Timeline for Number of homomorphisms from a group to $\mathrm{GL}_n(\mathbb{F}_q)$
Current License: CC BY-SA 4.0
13 events
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| Mar 10, 2022 at 0:15 | comment | added | Will Sawin | @Carl-FredrikNybergBrodda I think Bate-Gullon also include in their assumptions "Suppose that $\mathbb F_q$ is a splitting field for $A$." | |
| Mar 9, 2022 at 23:40 | history | edited | Dr. Evil | CC BY-SA 4.0 |
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| Mar 9, 2022 at 23:39 | comment | added | LSpice | Names of @Carl-FredrikNybergBrodda's references: Chigira, Takegahara, and Yoshida - On the Number of Homomorphisms from a Finite Group to a General Linear Group; Liebeck and Shalev - The Number of Homomorphisms from a Finite Group to a General Linear Group. | |
| Mar 9, 2022 at 23:38 | history | edited | Dr. Evil | CC BY-SA 4.0 |
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| Mar 9, 2022 at 23:35 | history | edited | LSpice | CC BY-SA 4.0 |
Names of papers; PDF -> abs; TeX
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| Mar 9, 2022 at 23:30 | comment | added | Dr. Evil | Thanks all for pointing out my fallacy. I have put in an edit at the end of the question relaxing the requirements. | |
| Mar 9, 2022 at 23:29 | history | edited | Dr. Evil | CC BY-SA 4.0 |
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| Mar 9, 2022 at 10:24 | comment | added | Jeremy Rickard | @Carl-FredrikNybergBrodda At the beginning of the "Preliminaries" section of Bate-Gullon, they say that their standing assumptions include that the order of the group is not divisible by the characteristic of the field. | |
| Mar 9, 2022 at 9:54 | history | edited | YCor | CC BY-SA 4.0 |
added more info on links
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| Mar 9, 2022 at 9:44 | comment | added | Carl-Fredrik Nyberg Brodda | (...cont) But then Bate-Gullon claims to prove things about the leading term of $c_n(q)$, which is not defined in the excluded case at all (such as observed by David E Speyer)! Either I am, or Bate-Gullon are, somewhat confused here; maybe I am misreading something. | |
| Mar 9, 2022 at 9:43 | comment | added | Carl-Fredrik Nyberg Brodda | It's strange. The Bate-Gullon article (linked in the question) claims that $c_n(q)$ (which they denote $X(n,q)$ is always a polynomial. But their reference for this is this article, specifically Proposition~4.1, and that article has the condition that (in OP's notation) $|\Gamma|$ is not a multiple of $p$, where $q = p^k$. The article also references this article, which deals with that excluded case by giving polynomial bounds on $c_n(q)$, but doesn't claim that it's polynomial. (cont...) | |
| Mar 9, 2022 at 9:29 | comment | added | David E Speyer | This is not a polynomial when $\Gamma$ is a cyclic group of order $k$ and $n=1$; the formula is $\text{GCD}(k, q-1)$. Perhaps you want to ask for a quasi-polynomial? en.wikipedia.org/wiki/Quasi-polynomial | |
| Mar 9, 2022 at 9:20 | history | asked | Dr. Evil | CC BY-SA 4.0 |