Timeline for Irreducible tensor product representations in finite simple groups
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| May 28 at 14:24 | comment | added | Sebastien Palcoux | @NickGill About your second last comment, the answer is yes. | |
| May 28 at 10:44 | comment | added | Nick Gill | BTW, they summarise the Bessenrodt -- Kleschchev result as follows: if $\alpha, \beta$ are complex irreducible characters of degree $>1$ for an alternating group $A_n$ ($n\geq 5$), then $\alpha\otimes\beta$ is irreducible if and only if $n=a^2$ for some $a\geq 3$, one of the characters $\alpha, \beta$ is the nontrivial irreducible component of the natural permuation character, and the other one corresponds to the partition $(a^a)$. | |
| May 28 at 10:42 | comment | added | Nick Gill | @SebastienPalcoux, Can I check what, exactly, you are checking: do I understand that the 9 sporadic groups that you list are the ones with the property that there exist $U, V$ irreducible reps of dim $>1$ such that $U\otimes V$ is also irreducible? If so, then, yes, this seems inconsistent with the publication I link to. That publication only mentions the sporadic groups as an aside... and their example for $BM$, for instance, seems odd -- it refers to a rep which exists for $2BM$ but not for $BM$. So I am confused. | |
| May 28 at 4:25 | comment | added | Sebastien Palcoux | @NickGill: I recalculated for the sporadic groups (see my edits), and the results differ from those mentioned in the review you referred to in your first comment above. | |
| May 28 at 4:22 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
Improved the script following Max Horn's answer. Improved the computation. For the sporadic groups, the result differs from the literature...
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| May 27 at 14:56 | comment | added | Geoff Robinson | Notice that the groups ${\rm PSL}(2,5)$ and ${\rm PSL}(2,9)$ are two examples of simple groups which have irreducible characters that factorize when considered as characters of covering groups. | |
| May 27 at 7:58 | answer | added | Max Horn | timeline score: 4 | |
| May 26 at 21:13 | answer | added | Geoff Robinson | timeline score: 5 | |
| May 26 at 18:33 | comment | added | Geoff Robinson | You get more examples if you work with quasimple groups instead of simple ones ( though the product characters might not be faithful, but contain (some of) the centre in their kernel. For example, the degree 4 irreducible characters of $A_{5}$ is a product of two characters of degree $2$ of ${\rm SL}(2,5).$ | |
| May 26 at 14:25 | comment | added | Nick Gill | The problem has been solved for the alternating groups by Bessenrodt and Kleshchev: zbmath.org/1009.20013 | |
| May 26 at 14:22 | comment | added | Nick Gill | Magaard and Tiep have studied this problem in several papers. For instance, zbmath.org/0992.20009 | |
| May 26 at 2:19 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
Distinguished the internal and the external tensor product, as suggested by LSpice in comment
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| May 26 at 0:32 | comment | added | Victor Ostrik | Here is an example: for $k=5$ representation $V_1$ is of dimension 24 and representation $V_2$ is of dimension $350574510$ (I used wonderful Young Diagram Calculator integral-domain.org/lwilliams/Applets/Math/YoungDiagrams.php ) | |
| May 26 at 0:25 | comment | added | Victor Ostrik | Irreducible tensor products do exist for alternating group $A_n$ where $n$ is a square. Namely take $V_1$ to be representation of dimension $n-1$ (just as you suggested) and $V_2$ to be representation corresponding to square Young diagram of size $k\times k$ (since the diagram is invariant under transposition, the corresponding representation of the symmetric group splits into 2 non-isomorphic representations of the alternating group; choose any of them to be $V_2$). Then tensor product $V_1\otimes V_2$ is isomorphic to representation corresponding to the diagram $(k+1,k,\ldots,k,k-1)$. | |
| May 25 at 15:09 | history | edited | YCor | CC BY-SA 4.0 |
formatting
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| May 25 at 15:00 | history | asked | Sebastien Palcoux | CC BY-SA 4.0 |