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

Abstract

Abstract. We study the set P(N B ) of all possible Jordan canonical forms of nilpotent matrices commuting with a given nilpotent matrix B. We describe P(N B ) in the special case when B has only one Jordan block and discuss some consequences. In the general case, we find the maximal possible index of nilpotency in the set of all nilpotent matrices commuting with a given nilpotent matrix. We consider several examples. Math. Subj. Class (2000): 15A04, 15A21, 15A27

On types and classes of commuting matrices over finite fields

“…Proposition 4.2 describes the nilpotent classes that commute with N (λ) when λ has a single part. This result has appeared previously in [12]; our Proposition 4.5 is similar to, but slightly stronger than, the result that appears there as Proposition 2.…”

“…The first possibility is ruled out since S * can have no component of dimension 16. The only possible component of S * of dimension 24 is 1 (12,12) , and so, if our supposition is correct, then the primary types 2 (8,4) and 1 (12,12) must commute over F 2 . By Theorem 2.12, this is the case only if 1 (8,4) and 1 (6,6) commute.…”

“…Let C be the similarity class of matrices over F 2 with cycle type p (12,12) q (2,2,2) r (3) s (1) and let D be the similarity class with cycle type r (7,5) t (2,2,1) . We shall prove that C commutes with D. By Theorem 2.8, this is equivalent to showing that the types…”

“…Proposition 4.5 is slightly more general than Proposition 2 in [12], which states that the nilpotent classes N (λ) and N (μ) commute if μ is an almost rectangular refinement of λ. It was noted in [12] that there exist examples of classes commuting that cannot be explained in this way. We remark that our Proposition 4.5 does not account for all commuting between classes either.…”

On pairs of commuting nilpotent matrices

“…3.29] it is easy to answer this question when D(λ) has at most two parts, i. e. when r B ≤ 2. Here we would like to remark that the proofs of Lemma 11 and Theorems 12 and 13 in [14] hold over any field of characteristic 0, while Basili and Iarrobino assume in [4] that the underlying field is algebraically closed of arbitrary characteristic. …”