Timeline for Irreducible Representation of A_5
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
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| Apr 7, 2021 at 7:34 | comment | added | Derek Holt | @SamHopkins Yes I can see that this question is a doubtful case! But I believe that closed questions are not so universally visible as others. Apparently a closed question is "viewable by the post author and users with the close/reopen votes privilege". | |
| Apr 7, 2021 at 2:06 | comment | added | Sam Hopkins | @DerekHolt: You are probably aware of this, but closing the question is not the same as deleting it: Geoff's work would not go away if the question were closed. Imo whether a question should be closed has to do with whether it is on-topic, not whether it has nice answers (which this one does). | |
| Apr 7, 2021 at 2:01 | history | reopened |
Derek Holt Yemon Choi Alex M. Arun Debray Alexey Ustinov |
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| Apr 6, 2021 at 16:17 | review | Reopen votes | |||
| Apr 7, 2021 at 2:01 | |||||
| Apr 6, 2021 at 16:02 | comment | added | Derek Holt | I am voting to reopen, mainly because Geoff has devoted some effort into answering this and related questions. | |
| Apr 6, 2021 at 14:35 | history | closed |
abx Derek Holt Benjamin Steinberg LeechLattice Sam Hopkins |
Not suitable for this site | |
| Apr 6, 2021 at 13:42 | comment | added | Derek Holt | For the specific example of $A_5$, the result remains true in characteristics $2$ and $3$, but not $5$. (It is never true when the characteristic divides the degree of the permutation representation.) | |
| Apr 6, 2021 at 13:16 | comment | added | Benjamin Steinberg | The definition of the permutation module is the same over any field and he does endomorphism rings equal centralizer rings. The argument @GeoffRobinson gives in characteristic zero works in any non modular characteristic. | |
| Apr 6, 2021 at 12:48 | comment | added | lambda | @GeoffRobinson Clifford theory could be relevant if the problem was somehow easier for $S_5$. | |
| Apr 6, 2021 at 12:31 | comment | added | Geoff Robinson | As a general comment, Clifford theory is only useful in the representation theory of finite groups in the presence of proper non-trivial normal subgroups., so is not so relevant for simple groups. | |
| Apr 6, 2021 at 12:24 | answer | added | Geoff Robinson | timeline score: 6 | |
| Apr 6, 2021 at 12:15 | comment | added | HIMANSHU | @BenjaminSteinberg In Issac's character theory book, they are taking the field of complex numbers to define permutation module. Why will the same reasoning be valid in non-modular case ? | |
| Apr 6, 2021 at 11:56 | comment | added | Benjamin Steinberg | You can find all these concepts in Isaac's character theory book | |
| Apr 6, 2021 at 10:52 | comment | added | HIMANSHU | Let us continue this discussion in chat. | |
| Apr 6, 2021 at 10:25 | comment | added | HIMANSHU | @Benjamin Please refer some book to me , where I can read these concepts and arrive at the result. | |
| Apr 6, 2021 at 10:16 | comment | added | Benjamin Steinberg | I'm suggesting a conceptual way to do this using permutation modules and centralizer algebras or endomorphism rings that would work for An for n\geq 4 and S_n. If you want just A5 by bare hands I think math stack exchange is a more appropriate site. | |
| Apr 6, 2021 at 10:09 | comment | added | HIMANSHU | @Benjamin Actually I am not familiar with the concepts you mentioned. Can I prove the irreducibilty of FA_5 module V (where V is standard permutation module) for the field of given characteristic not dividing order of A_5? by using module theory arguments, as we can prove in case of S_5 by using the transposition (12) ( which is not here ) . | |
| Apr 6, 2021 at 9:59 | comment | added | Benjamin Steinberg | In other words the argument that the dimension of the endomorphism algebra of a permutation module equals the number of orbits on pairs can be carried out in nonmodular characteristic without using characters by looking at hom set dimensions and then the result follows from there. | |
| Apr 6, 2021 at 9:50 | comment | added | Benjamin Steinberg | The standard argument on how to decompose the permutation character of a 2-transitive action can be made to work for representations in good characteristic of you reinterpret the character arguments as computing dimensions of him spaces. You will want to use that a permutation representation is equivalent to its contragredient. | |
| Apr 6, 2021 at 9:38 | comment | added | HIMANSHU | @AchimKrause I am actually asking the semi simple casein my question and you have answered my question, so Thanks . Can you refer me some article to support your argument that charactersitic zero and my case behave in same way. How ? | |
| Apr 6, 2021 at 9:27 | comment | added | Achim Krause | You write "whose characteristic does NOT divide the order", is that a typo? I'm asking since you also added the "modular representation theory" tag. Typically modular representation theory refers to the case where the characteristic divides the group order. In characteristic not dividing the group order, the representation theory behaves as in characteristic zero, in particular your representation is irreducible. | |
| Apr 6, 2021 at 8:26 | review | Close votes | |||
| Apr 6, 2021 at 14:41 | |||||
| Apr 6, 2021 at 7:34 | history | asked | HIMANSHU | CC BY-SA 4.0 |