Timeline for Representations of GL(n,2) over a field of characteristic 2
Current License: CC BY-SA 4.0
13 events
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| Oct 15, 2018 at 13:16 | comment | added | Mikko Korhonen | @TobiasKildetoft: Yes, the tables on Lübeck's website extend the ones given in his paper "Lübeck, F., Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math., 4 (2001), p. 135--169". The paper gives some explanation about the computations. They are based on using the contravariant form on the corresponding Weyl modules (using a result of Burgoyne) and the Jantzen sum formula. | |
| Oct 15, 2018 at 8:15 | comment | added | Tobias Kildetoft | @Mikko Great, I will take a look at that. Does it also include some explanation of how the calculations were done? | |
| Oct 15, 2018 at 7:53 | comment | added | Mikko Korhonen | @TobiasKildetoft: For $\operatorname{SL}_5$ and $\operatorname{SL}_6$, see computations done by Frank Lübeck here and here. I also think that for large $n$ the degrees are not known. | |
| S Oct 12, 2018 at 13:54 | history | suggested | Tobias Kildetoft |
Added further relevant tags.
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| Oct 12, 2018 at 13:18 | review | Suggested edits | |||
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| Oct 12, 2018 at 11:09 | comment | added | Tobias Kildetoft | $\dim(L(0,0,0)) = 1$, $\dim(L(1,0,0)) = \dim(L(0,0,1)) = 4$, $\dim(L(0,1,0)) = 6$, $\dim(L(1,1,0)) = \dim(L(0,1,1)) = 20$, $\dim(L(1,0,1)) = 14$, $\dim(L(1,1,1)) = 64$. Here the dimensions of all except $L(1,0,1)$ are given by Weyl's dimension formula, while the exception is one smaller than that due to the appearance of a copy of the trivial module in the induced module with that highest weight. | |
| Oct 12, 2018 at 11:06 | comment | added | Tobias Kildetoft | Just to add a bit more detail: For the case in 2) we have that $GL_n = SL_n$ so we don't need to worry about twisting with the determinant at all. The irreducible representations are labelled by $(n-1)$-tuples of $0$s and $1$s (by writing the weight in the fundamental basis), and denoted $L(a_1,a_2,\dots, a_{n-1})$. We then have the following dimensions for $n\leq 4$: $\dim(L(0)) = 1$, $\dim(L(1)) = 2$, $\dim(L(0,0)) = 1$, $\dim(L(1,0)) = \dim(L(0,1)) = 3$, $\dim(L(1,1)) = 8$ | |
| Oct 9, 2018 at 17:30 | comment | added | Tobias Kildetoft | Ahh, right, here $p=2$ is the characteristic. The description I mentioned is II.8.21 in Jantzen's "Representations of algebraic groups". II.8.23 from that book is also specifically about $GL_n$ and might be of interest. For $n\leq 4$ the questions are doable, but I think even $n=5$ would be quite hard to handle. | |
| Oct 9, 2018 at 16:59 | comment | added | Uep | Thanks, Tobias. When you said there are $p^{n-1}$ irreducible representations, what is $p$? what is the group and what is the representing field? Could you also tell me what book/paper of Jantzen you mentioned. | |
| Oct 9, 2018 at 16:50 | comment | added | Tobias Kildetoft | Once $n$ becomes large enough, I don't think the dimensions of the irreducible representations are known, even in characteristic $2$. The case of $SL_n$ is a bit easier to describe (though passing to $GL_n$ does not change much, as it just adds the possibility of twisting by powers of the determinant representation). Here there are $p^{n-1}$ irreducible representations, indexed by restricted weights. There is also a description due to Jantzen of which of these irreducibles have their dimension given by Weyl's formula, but this will be only some of them. | |
| Oct 9, 2018 at 16:17 | comment | added | Uep | To Tobias: I would like to know what are the irreducible representations and their degrees? Is there any description or construction of irreducible representations of GL(n,2) over F_2? | |
| Oct 9, 2018 at 15:54 | comment | added | Tobias Kildetoft | What sort of things should be covered, how advanced knowledge can it assume, and does it have to focus on this specific case? If the answers to the latter two are "quite" and "no" then the book by Humphreys titled "Representations on finite groups of Lie type" should cover pretty much any immediate need. | |
| Oct 9, 2018 at 13:33 | history | asked | Uep | CC BY-SA 4.0 |