Two-generated commutative subalgebras of \(M_ n(F)\)
Let \(A\) and \(B\) be a pair of commuting \(n\times n\)-matrices over a field \(K\). It is well known that \(\dim_ K K[A,B]\leq n\). The authors show that \(\dim_ K\text{Cent}_{M_ n(K)}(A,B)\geq n\), and equality holds in one of these inequalities if and only if it holds in the other. Further characterizations of the case where equality holds are also given. The space of column vectors of length \(n\) is a \(K[A,B]\) module \(M\) in the obvious way; write \(M=M_ 1\oplus\cdots\oplus M_ s\), where each \(M_ i\) is indecomposable. The complete characterization theorem now asserts that the following statements are equivalent: 1) \(\dim_ K K[A,B]=n\); 2) For \(1\leq i\leq s\), each \(M_ i\) is cyclic or cocyclic (i.e., its \(K\)-dual is cyclic), and \(\text{Hom}_{K[A,B]} (M_ i,M_ j)=0\) when \(i\neq j\); 3) \(K[A,B]\) is self-centralizing in \(M_ n(K)\) (i.e., \(K[A,B]\) is a maximal commutative subalgebra of \(M_ n(K)\), with respect to inclusion); 4) \((A,B)\) is a smooth point on the variety of pairs of commuting \(n\times n\) matrices over \(K\); and 5) \(\dim_ K\text{Cent}_{M_ n(K)} (A,B)=n\). In regard to the case where equality does not hold, \textit{M. Gerstenhaber} [Ann. Math., II. Ser. 73, 324-348 (1961; Zbl 0168.282)] showed that there is an \(n\)-dimensional commutative subalgebra \(G\) of \(M_ n(K)\) that contains \(K[A,B]\). The authors give a new proof of this result, and show by an example that the action of \(G\) on \(M\) need not be cyclic or cocyclic.
scientific article; zbMATH DE number 562454
scientific article