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

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([Al79] reference)
(Relations and references)
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1 & 0 & 1 & 1 & 1 & 0 & 0 \\
 
1 & 0 & 1 & 1 & 1 & 0 & 0 \\
 
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
 
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
−
\end{array}\right)</math>
+
\end{array}\right)</math><ref>Decomposition matrix taken from [http://www.math.rwth-aachen.de/~MOC/decomposition/]</ref>
 
|O-morita-frob = 1
 
|O-morita-frob = 1
 
|Pic-O = <math>C_3</math>;
 
|Pic-O = <math>C_3</math>;
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|k-derived-known? = Yes
 
|k-derived-known? = Yes
 
|k-derived = [[M(8,5,4)]]
 
|k-derived = [[M(8,5,4)]]
−
|O-derived-known? = Yes
+
|O-derived-known? = Yes<ref>A [[Glossary#Splendid equivalence|splendid]] Rickard equivalence is given in [[References|[Ro95, 2.3]]], which then lifts to <math>\mathcal{O}</math></ref>
−
|coveringblocks =
+
|coveringblocks = [[M(8,5,8)]]
−
|coveredblocks =
+
|coveredblocks = Potentially [[M(8,5,8)]]
 
}}
 
}}
  
−
The projective indecomposable modules of the <math>2</math>-blocks of the groups <math>SL_2(2^n)</math> were computed by Alperin in [[References|[Al79]]].
+
The projective indecomposable modules of the <math>2</math>-blocks of the groups <math>SL_2(2^n)</math> were computed by Alperin in [[References|[Al79]]] and the quiver and relations by Koshita in [[References|[Ko94]]]. A splendid derived equivalence with [[M(8,5,4)]] was constructed by Rouquier in [[References|[Ro95]]].
  
 
== Basic algebra ==
 
== Basic algebra ==
  
−
<!-- '''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>,  
+
'''Quiver:''' a<1,2>, b:<2,3>, c:<3,2>, d:<2,1>, e:<1,4>, f:<4,5>, g:<5,4>, h:<4,1>, i:<1,6>, j:<6,7>, k:<7,6>, l:<6,1>
−
<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>,
+
'''Relations w.r.t. <math>k</math>:''' <math>da=he=li=0</math>,  
−
<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>cb=gf=kj=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>bcd=dil</math>, <math>fgh=had</math>, <math>jkl=leh</math>,
−
<math>a_1b_2=b_1a_3</math>,
+
<math>abc=ila</math>, <math>efg=ade</math>, <math>ijk=ehi</math>,
−
<math>a_2b_3=b_2a_4</math>,
+
<math>cdi=gha=kle</math>,
−
<math>a_3b_4=b_3a_5</math>,
+
<math>lab=def=hij</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>1,2,3,4,5,6,7</math>, the projective indecomposable modules have Loewy structure as follows:
  
 
<math>\begin{array}{ccccccc}
 
<math>\begin{array}{ccccccc}
 
   \begin{array}{c}
 
   \begin{array}{c}
−
     S_1 \\
+
     1 \\
−
     S_2 S_3 S_5 \\
+
     2 \ 4 \ 6 \\
−
     S_4 S_6 S_7 \\   
+
     3 \ 1 \ 5 \ 1 \ 7 \ 1 \\
 +
    2 \ 6 \ 4 \ 2 \ 6 \ 4 \\
 +
    \\   
 
     S_1 \\
 
     S_1 \\
 
   \end{array},
 
   \end{array},
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All irreducible characters have height zero.
 
All irreducible characters have height zero.
 +
 +
== Notes ==
 +
<references />
  
 
[[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>]]

Revision as of 14:22, 12 October 2018

M(8,5,5) - [math]B_0(kSL_2(8))[/math]
M(8,5,5)quiver.png
Representative: [math]B_0(kSL_2(8))[/math]
Defect groups: [math]C_2 \times C_2 \times C_2[/math]
Inertial quotients: [math]C_7[/math]
[math]k(B)=[/math] 8
[math]l(B)=[/math] 7
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math]  
Cartan matrix: [math]\left( \begin{array}{ccccccc} 8 & 4 & 4 & 4 & 2 & 2 & 2 \\ 4 & 4 & 2 & 2 & 0 & 2 & 1 \\ 4 & 2 & 4 & 2 & 1 & 0 & 2 \\ 4 & 2 & 2 & 4 & 2 & 1 & 0 \\ 2 & 0 & 1 & 2 & 2 & 0 & 0 \\ 2 & 2 & 0 & 1 & 0 & 2 & 0 \\ 2 & 1 & 2 & 0 & 0 & 0 & 2 \\ \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}SL_2(8))[/math]
Decomposition matrices: [math]\left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ \end{array}\right)[/math][1]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] [math]C_3[/math];
[math]PI(B)=[/math] {{{PIgroup}}}
Source algebras known? No
Source algebra reps:  
[math]k[/math]-derived equiv. classes known? Yes
[math]k[/math]-derived equivalent to: M(8,5,4)
[math]\mathcal{O}[/math]-derived equiv. classes known? Yes[2]
[math]p'[/math]-index covering blocks: M(8,5,8)
[math]p'[/math]-index covered blocks: Potentially M(8,5,8)
Index [math]p[/math] covering blocks: {{{pcoveringblocks}}}

The projective indecomposable modules of the [math]2[/math]-blocks of the groups [math]SL_2(2^n)[/math] were computed by Alperin in [Al79] and the quiver and relations by Koshita in [Ko94]. A splendid derived equivalence with M(8,5,4) was constructed by Rouquier in [Ro95].

Basic algebra

Quiver: a<1,2>, b:<2,3>, c:<3,2>, d:<2,1>, e:<1,4>, f:<4,5>, g:<5,4>, h:<4,1>, i:<1,6>, j:<6,7>, k:<7,6>, l:<6,1>

Relations w.r.t. [math]k[/math]: [math]da=he=li=0[/math], [math]cb=gf=kj=0[/math], [math]bcd=dil[/math], [math]fgh=had[/math], [math]jkl=leh[/math], [math]abc=ila[/math], [math]efg=ade[/math], [math]ijk=ehi[/math], [math]cdi=gha=kle[/math], [math]lab=def=hij[/math]

Other notatable representatives

Projective indecomposable modules

Irreducible characters

All irreducible characters have height zero.

Notes

  1. ↑ Decomposition matrix taken from [1]
  2. ↑ A splendid Rickard equivalence is given in [Ro95, 2.3], which then lifts to [math]\mathcal{O}[/math]

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

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