Difference between revisions of "C7"
(Created page with "__NOTITLE__ == Blocks with defect group <math>C_7</math> == These are blocks with cyclic defect groups and so they are described by Brauer trees. All <math>k</math>...") |
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== Blocks with defect group <math>C_7</math> == | == Blocks with defect group <math>C_7</math> == | ||
| − | These are blocks with [[cyclic defect groups]] and so they are described by [[Brauer trees]]. | + | These are blocks with [[cyclic defect groups]] and so they are described by [[Brauer trees]]. There are candidate Brauer trees with no known block realising them. |
All <math>k</math>-Morita equivalence classes lift to <math>\mathcal{O}</math>-classes. | All <math>k</math>-Morita equivalence classes lift to <math>\mathcal{O}</math>-classes. | ||
| + | |||
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>''' | '''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>''' | ||
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| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,19)tree.png|45px]] | | || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,19)tree.png|45px]] | ||
|} | |} | ||
| + | |||
| + | To do: | ||
| + | *Check that [[M(7,1,11)]] is indeed realised by <math>B_{16}(k(2.J_2))</math>. Note that it shares its decomposition matrix with the potential class M(7,1,19). | ||
| + | *The tree labelling the potential class M(7,1,14) is believed to be realised, but an example has not been confirmed. | ||
Revision as of 15:46, 9 September 2018
Blocks with defect group [math]C_7[/math]
These are blocks with cyclic defect groups and so they are described by Brauer trees. There are candidate Brauer trees with no known block realising them.
All [math]k[/math]-Morita equivalence classes lift to [math]\mathcal{O}[/math]-classes.
CLASSIFICATION INCOMPLETE
| Class | Representative | [math]k(B)[/math] | [math]l(B)[/math] | Inertial quotients | [math]{\rm Pic}_\mathcal{O}(B)[/math] | [math]{\rm Pic}_k(B)[/math] | [math]{\rm mf_\mathcal{O}(B)}[/math] | [math]{\rm mf_k(B)}[/math] | Notes |
|---|---|---|---|---|---|---|---|---|---|
| M(7,1,1) | [math]kC_7[/math] | 7 | 1 | [math]1[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,2) | [math]kD_{15}[/math] | 5 | 2 | [math]C_2[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,3) | [math]B_0(kPSL_2(13))[/math] | 5 | 2 | [math]C_2[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,4) | [math]k(C_7:C_3)[/math] | 5 | 3 | [math]C_3[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,5) | [math]B_9(k(3.A_7))[/math] | 5 | 3 | [math]C_3[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,6) | [math]B_0(kA_7)[/math] | 5 | 3 | [math]C_3[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,7) | [math]B_{15}(k(6.A_7))[/math] | 5 | 3 | [math]C_3[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,8) | [math]k(C_7:C_6)[/math] | 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,9) | [math]B_0(k(S_7))[/math] | 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,10) | [math]B_0(kPSL_3(29^3).3)[/math] | 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,11) | [math]B_{16}(k(2.J_2))[/math] | 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small} CHECK
| ||
| M(7,1,12) | [math]B_{29}(k(2.Ru))[/math] | 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,13) | [math]B_0(k({}^2G_2(27)))[/math] | 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small}
| ||
| M(7,1,14) | 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small}
| |||
| 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small}
| ||||
| 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small}
| ||||
| 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small}
| ||||
| 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small}
| ||||
| 7 | 6 | [math]C_6[/math] | 1 | 1 | tree.png){kind=small}
|
To do:
- Check that M(7,1,11) is indeed realised by [math]B_{16}(k(2.J_2))[/math]. Note that it shares its decomposition matrix with the potential class M(7,1,19).
- The tree labelling the potential class M(7,1,14) is believed to be realised, but an example has not been confirmed.
