Difference between revisions of "C2xC2xC2"

From Block library
Jump to: navigation, search
(Switched M(8,5,2) and M(8,5,3))
 
(8 intermediate revisions by the same user not shown)
Line 3: Line 3:
 
== Blocks with defect group <math>C_2 \times C_2 \times C_2</math> ==
 
== Blocks with defect group <math>C_2 \times C_2 \times C_2</math> ==
  
Each of the eight <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The classification uses the CFSG.
+
These were classified in [[References|[Ea16]]] using the [[Glossary#CFSG|CFSG]]. The Picard groups with respect to <math>\mathcal{O}</math> are computed in [[References#E|[EL18c]]] with the exception of the principal block of <math>J_1</math>, which has been computed by Eisele.
  
 
{| class="wikitable"
 
{| class="wikitable"
Line 9: Line 9:
 
! scope="col"| Class
 
! scope="col"| Class
 
! scope="col"| Representative
 
! scope="col"| Representative
 +
! scope="col"| [[Glossary|# 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>
Line 19: Line 20:
  
 
|-
 
|-
|[[M(8,5,1)]] || <math>k(C_2 \times C_2 \times C_2)</math> ||8 ||1 ||<math>1</math> || || ||1 ||1 ||
+
|[[M(8,5,1)]] || <math>k(C_2 \times C_2 \times C_2)</math> || 1 ||8 ||1 ||<math>1</math> || <math>(C_2 \times C_2 \times C_2):GL_3(2)</math> || ||1 ||1 ||
 
|-
 
|-
|[[M(8,5,2)]] || <math>B_0(k(A_5 \times C_2))</math> ||8 ||3 ||<math>C_3</math> || || ||1 ||1 ||  
+
|[[M(8,5,2)]] || <math>B_0(k(A_5 \times C_2))</math> || 1 ||8 ||3 ||<math>C_3</math> || <math>C_2 \times C_2</math>|| ||1 ||1 ||  
 
|-
 
|-
|[[M(8,5,3)]] || <math>k(A_4 \times C_2)</math> ||8 ||3 ||<math>C_3</math> || || ||1 ||1 ||  
+
|[[M(8,5,3)]] || <math>k(A_4 \times C_2)</math> || 1 ||8 ||3 ||<math>C_3</math> || <math>S_3 \times C_2</math>|| ||1 ||1 ||  
 
|-
 
|-
|[[M(8,5,4)]] || <math>k((C_2 \times C_2 \times C_2):C_7)</math> ||8 ||7 ||<math>C_7</math> || || ||1 ||1 ||  
+
|[[M(8,5,4)]] || <math>k((C_2 \times C_2 \times C_2):C_7)</math> || 1 ||8 ||7 ||<math>C_7</math> || <math>C_7:C_3</math>|| ||1 ||1 ||  
 
|-
 
|-
|[[M(8,5,5)]] || <math>B_0(kSL_2(8))</math> ||8 ||7 ||<math>C_7</math> || || ||1 ||1 ||  
+
|[[M(8,5,5)]] || <math>B_0(kSL_2(8))</math> || 1 ||8 ||7 ||<math>C_7</math> || <math>C_3</math> || ||1 ||1 ||  
 
|-
 
|-
|[[M(8,5,6)]] || <math>k((C_2 \times C_2 \times C_2):(C_7:C_3))</math> ||8 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||  
+
|[[M(8,5,6)]] || <math>k((C_2 \times C_2 \times C_2):(C_7:C_3))</math> || 1 ||8 ||5 ||<math>C_7:C_3</math> || <math>C_3</math> || ||1 ||1 ||  
 
|-
 
|-
|[[M(8,5,7)]] || <math>B_0(kJ_1)</math> ||8 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||
+
|[[M(8,5,7)]] || <math>B_0(kJ_1)</math> || 1 ||8 ||5 ||<math>C_7:C_3</math> || <math>1</math> || ||1 ||1 ||
 
|-  
 
|-  
|[[M(8,5,8)]] || <math>B_0(k{\rm Aut}(SL_2(8)))</math> ||8 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||  
+
|[[M(8,5,8)]] || <math>B_0(k{\rm Aut}(SL_2(8)))</math> || 1 ||8 ||5 ||<math>C_7:C_3</math> || <math>C_3</math> || ||1 ||1 ||  
 
|}
 
|}
  
[[M(8,5,2)]] and [[M(8,5,3)]] are derived equivalent over <math>\mathcal{O}</math>.
+
[[M(8,5,2)]] and [[M(8,5,3)]] are derived equivalent over <math>\mathcal{O}</math>. This is a consequence of the derived equivalence between [[M(4,2,2)]] and [[M(4,2,3)]] (see [[References#R|[Ri96]]]).
  
[[M(8,5,4)]] and [[M(8,5,5)]] are derived equivalent over <math>\mathcal{O}</math>.
+
[[M(8,5,4)]] and [[M(8,5,5)]] are derived equivalent over <math>\mathcal{O}</math>. See [[References#R|[Ro95]]].
  
[[M(8,5,6)]], [[M(8,5,7)]] and [[M(8,5,8)]] are derived equivalent over <math>\mathcal{O}</math>.
+
[[M(8,5,6)]], [[M(8,5,7)]] and [[M(8,5,8)]] are derived equivalent over <math>\mathcal{O}</math>. See [[References#G|[Go97]]], [[References#O|[Ok97]]] and [[References#C|[CR13]]].

Latest revision as of 16:49, 23 September 2019

Blocks with defect group [math]C_2 \times C_2 \times C_2[/math]

These were classified in [Ea16] using the CFSG. The Picard groups with respect to [math]\mathcal{O}[/math] are computed in [EL18c] with the exception of the principal block of [math]J_1[/math], which has been computed by Eisele.

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(8,5,1) [math]k(C_2 \times C_2 \times C_2)[/math] 1 8 1 [math]1[/math] [math](C_2 \times C_2 \times C_2):GL_3(2)[/math] 1 1
M(8,5,2) [math]B_0(k(A_5 \times C_2))[/math] 1 8 3 [math]C_3[/math] [math]C_2 \times C_2[/math] 1 1
M(8,5,3) [math]k(A_4 \times C_2)[/math] 1 8 3 [math]C_3[/math] [math]S_3 \times C_2[/math] 1 1
M(8,5,4) [math]k((C_2 \times C_2 \times C_2):C_7)[/math] 1 8 7 [math]C_7[/math] [math]C_7:C_3[/math] 1 1
M(8,5,5) [math]B_0(kSL_2(8))[/math] 1 8 7 [math]C_7[/math] [math]C_3[/math] 1 1
M(8,5,6) [math]k((C_2 \times C_2 \times C_2):(C_7:C_3))[/math] 1 8 5 [math]C_7:C_3[/math] [math]C_3[/math] 1 1
M(8,5,7) [math]B_0(kJ_1)[/math] 1 8 5 [math]C_7:C_3[/math] [math]1[/math] 1 1
M(8,5,8) [math]B_0(k{\rm Aut}(SL_2(8)))[/math] 1 8 5 [math]C_7:C_3[/math] [math]C_3[/math] 1 1

M(8,5,2) and M(8,5,3) are derived equivalent over [math]\mathcal{O}[/math]. This is a consequence of the derived equivalence between M(4,2,2) and M(4,2,3) (see [Ri96]).

M(8,5,4) and M(8,5,5) are derived equivalent over [math]\mathcal{O}[/math]. See [Ro95].

M(8,5,6), M(8,5,7) and M(8,5,8) are derived equivalent over [math]\mathcal{O}[/math]. See [Go97], [Ok97] and [CR13].