Commuting pairs and triples of matrices and related varieties

Abstract

In this note, we show that the set of all commuting d-tuples of commuting n × n matrices that are contained is an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber's theorem on commuting pairs of matrices is a consequence of the irreducibility of the variety of commuting pairs. We show that the variety of commuting triples of 4 × 4 matrices is irreducible. We also study the variety of n-dimensional commutative subalgebras of M n (F), and show that it is irreducible of dimension n 2 − n for n ≤ 4, but reducible, of dimension greater than n 2 − n for n ≥ 7.

The upper bound for the index of nilpotency for a matrix commuting with a given nilpotent matrix

“…It was proved by Motzkin and Taussky [13] (see also Guralnick [7]), that the variety of pairs of commuting matrices was irreducible. It was Guralnick [7] who showed that this is no longer the case for the variety of triples of commuting matrices (see also Guralnick and Sethuraman [8], Holbrook and Omladič [11], Omladič [14], Han [10],Šivic [16]). Recently, it was proved that the variety of commuting pairs of nilpotent matrices was irreducible (Baranovsky [1], Basili [2]).…”

Commuting triples of matrices

“…So B + tX commutes with C for all t ∈ F. Moreover for t = 0, the matrix B + tX has more than one point in the its spectrum. By Lemma 2.4, the triple (A, B + tX, C) belongs to G (3,7) for all t ∈ F, t = 0.…”

“…Guralnick [2] had proved that C(3, n) is reducible for n ≥ 32, but irreducible for n ≤ 3. Using the results from [7], Guralnick and Sethuraman [3] had proved that C(3, n) is irreducible when n = 4.…”

“…They had proved that C (3, n) is reducible for n ≥ 30, but irreducible for n = 5. Recently using this perturbation technique Omladič [8] gave the answer to this problem for the case n = 6; i.e., He proved that the variety C (3,6) is irreducible if charF = 0.…”

“…In this paper we consider the affine variety C (3,7) and prove that C(3, 7) is also irreducible. Actually, the proofs of the theorems given in section 3, 4, 5, and 6 work also if the characteristic of the field F is nonzero.…”