CesareGArdito at 11:42, 9 December 2019

2019-12-09T11:42:59Z

CesareGArdito: Created page with "{{blockbox |title = M(32,51,20) - $k((C_2)^5 : (C_7:C_3))$ |image = |representative = $k((C_2)^5 : (C_7:C_3))$ |defect = (C2)%5E5| $(C_2)..."$

2019-12-09T11:32:18Z

Created page with "{{blockbox |title = M(32,51,20) - <math>k((C_2)^5 : (C_7:C_3))</math> |image = |representative = <math>k((C_2)^5 : (C_7:C_3))</math> |defect = (C2)%5E5|<math>(C_2)..."

New page

{{blockbox
|title = M(32,51,20) - <math>k((C_2)^5 : (C_7:C_3))</math>
|image =  
|representative = <math>k((C_2)^5 : (C_7:C_3))</math>
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
|inertialquotients = <math>C_7:C_3</math>
|k(B) = 16
|l(B) = 5
|k-morita-frob = 1
|Pic-k=  
|cartan = <math>\left( \begin{array}{ccccc}
4 & 2 & 2 & 4 & 4 \\
2 & 4 & 2 & 4 & 4 \\
2 & 2 & 4 & 4 & 4 \\
4 & 4 & 4 & 16 & 12 \\
4 & 4 & 4 & 12 & 16
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>\mathcal{O}((C_2)^5 : (C_7:C_3))</math>
|decomp = See below.
|O-morita-frob = 1
|Pic-O =
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(32,51,21)]]
|O-derived-known? = Yes
|coveringblocks =
|coveredblocks =
|pcoveringblocks =
}}

The action of the inertial quotient on the defect group is such that the subgroup [[C_3]] acts as in [[M(32,51,5)]].

== 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(32,51,20), then <math>B</math> is in [[M(32,51,6)]], M(32,51,17), [[M(32,51,24)]].

== Projective indecomposable modules ==

Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_5</math>, the projective indecomposable modules have Loewy structure as follows:

<math>\begin{array}{ccc}
\begin{array}{c}
S_{1} \\
S_{3} S_{2} S_{4} \\
S_{1} S_{4} S_{4} S_{5} \\
S_{1} S_{5} S_{5} S_{4} \\
S_{2} S_{3} S_{5} \\
S_{1} \\
\end{array}
&
\begin{array}{c}
S_{2} \\
S_{1} S_{3} S_{4} \\
S_{2} S_{4} S_{4} S_{5} \\
S_{2} S_{5} S_{5} S_{4} \\
S_{3} S_{1} S_{5} \\
S_{2} \\
\end{array}
&
\begin{array}{c}
S_{3} \\
S_{2} S_{1} S_{4} \\
S_{3} S_{4} S_{4} S_{5} \\
S_{3} S_{5} S_{5} S_{4} \\
S_{2} S_{1} S_{5} \\
S_{3} \\
\end{array}
\end{array}</math>

<br> <br>

<math>
\begin{array}{cc}
\begin{array}{c}
S_{4} \\
S_{5} S_{4} S_{5} S_{4} S_{4} \\
S_{3} S_{2} S_{1} S_{5} S_{5} S_{4} S_{5} S_{4} S_{5} S_{5} S_{4} S_{4} \\
S_{1} S_{2} S_{3} S_{2} S_{3} S_{1} S_{4} S_{4} S_{5} S_{4} S_{5} S_{5} S_{5} S_{4} \\
S_{2} S_{1} S_{3} S_{4} S_{5} S_{4} S_{4} \\
S_{4} \\
\end{array}
&
\begin{array}{c}
S_{5} \\
S_{1} S_{2} S_{3} S_{4} S_{5} S_{5} S_{5} \\
S_{2} S_{1} S_{3} S_{1} S_{2} S_{3} S_{5} S_{4} S_{4} S_{4} S_{5} S_{4} S_{5} S_{5} \\
S_{1} S_{3} S_{2} S_{5} S_{4} S_{5} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\
S_{5} S_{4} S_{4} S_{5} S_{5} \\
S_{5} \\
\end{array}
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

== Decomposition matrix ==

<math>\left( \begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 \\
1 & 0 & 0 & 1 & 1 \\
0 & 1 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 & 1 \\
1 & 1 & 1 & 3 & 3
\end{array}\right)</math>

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

@@ Line 34: / Line 34: @@
 }}
-The action of the inertial quotient on the defect group is such that the subgroup [[C_3]] acts as in [[M(32,51,5)]].
+The action of the inertial quotient on the defect group is such that the subgroup <math>C_3</math> acts as in [[M(32,51,5)]].
 == Basic algebra ==

M(32,51,20) - Revision history

CesareGArdito at 11:42, 9 December 2019

CesareGArdito: Created page with "{{blockbox |title = M(32,51,20) - k((C_2)^5 : (C_7:C_3)) |image = |representative = k((C_2)^5 : (C_7:C_3)) |defect = (C2)%5E5|(C_2)..."

CesareGArdito: Created page with "{{blockbox |title = M(32,51,20) - $k((C_2)^5 : (C_7:C_3))$ |image = |representative = $k((C_2)^5 : (C_7:C_3))$ |defect = (C2)%5E5| $(C_2)..."$