The smallest part of the generic partition of the nilpotent commutator of a nilpotent matrix

Abstract

Let k be an infinite field. Fix a Jordan nilpotent n × n matrix B = J P with entries in k and associated Jordan type P . Let Q(P ) be the Jordan type of a generic nilpotent matrix commuting with B. In this paper, we use the combinatorics of a poset associated to the partition P , to give an explicit formula for the smallest part of Q(P ), which is independent of the characteristic of k. This, in particular, leads to a complete description of Q(P ) when it has at most three parts.Let k be an infinite field and Mat n (k) be the set of all n × n matrices with entries in k.Suppose that B = J P ∈ {Mat n (k)} is a Jordan matrix with associated Jordan type -Jordan block partition-P n. Recall that the centralizer and the nilpotent centralizer of MSC 2010: 05E40, 06A11, 14L30, 15A21

The poset of the nilpotent commutator of a nilpotent matrix

“…The author is grateful to A. Iarrobino for invaluable discussions on the topic, as well as for his comments and suggestions on the paper. The author is also thankful to Bart Van Steirteghem and to Tomaž Košir for their thorough comments on a draft of this paper.In [9] we further study the poset D P and the partition λ U (P ) and give an explicit formula for its smallest part µ(P ). By enumerating the disjoint maximum antichains in D P and use of results from [11] and [8], we prove that the smallest part of Q(P ) is µ(P ) as well.…”

Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit

“…T. Košir and P. Oblak showed that if the characteristic of k is 0 then Q(P ) has parts that differ pairwise by at least two (Theorem 2.6). Even in cases where the Oblak recursive conjecture had been shown some time ago, (as r P = 2 [27], or r P = 3 [26]) the set Q −1 (Q) remained mysterious. In 2012 P. Oblak made a second conjecture: when Q = (u, u − r) with u > r ≥ 2, then the cardinality |Q −1 (Q)| = (r − 1)(u − r) [35,Remark 2].…”

“…Then Q(P ) ≥ λ U (P ).L. Khatami studied the smallest part of Q(P ) and defined a somewhat subtle combinatorial invariant µ(P )[26, Definition 2.6]. Using a study of the antichains of D P she showed…”

“…[26, Theorem 4.1] Let P be a partition of n and let k be an infinite field. The three partitions λ(D P ), λ U (P ) and Q(P ) have the same smallest part, which is equal to µ(P ).Together with P. Oblak's Index Theorem 2.8, this implies the following fact we will use in Section 4.3.…”

“…Summary of results on the Oblak Recursive Conjecture). Thus, the cases r P = 2[5,27,34,46] and r P = 3[26] of the Conjecture have been known since 2008 and 2012, respectively. Theorem 4.4 of the first and second authors then showed "half" the Conjecture in all characteristics.…”