M(32,51,25) - Revision history

CesareGArdito: Created page with "{{blockbox |title = M(32,51,25) - $B_0(k(((C_2)^3 : (C_7:C_3)) \times A_5))$ |image = |representative = $B_0(k(((C_2)^3 : (C_7:C_3)) \times A_5))$
2019-12-09T13:08:13Z

Created page with "{{blockbox |title = M(32,51,25) - <math>B_0(k(((C_2)^3 : (C_7:C_3)) \times A_5))</math> |image = |representative = <math>B_0(k(((C_2)^3 : (C_7:C_3)) \times A_5))</ma..."

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{{blockbox
|title = M(32,51,25) - <math>B_0(k(((C_2)^3 : (C_7:C_3)) \times A_5))</math>
|image =  
|representative = <math>B_0(k(((C_2)^3 : (C_7:C_3)) \times A_5))</math>
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
|inertialquotients = <math>(C_{7}:C_3) \times C_3</math>
|k(B) = 32
|l(B) = 15
|k-morita-frob = 1
|Pic-k=  
|cartan = See below.
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O}(((C_2)^3 : (C_7:C_3)) \times A_5))</math>
|decomp = See below.
|O-morita-frob = 1
|Pic-O =
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(32,51,24)]], [[M(32,51,26)]], [[M(32,51,27)]], [[M(32,51,28)]], [[M(32,51,29)]]
|O-derived-known? = Yes
|coveringblocks =
|coveredblocks =
|pcoveringblocks =
}}

== Basic algebra ==

== Other notatable representatives ==

== Covering blocks and covered blocks ==

Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.

If <math>b</math> is in M(32,51,25), then <math>B</math> is in [[M(32,51,14)]] or M(32,51,25).

== Projective indecomposable modules ==

Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{15}</math>, the projective indecomposable modules have Loewy structure as follows:

<math>\begin{array}{ccc}
\begin{array}{c}
S_{1} \\
S_{6} S_{7} S_{10} \\
S_{1} S_{1} S_{11} S_{14} S_{15} \\
S_{1} S_{7} S_{6} S_{10} S_{10} S_{12} S_{13} \\
S_{1} S_{7} S_{6} S_{11} S_{11} S_{14} S_{15} \\
S_{1} S_{1} S_{10} S_{12} S_{13} \\
S_{7} S_{6} S_{11} \\
S_{1} \\
\end{array}
&
\begin{array}{c}
S_{2} \\
S_{9} S_{5} S_{10} \\
S_{2} S_{2} S_{11} S_{15} S_{14} \\
S_{2} S_{5} S_{9} S_{10} S_{10} S_{13} S_{12} \\
S_{2} S_{5} S_{9} S_{11} S_{11} S_{15} S_{14} \\
S_{2} S_{2} S_{10} S_{13} S_{12} \\
S_{9} S_{5} S_{11} \\
S_{2} \\
\end{array}
&
\begin{array}{c}
S_{3} \\
S_{4} S_{8} S_{10} \\
S_{3} S_{3} S_{11} S_{14} S_{15} \\
S_{3} S_{8} S_{4} S_{10} S_{10} S_{13} S_{12} \\
S_{3} S_{4} S_{8} S_{11} S_{11} S_{15} S_{14} \\
S_{3} S_{3} S_{10} S_{13} S_{12} \\
S_{8} S_{4} S_{11} \\
S_{3} \\
\end{array}
\end{array}</math>

<br> <br>

<math>
\begin{array}{ccc}
\begin{array}{c}
S_{4} \\
S_{3} S_{14} \\
S_{8} S_{10} S_{12} \\
S_{3} S_{4} S_{11} S_{15} \\
S_{3} S_{4} S_{10} S_{13} \\
S_{8} S_{11} S_{14} \\
S_{3} S_{12} \\
S_{4} \\
\end{array}
&
\begin{array}{c}
S_{5} \\
S_{2} S_{15} \\
S_{9} S_{10} S_{13} \\
S_{2} S_{5} S_{11} S_{14} \\
S_{2} S_{5} S_{10} S_{12} \\
S_{9} S_{11} S_{15} \\
S_{2} S_{13} \\
S_{5} \\
\end{array}
&
\begin{array}{c}
S_{6} \\
S_{1} S_{15} \\
S_{7} S_{10} S_{13} \\
S_{1} S_{6} S_{11} S_{14} \\
S_{1} S_{6} S_{10} S_{12} \\
S_{7} S_{11} S_{15} \\
S_{1} S_{13} \\
S_{6} \\
\end{array}
\end{array}</math>

<br> <br>

<math>
\begin{array}{ccc}
\begin{array}{c}
S_{7} \\
S_{1} S_{14} \\
S_{6} S_{10} S_{12} \\
S_{1} S_{7} S_{11} S_{15} \\
S_{1} S_{7} S_{10} S_{13} \\
S_{6} S_{11} S_{14} \\
S_{1} S_{12} \\
S_{7} \\
\end{array}
&
\begin{array}{c}
S_{8} \\
S_{3} S_{15} \\
S_{4} S_{10} S_{13} \\
S_{3} S_{8} S_{11} S_{14} \\
S_{3} S_{8} S_{10} S_{12} \\
S_{4} S_{11} S_{15} \\
S_{3} S_{13} \\
S_{8} \\
\end{array}
&
\begin{array}{c}
S_{9} \\
S_{2} S_{14} \\
S_{5} S_{10} S_{12} \\
S_{2} S_{9} S_{11} S_{15} \\
S_{2} S_{9} S_{10} S_{13} \\
S_{5} S_{11} S_{14} \\
S_{2} S_{12} \\
S_{9} \\
\end{array}
\end{array}</math>

<br> <br>

<math>
\begin{array}{ccc}
\begin{array}{c}
S_{10} \\
S_{10} S_{11} S_{11} S_{14} S_{15} \\
S_{2} S_{3} S_{1} S_{11} S_{10} S_{10} S_{10} S_{15} S_{14} S_{12} S_{12} S_{13} S_{13} \\
S_{9} S_{4} S_{8} S_{7} S_{6} S_{5} S_{11} S_{11} S_{11} S_{10} S_{10} S_{10} S_{11} S_{14} S_{15} S_{13} S_{14} S_{15} S_{12} \\
S_{1} S_{3} S_{3} S_{2} S_{2} S_{1} S_{10} S_{10} S_{11} S_{11} S_{10} S_{12} S_{15} S_{13} S_{12} S_{14} S_{15} S_{13} S_{14} \\
S_{9} S_{6} S_{7} S_{5} S_{8} S_{4} S_{11} S_{10} S_{10} S_{10} S_{11} S_{14} S_{15} S_{12} S_{13} \\
S_{3} S_{1} S_{2} S_{11} S_{10} S_{15} S_{14} \\
S_{10} \\
\end{array}
&
\begin{array}{c}
S_{11} \\
S_{2} S_{1} S_{3} S_{11} S_{10} S_{12} S_{13} \\
S_{8} S_{5} S_{4} S_{9} S_{7} S_{6} S_{11} S_{10} S_{11} S_{11} S_{10} S_{15} S_{13} S_{14} S_{12} \\
S_{1} S_{3} S_{2} S_{3} S_{1} S_{2} S_{11} S_{11} S_{10} S_{10} S_{11} S_{13} S_{14} S_{12} S_{12} S_{15} S_{13} S_{14} S_{15} \\
S_{5} S_{9} S_{6} S_{8} S_{7} S_{4} S_{11} S_{10} S_{10} S_{11} S_{10} S_{10} S_{11} S_{13} S_{12} S_{14} S_{15} S_{12} S_{13} \\
S_{2} S_{1} S_{3} S_{10} S_{11} S_{11} S_{11} S_{15} S_{15} S_{14} S_{14} S_{13} S_{12} \\
S_{11} S_{10} S_{10} S_{12} S_{13} \\
S_{11} \\
\end{array}
&
\begin{array}{c}
S_{12} \\
S_{9} S_{4} S_{7} S_{11} S_{12} S_{14} \\
S_{1} S_{2} S_{3} S_{10} S_{11} S_{12} S_{14} S_{13} S_{14} \\
S_{6} S_{8} S_{5} S_{10} S_{11} S_{11} S_{10} S_{13} S_{12} S_{15} \\
S_{2} S_{1} S_{3} S_{11} S_{11} S_{10} S_{15} S_{15} S_{13} S_{12} \\
S_{7} S_{9} S_{4} S_{11} S_{10} S_{10} S_{12} S_{14} S_{13} \\
S_{11} S_{12} S_{14} S_{14} \\
S_{12} \\
\end{array}
\end{array}
</math>

<br> <br>

<math>
\begin{array}{ccc}
\begin{array}{c}
S_{13} \\
S_{8} S_{5} S_{6} S_{11} S_{13} S_{15} \\
S_{1} S_{2} S_{3} S_{10} S_{11} S_{12} S_{13} S_{15} S_{15} \\
S_{9} S_{7} S_{4} S_{11} S_{11} S_{10} S_{10} S_{14} S_{13} S_{12} \\
S_{1} S_{3} S_{2} S_{11} S_{10} S_{11} S_{13} S_{12} S_{14} S_{14} \\
S_{8} S_{6} S_{5} S_{11} S_{10} S_{10} S_{15} S_{13} S_{12} \\
S_{11} S_{15} S_{15} S_{13} \\
S_{13} \\
\end{array}
&
\begin{array}{c}
S_{14} \\
S_{10} S_{14} S_{12} S_{12} \\
S_{4} S_{7} S_{9} S_{10} S_{11} S_{11} S_{15} S_{14} S_{12} \\
S_{3} S_{2} S_{1} S_{10} S_{11} S_{10} S_{14} S_{15} S_{13} S_{13} \\
S_{5} S_{8} S_{6} S_{10} S_{10} S_{11} S_{11} S_{13} S_{15} S_{14} \\
S_{3} S_{2} S_{1} S_{11} S_{10} S_{14} S_{15} S_{12} S_{12} \\
S_{4} S_{7} S_{9} S_{10} S_{12} S_{14} \\
S_{14} \\
\end{array}
&
\begin{array}{c}
S_{15} \\
S_{10} S_{15} S_{13} S_{13} \\
S_{5} S_{6} S_{8} S_{10} S_{11} S_{11} S_{13} S_{14} S_{15} \\
S_{3} S_{2} S_{1} S_{10} S_{11} S_{10} S_{14} S_{15} S_{12} S_{12} \\
S_{4} S_{9} S_{7} S_{10} S_{10} S_{11} S_{11} S_{14} S_{15} S_{12} \\
S_{3} S_{2} S_{1} S_{10} S_{11} S_{15} S_{14} S_{13} S_{13} \\
S_{5} S_{6} S_{8} S_{10} S_{15} S_{13} \\
S_{15} \\
\end{array}
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

== Cartan matrix ==
<math>\left( \begin{array}{ccccccccccccccccccccc}
8 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 0 & 4 & 4 & 2 & 2 & 2 & 2 \\
0 & 8 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 4 & 4 & 2 & 2 & 2 & 2 \\
0 & 0 & 8 & 4 & 0 & 0 & 0 & 4 & 0 & 4 & 4 & 2 & 2 & 2 & 2 \\
0 & 0 & 4 & 4 & 0 & 0 & 0 & 2 & 0 & 2 & 2 & 2 & 1 & 2 & 1 \\
0 & 4 & 0 & 0 & 4 & 0 & 0 & 0 & 2 & 2 & 2 & 1 & 2 & 1 & 2 \\
4 & 0 & 0 & 0 & 0 & 4 & 2 & 0 & 0 & 2 & 2 & 1 & 2 & 1 & 2 \\
4 & 0 & 0 & 0 & 0 & 2 & 4 & 0 & 0 & 2 & 2 & 2 & 1 & 2 & 1 \\
0 & 0 & 4 & 2 & 0 & 0 & 0 & 4 & 0 & 2 & 2 & 1 & 2 & 1 & 2 \\
0 & 4 & 0 & 0 & 2 & 0 & 0 & 0 & 4 & 2 & 2 & 2 & 1 & 2 & 1 \\
4 & 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 16 & 12 & 6 & 6 & 8 & 8 \\
4 & 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 12 & 16 & 8 & 8 & 6 & 6 \\
2 & 2 & 2 & 2 & 1 & 1 & 2 & 1 & 2 & 6 & 8 & 8 & 4 & 6 & 3 \\
2 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 1 & 6 & 8 & 4 & 8 & 3 & 6 \\
2 & 2 & 2 & 2 & 1 & 1 & 2 & 1 & 2 & 8 & 6 & 6 & 3 & 8 & 4 \\
2 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 1 & 8 & 6 & 3 & 6 & 4 & 8
\end{array}\right)</math>

== Decomposition matrix ==

<math>\left( \begin{array}{ccccccccccccccccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1
\end{array}\right)</math>

[[(C2)%5E5|Back to <math>(C_2)^5</math>]]