Difference between revisions of "C7"

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(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>...")
 
(Blocks with defect group C_7)
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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]].
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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]]
 
|}
 
|}
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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).
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*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 M(7,1,1)tree.png
M(7,1,2) [math]kD_{15}[/math] 5 2 [math]C_2[/math] 1 1 M(7,1,2)tree.png
M(7,1,3) [math]B_0(kPSL_2(13))[/math] 5 2 [math]C_2[/math] 1 1 M(7,1,3)tree.png
M(7,1,4) [math]k(C_7:C_3)[/math] 5 3 [math]C_3[/math] 1 1 M(7,1,4)tree.png
M(7,1,5) [math]B_9(k(3.A_7))[/math] 5 3 [math]C_3[/math] 1 1 M(7,1,5)tree.png
M(7,1,6) [math]B_0(kA_7)[/math] 5 3 [math]C_3[/math] 1 1 M(7,1,6)tree.png
M(7,1,7) [math]B_{15}(k(6.A_7))[/math] 5 3 [math]C_3[/math] 1 1 M(7,1,7)tree.png
M(7,1,8) [math]k(C_7:C_6)[/math] 7 6 [math]C_6[/math] 1 1 M(7,1,8)tree.png
M(7,1,9) [math]B_0(k(S_7))[/math] 7 6 [math]C_6[/math] 1 1 M(7,1,9)tree.png
M(7,1,10) [math]B_0(kPSL_3(29^3).3)[/math] 7 6 [math]C_6[/math] 1 1 M(7,1,10)tree.png
M(7,1,11) [math]B_{16}(k(2.J_2))[/math] 7 6 [math]C_6[/math] 1 1 M(7,1,11)tree.png CHECK
M(7,1,12) [math]B_{29}(k(2.Ru))[/math] 7 6 [math]C_6[/math] 1 1 M(7,1,12)tree.png
M(7,1,13) [math]B_0(k({}^2G_2(27)))[/math] 7 6 [math]C_6[/math] 1 1 M(7,1,13)tree.png
M(7,1,14) 7 6 [math]C_6[/math] 1 1 M(7,1,14)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,15)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,16)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,17)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,18)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,19)tree.png

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.