Bound on the Jordan type of a generic nilpotent matrix commuting with a given matrix

The poset of the nilpotent commutator of a nilpotent matrix

Let $B$ be an $n \times n$ nilpotent matrix with entries in an infinite field $\k$. Assume that $B$ is in Jordan canonical form with the associated Jordan block partition $P$. In this paper, we study a poset $\mathcal{D}_P$ associated to the nilpotent commutator of $B$ and a certain partition of $n$, denoted by $λ_U(P)$, defined in terms of the lengths of unions of special chains in $\mathcal{D}_P$. Polona Oblak associated to a given partition $P$ another partition $Ob(P)$ resulting from a recursive process. She conjectured that $Ob(P)$ is the same as the Jordan partition $Q(P)$ of a generic e…

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

Let $k$ be an infinite field. Fix a Jordan nilpotent $n$ by $n$ 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.

A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs

In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if $G$ and $H$ are finite groups with isomorphic non-commuting graphs and $G$ is nilpotent, then $H$ must be nilpotent as well (Conjecture 2). We pose a new conjecture (Conjecture 3) that, together with the assumption $|Z(G)|\geq|Z(H)|$, implies Conjecture 2 and we prove it for groups in which all centralizers of non-central elements are abelian.

$A_n$ 型箭图的路模的典范 Jordan 可恢复子范畴

Canonically Jordan recoverable categories for modules over the path algebra of $A_n$ type quivers

本文研究 $A_n$ 型箭图表示的子范畴的典范 Jordan 可恢复性，通过组合刻画给出了这些子范畴的特征。

识别具有最大交换轨道$Q=(u,u-r)$的分拆

Identifying Partitions with maximum commuting orbit $Q=(u,u-r)$

该研究证明[IKVZ2]中定义的分拆$P_{k,l}(Q)$与[BIK]中通过Burge对应定义的分拆$P_{k,l}^Q$相同，解决了两类极大交换轨道分拆的等价性问题。