Finite nilpotent orbits: GL(n,q)-conjugacy classes and a partial order on partitions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T03:10:01Z http://mathoverflow.net/feeds/question/97439 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97439/finite-nilpotent-orbits-gln-q-conjugacy-classes-and-a-partial-order-on-partiti Finite nilpotent orbits: GL(n,q)-conjugacy classes and a partial order on partitions Glasby 2012-05-19T21:56:31Z 2012-05-20T12:58:04Z <p>I have a question regarding a partial order <code>$&lt;$</code> on the set <code>${\rm Part}(n)$</code> 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, <code>$J_{(3,2,1)}=\left(\begin{smallmatrix}0&amp;1&amp;0&amp;&amp;&amp;\\0&amp;0&amp;1&amp;&amp;&amp;\\0&amp;0&amp;0&amp;&amp;&amp;\\&amp;&amp;&amp;0&amp;1&amp;\\&amp;&amp;&amp;0&amp;0&amp;\\&amp;&amp;&amp;&amp;&amp;0\end{smallmatrix}\right)$</code>. 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 <code>$\mathcal{O}_\lambda:=J_\lambda^{{\rm GL}(n,F)}$</code> for a unique $\lambda\in{\rm Part}(n)$. (Nilpotent means $X^n=0$.)</p> <p>If <code>$F=\mathbb{F}_q$</code> is a finite field, then set <code>$n_\lambda:=|J_\lambda^{{\rm GL}(n,q)}|$</code>. A formula for <code>$n_\lambda$</code> is given in Fulman, Cycle indices for finite classical groups. It turns out that <code>$n_\lambda=n_\lambda(q)$</code> is a polynomial in $q$ with integer coefficients. Define a partial order <code>$&lt;$</code> on <code>${\rm Part}(n)$</code> as follows: <code>$\lambda&lt;\mu$</code> if and only if <code>$n_\lambda(q)$</code> divides <code>$n_\mu(q)$</code>. I call this the <em>divisibility</em> partial order.</p> <p>When $F$ is the complex field $\mathbb{C}$, define $\lambda\triangleleft\mu$ if <code>$\overline{\mathcal{O}_\lambda}\subset\overline{\mathcal{O}_\mu}$</code> where <code>$\overline{\mathcal{O}_\lambda}$</code> denotes the Zariski closure of <code>$\mathcal{O}_\lambda$</code>. It is shown in Collingwood and McGovern, Nilpotent orbits of semisimple Lie algebras, pp 93--95, that $\triangleleft$ is the <em>dominance</em> partial order on ${\rm Part}(n)$. That is, $\lambda\triangleleft\mu$ if and only if <code>$\sum_{i=1}^{k-1}\lambda_i=\sum_{i=1}^{k-1}\mu_i$</code> and <code>$\lambda_k&lt;\mu_k$</code> for some $k\geq1$.</p> <p>If $n\leq5$, then the partial orders <code>$&lt;$</code> and <code>$\triangleleft$</code> are identical and are total orders. However, when $n=6$ the partition $(3,2,1)$ of 6 has three partitions <em>divisibility larger</em>, and has five partitions <em>dominance larger</em>.</p> <p>Does anyone have any insight into divisibility partial order? or know of its appearance in the literature? (I have not found a reference to <code>$&lt;$</code> 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 <code>$\lambda&lt;\mu$</code> for <em>specific</em> $\lambda$ and $\mu$, but I have few global results.</p> http://mathoverflow.net/questions/97439/finite-nilpotent-orbits-gln-q-conjugacy-classes-and-a-partial-order-on-partiti/97441#97441 Answer by Will Sawin for Finite nilpotent orbits: GL(n,q)-conjugacy classes and a partial order on partitions Will Sawin 2012-05-19T22:31:12Z 2012-05-20T03:21:09Z <p>I think the right way is to factor $n_\lambda(q)$ in general. In particular, it is obviously a quotient of the order of $GL_n(q)$. The formula for the order of $GL_n(q)$ does not have very many prime factors: just $q$ and the first $n$ cyclotomic polynomials.</p> <p>One could consider an alternate question, the order of the centralizer of $J$ in $GL(n,\mathbb F_q)$. This gives the reverse of the partial order, since the divisibility relation is reversed. The centralizer is the automorphism group of the corresponding $\mathbb F_q[x]/x^n$-module, $M$. The subgroup that fixes $M/x$ is a $q$-group, since it is unipotent. Its quotient is a product of copies of $GL(k,\mathbb F_q)$: one for each type of block, with $k$ equal to the number of that block that appears. </p> <p>The number of times that the $k$th elementary cyclotomic polynomial appears in the size of the centralizer is just the sum over all sizes $n$ of the floor of $a_n/k$, where $a_n$ is the number of blocks of size $n$.</p> <p>This function behaves very erratically, so we can conclude that the partial order behaves erratically as well.</p>