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

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{{blockbox
 
{{blockbox
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|title = M(8,5,8) - <math>B_0(k(\rm Aut (SL_2(8)))</math>  
+
|title = M(8,5,8) - <math>B_0(k(\rm Aut (SL_2(8))))</math>  
 
|image = M(8,5,8)quiver.png
 
|image = M(8,5,8)quiver.png
−
|representative =  <math>B_0(k(\rm Aut (SL_2(8)))</math>
+
|representative =  <math>B_0(k(\rm Aut (SL_2(8))))</math>
 
|defect = [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]]
 
|defect = [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]]
 
|inertialquotients = <math>C_7:C_3</math>
 
|inertialquotients = <math>C_7:C_3</math>
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|inertial-morita-inv? = Yes
 
|inertial-morita-inv? = Yes
 
|O-morita? = Yes
 
|O-morita? = Yes
−
|O-morita = <math>\mathcal{O} ((C_2 \times C_2 \times C_2):(C_7:C_3))</math>
+
|O-morita = <math>B_0(\mathcal{O}(\rm Aut (SL_2(8))))</math>
 
|decomp = <math>\left( \begin{array}{ccccc}
 
|decomp = <math>\left( \begin{array}{ccccc}
 
1 & 0 & 0 & 0 & 0 \\
 
1 & 0 & 0 & 0 & 0 \\
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1 & 1 & 1 & 1 & 1 \\
 
1 & 1 & 1 & 1 & 1 \\
 
1 & 1 & 1 & 2 & 1 \\
 
1 & 1 & 1 & 2 & 1 \\
−
\end{array}\right)</math>
+
\end{array}\right)</math><ref>Decomposition matrix taken from [http://www.math.rwth-aachen.de/~MOC/decomposition/], although it was first determined in [[References#L|[LM80]]] following partial results of Fong</ref>
 
|O-morita-frob = 1
 
|O-morita-frob = 1
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|Pic-O =
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|Pic-O = <math>C_3</math><ref>See [[References#E|[EL18c]]]</ref>
 +
|PIgroup =
 
|source? = No
 
|source? = No
 
|sourcereps =
 
|sourcereps =
 
|k-derived-known? = Yes
 
|k-derived-known? = Yes
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|k-derived = [[M(8,5,6)]], [[M(8,5,7)]]
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|k-derived = [[M(8,5,6)]], [[M(8,5,7)]]<ref>Derived equivalent by [[References#G|[GO97]]]</ref>
−
|O-derived-known? = Yes
+
|O-derived-known? = Yes <ref>As noted in [[References#C|[CR13]]] the derived equivalences in [[References#G|[GO97]]] are [[Glossary#Splendid equivalence|splendid]] and so lift to <math>\mathcal{O}</math></ref>
−
|coveringblocks =
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|coveringblocks = Potentially [[M(8,5,5)]]
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|coveredblocks =
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|coveredblocks = [[M(8,5,5)]]
 +
|pcoveringblocks =
 
}}
 
}}
 +
 +
The principal 2-blocks of all Ree groups <math>{}^2G_2(3^{2m+1})</math> belong to this Morita equivalence class.
  
 
== Basic algebra ==
 
== Basic algebra ==
  
−
'''Quiver:'''  
+
'''Quiver:''' a:<4,1>, b:<1,4>, c:<2,4>, d:<4,2>, e:<4,3>, f:<3,4>, g:<4,5>, h:<5,4><ref>Computed using [http://magma.maths.usyd.edu.au/magma/ MAGMA]</ref>
  
 
'''Relations w.r.t. <math>k</math>:'''  
 
'''Relations w.r.t. <math>k</math>:'''  
  
 
== Other notatable representatives ==
 
== Other notatable representatives ==
 +
 +
<math>{\rm Aut (SL_2(8))} \cong {}^2G_2(3)</math>, and the blocks <math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math> are Morita equivalent for all <math>m</math>. This follows from Example 3.3 of [[References#O|[Ok97]]], where, as noted in 6.2.2 of [[References#C|[CR13]]] the Morita equivalence is [[Glossary#Splendid equivalence|splendid]] and so lifts to <math>\mathcal{O}</math>
 +
 +
By [[References#E|[Ea16]]] the principal block of each subgroup of <math>{\rm Aut}({}^2G_2(3^{2m+1}))</math> containing <math>{}^2G_2(3^{2m+1})</math> is in this Morita equivalence class.
  
 
== 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<ref>The structure of the projective indecomposable modules was first given in [[References#L|[LM80]]], although with a mistake corrected in [[References#G|[GO97]]]</ref>:
  
−
<math>\begin{array}{ccccccc}
+
<math>\begin{array}{ccccc}
 
   \begin{array}{c}
 
   \begin{array}{c}
−
     S_1 \\
+
     1 \\
−
    S_2 S_3 S_5 \\
+
    4 \\
−
    S_4 S_6 S_7 \\   
+
    1 \ 2 \ 3 \ 5 \\
−
     S_1 \\
+
    4 \ 4 \\
 +
    1 \ 2 \ 3 \ 5 \\
 +
    4 \\
 +
    1 \\
 +
  \end{array},
 +
&
 +
\begin{array}{c}
 +
    2 \\
 +
    4 \\
 +
    1 \ 2 \ 3 \ 5 \\
 +
    4 \ 4 \\
 +
    1 \ 2 \ 3 \ 5 \\
 +
    4 \\
 +
    2 \\
 +
  \end{array},
 +
&
 +
  \begin{array}{c}
 +
    3 \\
 +
    4 \\
 +
    1 \ 2 \ 3 \ 5 \\
 +
    4 \ 4 \\
 +
    1 \ 2 \ 3 \ 5 \\
 +
    4 \\
 +
     3 \\
 
   \end{array},
 
   \end{array},
 
&
 
&
−
  \begin{array}{ccc}
+
\begin{array}{c}
−
     S_2 \\
+
     4 \\
−
    S_3 S_4 S_6 \\
+
    1 \ 2 \ 3 \ 5 \\
−
    S_1 S_5 S_7 \\
+
    4 \ 4 \ 4 \\
−
     S_2 \\  
+
     1 \ 1 \ 2 \ 2 \ 3 \ 3 \ 5 \\
−
   \end{array},
+
    4 \ 4 \ 4 \\
−
&  
+
    1 \ 2 \ 3 \ 5 \\
−
  \begin{array}{ccc}
+
    4 \\
−
     S_3 \\
+
   \end{array},
−
    S_4 S_5 S_7 \\
+
&
−
    S_1 S_2 S_6 \\
+
\begin{array}{c}
−
     S_3 \\  
+
     5 \\
 +
    4 \\
 +
    1 \ 2 \ 3 \\
 +
    4 \\
 +
    1 \ 2 \ 3 \\
 +
    4 \\
 +
     5 \\
 
   \end{array}
 
   \end{array}
−
, 
 
−
&
 
−
  \begin{array}{ccc}
 
−
    S_4 \\
 
−
    S_5 S_6 S_1 \\
 
−
    S_2 S_3 S_7 \\ 
 
−
    S_4 \\
 
−
  \end{array}, 
 
−
&
 
−
  \begin{array}{ccc}
 
−
    S_5 \\
 
−
    S_6 S_7 S_2 \\
 
−
    S_1 S_3 S_4 \\ 
 
−
    S_3 \\
 
−
  \end{array}, 
 
−
&
 
−
  \begin{array}{ccc}
 
−
    S_6 \\
 
−
    S_7 S_1 S_3 \\
 
−
    S_2 S_4 S_5 \\ 
 
−
    S_6 \\
 
−
  \end{array}, 
 
−
&
 
−
  \begin{array}{ccc}
 
−
    S_7 \\
 
−
    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 ==
  
 
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>]]

Latest revision as of 14:22, 12 April 2019

M(8,5,8) - [math]B_0(k(\rm Aut (SL_2(8))))[/math]
M(8,5,8)quiver.png
Representative: [math]B_0(k(\rm Aut (SL_2(8))))[/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}{ccccccc} 4 & 2 & 2 & 4 & 2 \\ 2 & 4 & 2 & 4 & 2 \\ 2 & 2 & 4 & 4 & 2 \\ 4 & 4 & 4 & 8 & 3 \\ 2 & 2 & 2 & 3 & 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}(\rm Aut (SL_2(8))))[/math]
Decomposition matrices: [math]\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ \end{array}\right)[/math][1]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] [math]C_3[/math][2]
[math]PI(B)=[/math]
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(8,5,7)[3]
[math]\mathcal{O}[/math]-derived equiv. classes known? Yes [4]
[math]p'[/math]-index covering blocks: Potentially M(8,5,5)
[math]p'[/math]-index covered blocks: M(8,5,5)
Index [math]p[/math] covering blocks:

The principal 2-blocks of all Ree groups [math]{}^2G_2(3^{2m+1})[/math] belong to this Morita equivalence class.

Basic algebra

Quiver: a:<4,1>, b:<1,4>, c:<2,4>, d:<4,2>, e:<4,3>, f:<3,4>, g:<4,5>, h:<5,4>[5]

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

Other notatable representatives

[math]{\rm Aut (SL_2(8))} \cong {}^2G_2(3)[/math], and the blocks [math]B_0(\mathcal{O}({}^2G_2(3^{2m+1})))[/math] are Morita equivalent for all [math]m[/math]. This follows from Example 3.3 of [Ok97], where, as noted in 6.2.2 of [CR13] the Morita equivalence is splendid and so lifts to [math]\mathcal{O}[/math]

By [Ea16] the principal block of each subgroup of [math]{\rm Aut}({}^2G_2(3^{2m+1}))[/math] containing [math]{}^2G_2(3^{2m+1})[/math] is in this Morita equivalence class.

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[6]:

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

Irreducible characters

All irreducible characters have height zero.

Notes

  1. ↑ Decomposition matrix taken from [1], although it was first determined in [LM80] following partial results of Fong
  2. ↑ See [EL18c]
  3. ↑ Derived equivalent by [GO97]
  4. ↑ As noted in [CR13] the derived equivalences in [GO97] are splendid and so lift to [math]\mathcal{O}[/math]
  5. ↑ Computed using MAGMA
  6. ↑ The structure of the projective indecomposable modules was first given in [LM80], although with a mistake corrected in [GO97]

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

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