalgebras of the algebra Homk(V, V), where dim, V = n. It had been conjccturcd that dim, R > n for all R E V,(k). However, Courter [l, and 21 showed that the ratio min{dim,R / R E V,(k)}/n