Difference between revisions of "C7"

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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]]. There are candidate Brauer trees with no known block realising them.  
+
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.
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! scope="col"| Class
 
! scope="col"| Class
 
! scope="col"| Representative
 
! scope="col"| Representative
 +
! scope="col"| # lifts / <math>\mathcal{O}</math>
 
! scope="col"| <math>k(B)</math>
 
! scope="col"| <math>k(B)</math>
 
! scope="col"| <math>l(B)</math>
 
! scope="col"| <math>l(B)</math>
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|-
 
|-
−
|[[M(7,1,1)]] || <math>kC_7</math> ||7 ||1 ||<math>1</math> || || ||1 ||1 || [[Image:M(7,1,1)tree.png|45px]]
+
|[[M(7,1,1)]] || <math>kC_7</math> || 1 ||7 ||1 ||<math>1</math> || <math>C_7:C_6</math> || ||1 ||1 || [[Image:M(7,1,1)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,2)]] || <math>kD_{15}</math> ||5 ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(7,1,2)tree.png|45px]]
+
|[[M(7,1,2)]] || <math>kD_{14}</math> || 1 ||5 ||2 ||<math>C_2</math> || <math>C_2 \times C_3</math> || ||1 ||1 || [[Image:M(7,1,2)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,3)]] || <math>B_0(kPSL_2(13))</math> ||5 ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(7,1,3)tree.png|45px]]
+
|[[M(7,1,3)]] || <math>B_0(kPSL_2(13))</math> || 1 ||5 ||2 ||<math>C_2</math> || <math>C_3</math> || ||1 ||1 || [[Image:M(7,1,3)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,4)]] || <math>k(C_7:C_3)</math> ||5 ||3 ||<math>C_3</math> || || ||1 ||1 || [[Image:M(7,1,4)tree.png|45px]]
+
|[[M(7,1,4)]] || <math>k(C_7:C_3)</math> || 1 ||5 ||3 ||<math>C_3</math> || <math>C_3 \times C_2</math> || ||1 ||1 || [[Image:M(7,1,4)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,5)]] || <math>B_9(k(3.A_7))</math> ||5 ||3 ||<math>C_3</math> || || ||1 ||1 || [[Image:M(7,1,5)tree.png|45px]]
+
|[[M(7,1,5)]] || <math>B_9(k(3.A_7))</math> || 1 ||5 ||3 ||<math>C_3</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(7,1,5)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,6)]] || <math>B_0(kA_7)</math> ||5 ||3 ||<math>C_3</math> || || ||1 ||1 || [[Image:M(7,1,6)tree.png|45px]]
+
|[[M(7,1,6)]] || <math>B_0(kA_7)</math> || 1 ||5 ||3 ||<math>C_3</math> ||<math>C_2</math> || ||1 ||1 || [[Image:M(7,1,6)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,7)]] || <math>B_{15}(k(6.A_7))</math> ||5 ||3 ||<math>C_3</math> |||| ||1 ||1 || [[Image:M(7,1,7)tree.png|45px]]
+
|[[M(7,1,7)]] || <math>B_{15}(k(6.A_7))</math> || 1 ||5 ||3 ||<math>C_3</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(7,1,7)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,8)]] || <math>k(C_7:C_6)</math> ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,8)tree.png|45px]]
+
|[[M(7,1,8)]] || <math>k(C_7:C_6)</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>C_6</math> || ||1 ||1 || [[Image:M(7,1,8)tree.png|45px]]
 +
|-,
 +
|[[M(7,1,9)]] || <math>B_0(k(S_7))</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>C_2</math> || ||1 ||1 || [[Image:M(7,1,9)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,9)]] || <math>B_0(k(S_7))</math> ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,9)tree.png|45px]]
+
|[[M(7,1,10)]] || <math>B_0(kPSL_3(29^3).3)</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>C_3</math> || ||1 ||1 || [[Image:M(7,1,10)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,10)]] || <math>B_0(kPSL_3(29^3).3)</math> ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,10)tree.png|45px]]
+
|[[M(7,1,11)]] || <math>B_{16}(k(2.J_2))</math> || 1 ||7 ||6 ||<math>C_6</math> || ? || ||1 ||1 || [[Image:M(7,1,11)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,11)]] || <math>B_{16}(k(2.J_2))</math> ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,11)tree.png|45px]]
+
|[[M(7,1,12)]] || <math>B_{29}(k(2.Ru))</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>1</math> || ||1 ||1 || [[Image:M(7,1,12)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,12)]] || <math>B_{29}(k(2.Ru))</math> ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,12)tree.png|45px]]
+
|[[M(7,1,13)]] || <math>B_0(k({}^2G_2(27)))</math> || 1 ||7 ||6 ||<math>C_6</math> || <math>1</math> || ||1 ||1 || [[Image:M(7,1,13)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,13)]] || <math>B_0(k({}^2G_2(27)))</math> ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,13)tree.png|45px]]
+
|[[M(7,1,14)]] ||<math>B_0(k({}^2G_2(243)))</math><ref>Theorem 3.8 of [[References|[Du14]]] gives existence for <math>q=3^{2m+1}</math> such that <math>(3^{2m+1}-3^{m+1}+1)_7=7</math></ref> || 1 ||7 ||6 ||<math>C_6</math> ||<math>1</math> || ||1 ||1 || [[Image:M(7,1,14)tree.png|45px]]
 
|-
 
|-
−
|[[M(7,1,14)]] || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,14)tree.png|45px]]
+
| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,15)tree.png|45px]]
 
|-
 
|-
−
| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,15)tree.png|45px]]
+
| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,16)tree.png|45px]]
 
|-
 
|-
−
| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,16)tree.png|45px]]
+
| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,17)tree.png|45px]]
 
|-
 
|-
−
| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,17)tree.png|45px]]
+
| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,18)tree.png|45px]]
 
|-
 
|-
−
| || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,18)tree.png|45px]]
+
| || || ||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]]
+
| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,20)tree.png|45px]]
 +
|-
 +
| || || ||7 ||6 ||<math>C_6</math> |||| ||1 ||1 || [[Image:M(7,1,21)tree.png|45px]]
 
|}
 
|}
  
 
To do:  
 
To do:  
−
*The tree labelling the potential class M(7,1,14) is believed to be realised, but an example has not been confirmed.
+
*Picard group for [[M(7,1,11)]] trivial or <math>C_2</math>?
 +
 
 +
== Notes ==
 +
<references />

Latest revision as of 11:58, 22 November 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 # lifts / [math]\mathcal{O}[/math] [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] 1 7 1 [math]1[/math] [math]C_7:C_6[/math] 1 1 M(7,1,1)tree.png
M(7,1,2) [math]kD_{14}[/math] 1 5 2 [math]C_2[/math] [math]C_2 \times C_3[/math] 1 1 M(7,1,2)tree.png
M(7,1,3) [math]B_0(kPSL_2(13))[/math] 1 5 2 [math]C_2[/math] [math]C_3[/math] 1 1 M(7,1,3)tree.png
M(7,1,4) [math]k(C_7:C_3)[/math] 1 5 3 [math]C_3[/math] [math]C_3 \times C_2[/math] 1 1 M(7,1,4)tree.png
M(7,1,5) [math]B_9(k(3.A_7))[/math] 1 5 3 [math]C_3[/math] [math]C_2[/math] 1 1 M(7,1,5)tree.png
M(7,1,6) [math]B_0(kA_7)[/math] 1 5 3 [math]C_3[/math] [math]C_2[/math] 1 1 M(7,1,6)tree.png
M(7,1,7) [math]B_{15}(k(6.A_7))[/math] 1 5 3 [math]C_3[/math] [math]C_2[/math] 1 1 M(7,1,7)tree.png
M(7,1,8) [math]k(C_7:C_6)[/math] 1 7 6 [math]C_6[/math] [math]C_6[/math] 1 1 M(7,1,8)tree.png
M(7,1,9) [math]B_0(k(S_7))[/math] 1 7 6 [math]C_6[/math] [math]C_2[/math] 1 1 M(7,1,9)tree.png
M(7,1,10) [math]B_0(kPSL_3(29^3).3)[/math] 1 7 6 [math]C_6[/math] [math]C_3[/math] 1 1 M(7,1,10)tree.png
M(7,1,11) [math]B_{16}(k(2.J_2))[/math] 1 7 6 [math]C_6[/math]  ? 1 1 M(7,1,11)tree.png
M(7,1,12) [math]B_{29}(k(2.Ru))[/math] 1 7 6 [math]C_6[/math] [math]1[/math] 1 1 M(7,1,12)tree.png
M(7,1,13) [math]B_0(k({}^2G_2(27)))[/math] 1 7 6 [math]C_6[/math] [math]1[/math] 1 1 M(7,1,13)tree.png
M(7,1,14) [math]B_0(k({}^2G_2(243)))[/math][1] 1 7 6 [math]C_6[/math] [math]1[/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
7 6 [math]C_6[/math] 1 1 M(7,1,20)tree.png
7 6 [math]C_6[/math] 1 1 M(7,1,21)tree.png

To do:

  • Picard group for M(7,1,11) trivial or [math]C_2[/math]?

Notes

  1. ↑ Theorem 3.8 of [Du14] gives existence for [math]q=3^{2m+1}[/math] such that [math](3^{2m+1}-3^{m+1}+1)_7=7[/math]
Retrieved from ‘http://wiki.manchester.ac.uk/blocks/index.php?title=C7&oldid=699’