Difference between revisions of "M(8,5,7)"

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(Created page with "{{blockbox |title = M(8,5,7) - <math>B_0(kJ_1)</math> |image = M(8,5,7)quiver.png |representative = <math>B_0(kJ_1)</math> |defect = C2xC2xC2|<math>C_2 \times C_2 \times C...")
 
 
(5 intermediate revisions by the same user not shown)
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\end{array}\right)</math>
 
\end{array}\right)</math>
 
|O-morita-frob = 1
 
|O-morita-frob = 1
−
|Pic-O =
+
|Pic-O = <math>1</math><ref>Shown by Eisele, using [[References|[Ne02]]].</ref>
 +
|PIgroup = <math>(C_2 \wr S_4) \times C_2</math><ref>See 8.7 of [[References|[Ru11]]], which uses GAP.</ref>
 
|source? = No
 
|source? = No
 
|sourcereps = &nbsp;
 
|sourcereps = &nbsp;
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|k-derived = [[M(8,5,6)]], [[M(8,5,8)]]
 
|k-derived = [[M(8,5,6)]], [[M(8,5,8)]]
 
|O-derived-known? = Yes
 
|O-derived-known? = Yes
−
|coveringblocks =
+
|coveringblocks = M(8,5,7) (complete)
−
|coveredblocks =
+
|coveredblocks = M(8,5,7) (complete)
 +
|pcoveringblocks = [[M(16,14,14)]] (complete)
 
}}
 
}}
  
 +
== Basic algebra ==
  
 +
'''Quiver:''' a:<1,4>, b:<4,2>, c:<2,3>, d:<3,1>, e:<1,5>, f:<5,5>, g:<5,1>, h:<1,3>, i:<3,2>, j:<2,4>, k:<4,1>
  
−
== Basic algebra ==
+
'''Relations w.r.t. <math>k</math>:'''
−
 
 
−
<!-- '''Quiver:''' <math>a_1:<1,2>, a_2:<2,3>, a_3:<3,4>, a_4:<4,5>, a_5:<5,6>, a_6:<6,7>, a_7:<7,1></math>,
 
−
<math>b_1:<1,3>, b_2:<2,4>, b_3:<3,5>, b_4:<4,6>, b_5:<5,7>, b_6:<6,1>, b_7:<7,2></math>,
 
−
<math>c_1:<1,5>, c_2:<2,6>, c_3:<3,7>, c_4:<4,1>, c_5:<5,2>, c_6:<6,3>, c_7:<7,4></math>
 
  
−
'''Relations w.r.t. <math>k</math>:''' <math>a_1a_2=a_2a_3=a_3a_4=a_4a_5=a_5a_6=a_6a_7=a_7a_1=0</math>,
+
The basic algebra for the block defined over <math>\mathcal{O}</math> is described in [[References|[Ne02]]].
−
<math>b_1b_3=b_2b_4=b_3b_5=b_4b_6=b_5b_7=b_6b_1=b_7b_2=0</math>,
 
−
<math>c_1c_5=c_2c_6=c_3c_7=c_4c_1=c_5c_2=c_6c_3=c_7c_4=0</math>,
 
−
<math>a_1b_2=b_1a_3</math>,
 
−
<math>a_2b_3=b_2a_4</math>,
 
−
<math>a_3b_4=b_3a_5</math>,
 
−
<math>a_4b_5=b_4a_6</math>,
 
−
<math>a_5b_6=b_5a_7</math>,
 
−
<math>a_6b_7=b_6a_1</math>,
 
−
<math>a_7b_1=b_7a_2</math>,
 
−
<math>a_1c_2=c_1a_5</math>,
 
−
<math>a_2c_3=c_2a_6</math>,
 
−
<math>a_3c_4=c_3a_7</math>,
 
−
<math>a_4c_5=c_4a_1</math>,
 
−
<math>a_5c_6=c_5a_2</math>,
 
−
<math>a_6c_7=c_6a_3</math>,
 
−
<math>a_7c_1=c_7a_4</math>,
 
−
<math>b_1c_3=c_1b_5</math>,
 
−
<math>b_2c_4=c_2b_6</math>,
 
−
<math>b_3c_5=c_3b_7</math>,
 
−
<math>b_4c_6=c_4b_1</math>,
 
−
<math>b_5c_7=c_5b_2</math>,
 
−
<math>b_6c_1=c_6b_3</math>,
 
−
<math>b_7c_2=c_7b_4</math>
 
−
<math>a_1b_2c_4=b_1c_3a_7=c_1a_5b_6</math>
 
−
<math>a_2b_3c_5=b_2c_4a_1=c_2a_6b_7</math>
 
−
<math>a_3b_4c_6=b_3c_5a_2=c_3a_7b_1</math>
 
−
<math>a_4b_5c_7=b_4c_6a_3=c_4a_1b_2</math>
 
−
<math>a_5b_6c_1=b_5c_7a_4=c_5a_2b_3</math>
 
−
<math>a_6b_7c_2=b_6c_1a_5=c_6a_3b_4</math>
 
−
<math>a_7b_1c_3=b_7c_2a_6=c_7a_4b_5</math>
 
  
 
== Other notatable representatives ==
 
== Other notatable representatives ==
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== Projective indecomposable modules ==
 
== Projective indecomposable modules ==
  
−
Labelling the simple <math>B</math>-modules by <math>S_1, S_2, S_3</math>, the projective indecomposable modules have Loewy structure as follows:
+
Labelling the simple <math>B</math>-modules by <math>1,2,3,4,5</math>, the projective indecomposable modules have Loewy structure as follows:
  
−
<math>\begin{array}{ccccccc}
+
<math>\begin{array}{ccccc}
 
   \begin{array}{c}
 
   \begin{array}{c}
−
     S_1 \\
+
     1 \\
−
     S_2 S_3 S_5 \\
+
     3 \ 4 \ 5 \\
−
     S_4 S_6 S_7 \\   
+
     1 \ 2 \ 1 \ 2 \ 1 \\   
−
     S_1 \\
+
    3 \ 3 \ 4 \ 4 \ 5 \ 5 \\
 +
    1 \ 2 \ 1 \ 2 \ 1 \\
 +
    3 \ 4 \ 5 \\
 +
     1 \\
 
   \end{array},
 
   \end{array},
 
&
 
&
−
   \begin{array}{ccc}
+
   \begin{array}{c}
−
     S_2 \\
+
     2 \\
−
     S_3 S_4 S_6 \\
+
     3 \ 4 \\
−
     S_1 S_5 S_7 \\   
+
     1 \ 2 \ 1 \\   
−
     S_2 \\  
+
    3 \ 4 \ 5 \\
 +
    1 \ 2 \ 1 \\
 +
    3 \ 4 \\
 +
     2 \\
 
   \end{array},   
 
   \end{array},   
 
&  
 
&  
−
   \begin{array}{ccc}
+
   \begin{array}{c}
−
     S_3 \\
+
     3 \\
−
     S_4 S_5 S_7 \\
+
     1 \ 2 \\
−
     S_1 S_2 S_6 \\   
+
     3 \ 4 \ 5 \\   
−
     S_3 \\  
+
    1 \ 2 \ 1 \\
 +
    3 \ 4 \ 5 \\
 +
    1 \ 2 \\
 +
     3 \\
 
   \end{array}
 
   \end{array}
 
,   
 
,   
 
&  
 
&  
−
   \begin{array}{ccc}
+
   \begin{array}{c}
−
     S_4 \\
+
     4 \\
−
     S_5 S_6 S_1 \\
+
     1 \ 2 \\
−
     S_2 S_3 S_7 \\   
+
     3 \ 4 \ 5 \\   
−
    S_4 \\  
+
    1 \ 2 \ 1 \\  
−
  \end{array}, 
+
    3 \ 4 \ 5 \\
−
&
+
     1 \ 2 \\
−
  \begin{array}{ccc}
+
     4 \\
−
    S_5 \\
 
−
    S_6 S_7 S_2 \\
 
−
     S_1 S_3 S_4 \\
 
−
     S_3 \\  
 
 
   \end{array},   
 
   \end{array},   
 
&  
 
&  
 
   \begin{array}{ccc}
 
   \begin{array}{ccc}
−
     S_6 \\
+
     5 \\
−
     S_7 S_1 S_3 \\
+
     1 \ 5 \\
−
     S_2 S_4 S_5 \\   
+
     3 \ 4 \\   
−
    S_6 \\  
+
    1 \ 2 \ 1 \\  
−
  \end{array}, 
+
    3 \ 4 \ 5 \\
−
&
+
    1 \\
−
  \begin{array}{ccc}
+
     5 \\
−
    S_7 \\
+
   \end{array}  
−
    S_1 S_2 S_4 \\
 
−
    S_3 S_5 S_6 \\
 
−
     S_7 \\  
 
−
   \end{array}  
 
 
\end{array}
 
\end{array}
−
</math> -->
+
</math>
  
 
== Irreducible characters ==
 
== Irreducible characters ==
Line 146: Line 117:
  
 
[[C2xC2xC2|Back to <math>C_2 \times C_2 \times C_2</math>]]
 
[[C2xC2xC2|Back to <math>C_2 \times C_2 \times C_2</math>]]
 +
 +
== Notes ==
 +
 +
<references />

Latest revision as of 09:10, 5 June 2019

M(8,5,7) - [math]B_0(kJ_1)[/math]
M(8,5,7)quiver.png
Representative: [math]B_0(kJ_1)[/math]
Defect groups: [math]C_2 \times C_2 \times C_2[/math]
Inertial quotients: [math]C_7:C_3[/math]
[math]k(B)=[/math] 8
[math]l(B)=[/math] 5
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math]  
Cartan matrix: [math]\left( \begin{array}{ccccc} 8 & 4 & 4 & 4 & 4 \\ 4 & 4 & 3 & 3 & 1 \\ 4 & 3 & 4 & 2 & 2 \\ 4 & 3 & 2 & 4 & 2 \\ 4 & 1 & 2 & 2 & 4 \\ \end{array} \right)[/math]
Defect group Morita invariant? Yes
Inertial quotient Morita invariant? Yes
[math]\mathcal{O}[/math]-Morita classes known? Yes
[math]\mathcal{O}[/math]-Morita classes: [math]B_0(\mathcal{O}J_1)[/math]
Decomposition matrices: [math]\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] [math]1[/math][1]
[math]PI(B)=[/math] [math](C_2 \wr S_4) \times C_2[/math][2]
Source algebras known? No
Source algebra reps:  
[math]k[/math]-derived equiv. classes known? Yes
[math]k[/math]-derived equivalent to: M(8,5,6), M(8,5,8)
[math]\mathcal{O}[/math]-derived equiv. classes known? Yes
[math]p'[/math]-index covering blocks: M(8,5,7) (complete)
[math]p'[/math]-index covered blocks: M(8,5,7) (complete)
Index [math]p[/math] covering blocks: M(16,14,14) (complete)

Basic algebra

Quiver: a:<1,4>, b:<4,2>, c:<2,3>, d:<3,1>, e:<1,5>, f:<5,5>, g:<5,1>, h:<1,3>, i:<3,2>, j:<2,4>, k:<4,1>

Relations w.r.t. [math]k[/math]:

The basic algebra for the block defined over [math]\mathcal{O}[/math] is described in [Ne02].

Other notatable representatives

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]1,2,3,4,5[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccccc} \begin{array}{c} 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \ 2 \ 1 \\ 3 \ 3 \ 4 \ 4 \ 5 \ 5 \\ 1 \ 2 \ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 3 \ 4 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \\ 2 \\ \end{array}, & \begin{array}{c} 3 \\ 1 \ 2 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \\ 3 \\ \end{array} , & \begin{array}{c} 4 \\ 1 \ 2 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \\ 4 \\ \end{array}, & \begin{array}{ccc} 5 \\ 1 \ 5 \\ 3 \ 4 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \\ 5 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_2 \times C_2 \times C_2[/math]

Notes

  1. ↑ Shown by Eisele, using [Ne02].
  2. ↑ See 8.7 of [Ru11], which uses GAP.
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