2 improve formatting

I have a question regarding a partial order $$J_{(3,2,1)}=\left(\begin{smallmatrix}0&1&0&&&\\0&0&1&&&\\0&0&0&&&\\&&&0&1&\\&&&0&0&\\&&&&&0\end{smallmatrix}\right). < on the set {\rm Part}(n) of partitions of n. Given \lambda=(\lambda_1,\lambda_2,\ldots)\in{\rm Part}(n) with \sum_{i\geq1} \lambda_i=n and \lambda_1\geq\lambda_2\geq\cdots\geq0, let J_\lambda denote the n\times n block diagonal matrix \bigoplus_{i\geq1}J_{\lambda_i}. For example, J_{(3,2,1)}=\left(\begin{smallmatrix}0&1&0&&&\\0&0&1&&&\\0&0&0&&&\\&&&0&1&\\&&&0&0&\\&&&&&0\end{smallmatrix}\right). Consider the {\rm GL}(n,F)-conjugacy classes of the set {\rm M}(n,F) of all n\times n matrices over a field F. A nilpotent matrix X\in{\rm M}(n,F) lies in a conjugacy classes \mathcal{O}_\lambda:=J_\lambda^{{\rm GL}(n,F)} for a unique \lambda\in{\rm Part}(n). (Nilpotent means X^n=0.) If F=\mathbb{F}_q is a finite field, then set n_\lambda:=|J_\lambda^{{\rm GL}(n,q)}|. A formula for n_\lambda is given in Fulman, Cycle indices for finite classical groups. It turns out that n_\lambda=n_\lambda(q) is a polynomial in q with integer coefficients. Define a partial order $$n_\lambda(q)$<$ on ${\rm Part}(n)$ as follows: $\lambda<\mu$ if and only if $n_\lambda(q)$ divides $n_\mu(q)$. I call this the divisibility partial order.

When $F$ is the complex field $\mathbb{C}$, define $\lambda\triangleleft\mu$ if $\overline{\mathcal{O}_\lambda}\subset\overline{\mathcal{O}_\mu}$ where $\overline{\mathcal{O}_\lambda}$ denotes the Zariski closure of $\mathcal{O}_\lambda$. It is shown in Collingwood and McGovern, Nilpotent orbits of semisimple Lie algebras, pp 93--95, that $\triangleleft$ is the dominance partial order on ${\rm Part}(n)$. That is, $\lambda\triangleleft\mu$ if and only if $\sum_{i=1}^{k-1}\lambda_i=\sum_{i=1}^{k-1}\mu_i$ and $\lambda_k<\mu_k$ for some $k\geq1$.

If $n\leq5$, then the partial orders $<$ and $\triangleleft$ are identical and are total orders. However, when $n=6$ the partition $(3,2,1)$ of 6 has three partitions divisibility larger, and has five partitions dominance larger.

Does anyone have any insight into divisibility partial order? or know of its appearance in the literature? (I have not found a reference to $<$ in Roger Carter's book Finite groups of Lie type: conjugacy classes and complex characters, but $\triangleleft$ appears in 5.5 and 5.11.) For specific $\lambda$, I can (theoretically) factor $n_\lambda(q)$ and so can determined whether $\lambda<\mu$ for specific $\lambda$ and $\mu$, but I have few global results.

1

# Finite nilpotent orbits: GL(n,q)-conjugacy classes and a partial order on partitions

I have a question regarding a partial order $$J_{(3,2,1)}=\left(\begin{smallmatrix}0&1&0&&&\\0&0&1&&&\\0&0&0&&&\\&&&0&1&\\&&&0&0&\\&&&&&0\end{smallmatrix}\right). Consider the {\rm GL}(n,F)-conjugacy classes of the set {\rm M}(n,F) of all n\times n matrices over a field F. A nilpotent matrix X\in{\rm M}(n,F) lies in a conjugacy classes \mathcal{O}_\lambda:=J_\lambda^{{\rm GL}(n,F)} for a unique \lambda\in{\rm Part}(n). (Nilpotent means X^n=0.) If F=\mathbb{F}_q is a finite field, then set n_\lambda:=|J_\lambda^{{\rm GL}(n,q)}|. A formula for n_\lambda is given in Fulman, Cycle indices for finite classical groups. It turns out that n_\lambda=n_\lambda(q) is a polynomial in q with integer coefficients. Define a partial order$$n_\lambda(q)$divides $n_\mu(q)$. I call this the divisibility partial order. When$F$is the complex field$\mathbb{C}$, define$\lambda\triangleleft\mu$if $\overline{\mathcal{O}_\lambda}\subset\overline{\mathcal{O}_\mu}$ where $\overline{\mathcal{O}_\lambda}$ denotes the Zariski closure of $\mathcal{O}_\lambda$. It is shown in Collingwood and McGovern, Nilpotent orbits of semisimple Lie algebras, pp 93--95, that$\triangleleft$is the dominance partial order on${\rm Part}(n)$. That is,$\lambda\triangleleft\mu$if and only if $\sum_{i=1}^{k-1}\lambda_i=\sum_{i=1}^{k-1}\mu_i$ and $\lambda_k<\mu_k$ for some$k\geq1$. If$n\leq5$, then the partial orders$divisibility larger, and has five partitions dominance larger.

Does anyone have any insight into divisibility partial order? or know of its appearance in the literature? (I have not found a reference to $specific$\lambda$and$\mu\$, but I have few global results.