Timeline for Counting representations of $k[x,y]$ when $k$ is finite
Current License: CC BY-SA 3.0
16 events
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| Jul 15, 2014 at 3:17 | comment | added | Amritanshu Prasad | @RichardStanley I don't know, but here is another interesting observation: if $r_{n,k}$ is the number of iso classes of $n$-dimensional $\mathbf F_q[x_1,\dotsc,x_k]$-modules, then $r_{2,k} = q^k\binom{k+1}1_q$ and $r_{3,k} = q^k[\binom{k+2}2_q + \binom k2_q]$. | |
| Jul 14, 2014 at 23:57 | comment | added | Richard Stanley | Is it just a coincidence that $r_n(1)$ is the middle coefficient of $(1+x+x^2)^n$ for $1\leq n\leq 4$? | |
| Sep 15, 2013 at 3:06 | comment | added | Amritanshu Prasad | @JimBryan The difficulty here is finding a uniform way to deal with all $\lambda$ in a uniform manner. | |
| Sep 13, 2013 at 22:07 | comment | added | Jim Bryan | orbit space will be tricky to figure out. Still, maybe one could at least show it is paved by affines (or tori). | |
| Sep 13, 2013 at 22:06 | comment | added | Jim Bryan | I kind of agree with Allen's assessment. I did look at what would happen if you apply the power structure methods of my paper with Morrison to this problem and it boils down to computing the class of $Stab_{J_{\lambda}}(Gl(n))/Stab_{J_\lambda}(Gl(n))$ in the Grothendieck group. Here $\lambda$ is a partition of $n$ and $J_\lambda$ is the nilpotent matrix in Jordan form with blocks of size $\lambda_i$, and $Stab_{J_\lambda}(Gl(n))$ is the stabilizer of $J_\lambda$ in $Gl(n)$ acting on itself by conjugation. The class of the stack quotient is just 1, but the class of the | |
| Sep 13, 2013 at 20:11 | comment | added | Allen Knutson | It might well be easier (and good enough) to pave by tori instead of affine spaces. But anyway, this begins to sound too much like the wild problem of classifying representations of $k[x,y]$. I doubt you're going to get results for general $n$. | |
| Sep 13, 2013 at 14:48 | comment | added | Jim Bryan | You might want to edit your question so that the tags "commuting variety" and/or "algebraic geometry" appear. The question is really about the class of the coarse space $C(n)/Gl(n)$ in the Grothendieck group of varieties --- there is a lemma of Katz which says that a class in the Grothendieck group will be a polynomial in $[\mathbb{A}^1]$ iff the point count over $\mathbf{F}_q$ is polynomial in $q$. So you should look for an affine paving of $C(n)/Gl(n)$, i.e. a stratification where the strata are all affine spaces. You could find this over $\mathbb{C}$ and that would suffice. | |
| Sep 13, 2013 at 3:52 | comment | added | Amritanshu Prasad | @JimBryan I feel that the original question should have a very nice answer because I could do explicit calculations for $n\leq 4$ (see my edit). | |
| Sep 12, 2013 at 15:02 | comment | added | Jim Bryan | @Amritanshu : Just to clarify, the number you are asking for is $|C(n)/Gl(n)|$, which is the number of isomorphism classes of modules, which is the cardinality of the coarse moduli space of the stack $[C(n)/Gl(n)]$. The number that is given by the above formula is $|C(n)|/|Gl(n)|$, which is the number of isomorphism classes of modules counted by the reciprocal number of automorphisms, which is (in some sense) the cardinality of the stack $[C(n)/Gl(n)]$. The stack is often better behaved than its coarse space so maybe your original question doesn't have a good answer, but I don't know. | |
| Sep 12, 2013 at 8:01 | comment | added | Amritanshu Prasad | But the arxiv preprint looks very interesting. Thanks, @JimBryan for that. | |
| Sep 12, 2013 at 7:29 | comment | added | Dan Petersen | Yes, that's why he's dividing by $GL(n)$ acting by conjugation! | |
| Sep 12, 2013 at 7:21 | comment | added | Amritanshu Prasad | The number is seek is not however, the number of pairs of commuting matrices. It is the number of simulateneous similarity classes of pairs of commuting matrices. | |
| Sep 12, 2013 at 7:17 | vote | accept | Amritanshu Prasad | ||
| Sep 12, 2013 at 7:20 | |||||
| Sep 12, 2013 at 6:40 | comment | added | Jim Bryan | No, the stack is not DM. For example the trivial module of length n (corresponding to the 0 matrices in the commuting variety) has the full $Gl(n)$ as its automorphism group. | |
| Sep 12, 2013 at 6:38 | comment | added | Dan Petersen | If the stack is Deligne-Mumford, the number of $\mathbf F_q$-points is the same for the stack and the coarse space. Is this true here? | |
| Sep 12, 2013 at 6:36 | history | answered | Jim Bryan | CC BY-SA 3.0 |