M(32,51,11) - Revision history

CesareGArdito at 15:55, 9 December 2019

2019-12-09T15:55:16Z

CesareGArdito at 15:53, 9 December 2019

2019-12-09T15:53:28Z

CesareGArdito at 15:52, 9 December 2019

2019-12-09T15:52:51Z

CesareGArdito at 15:06, 9 December 2019

2019-12-09T15:06:50Z

CesareGArdito at 11:50, 9 December 2019

2019-12-09T11:50:04Z

CesareGArdito at 12:53, 6 December 2019

2019-12-06T12:53:50Z

CesareGArdito: Created page with "{{blockbox |title = M(32,51,11) - $k((C_2)^4 : C_{15}) \times C_2)$ |image = |representative = $k((C_2)^4 : C_{15}) \times C_2)$ |defect = (C..."

2019-12-06T12:49:46Z

Created page with "{{blockbox |title = M(32,51,11) - <math>k((C_2)^4 : C_{15}) \times C_2)</math> |image = |representative = <math>k((C_2)^4 : C_{15}) \times C_2)</math> |defect = (C..."

New page

{{blockbox
|title = M(32,51,11) - <math>k((C_2)^4 : C_{15}) \times C_2)</math>
|image =  
|representative = <math>k((C_2)^4 : C_{15}) \times C_2)</math>
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
|inertialquotients = <math>C_{15}</math>
|k(B) = 16
|l(B) = 5
|k-morita-frob = 1
|Pic-k=  
|cartan = See below.
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>\mathcal{O} ((C_2)^4 : C_{15}) \times C_2)</math>
|decomp = See below.
|O-morita-frob = 1
|Pic-O =
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(32,51,12)]]
|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(32,51,11), then <math>B</math> is in [[M(32,51,2)]], [[M(32,51,5)]], or M(32,51,11).

== 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}{ccccc}
\begin{array}{c}
S_1 \\
S_1 S_{10} S_{11} S_9 S_4 \\
S_{11} S_9 S_4 S_{10} S_{12} S_3 S_{14} S_{15} S_2 S_{13} \\
S_3 S_{12} S_{14} S_2 S_{15} S_{13} S_5 S_6 S_8 S_7 \\
S_6 S_5 S_7 S_8 S_1 \\
S_1 \\
\end{array}
&
\begin{array}{c}
S_2 \\
S_{12} S_{14} S_5 S_7 S_2 \\
S_{10} S_7 S_{14} S_6 S_8 S_4 S_1 S_5 S_{12} S_{15} \\
S_{15} S_6 S_1 S_8 S_4 S_{10} S_3 S_9 S_{13} S_{11} \\
S_9 S_{13} S_3 S_{11} S_2 \\
S_2 \\
\end{array}
&
\begin{array}{c}
S_3 \\
S_3 S_7 S_2 S_6 S_{11} \\
S_{12} S_7 S_{11} S_5 S_2 S_4 S_6 S_1 S_{14} S_{13} \\
S_{14} S_4 S_1 S_5 S_{12} S_{13} S_{15} S_9 S_8 S_{10} \\
S_9 S_8 S_{10} S_{15} S_3 \\
S_3 \\
\end{array}
&
\begin{array}{c}
S_4 \\
S_{13} S_9 S_{14} S_{15} S_4 \\
S_{14} S_2 S_8 S_6 S_3 S_{10} S_{15} S_5 S_9 S_{13} \\
S_8 S_{10} S_1 S_5 S_2 S_7 S_3 S_6 S_{12} S_{11} \\
S_7 S_1 S_{12} S_{11} S_4 \\
S_4 \\
\end{array}
&
\begin{array}{c}
S_5 \\
S_1 S_{10} S_{12} S_5 S_8 \\
S_4 S_7 S_1 S_{10} S_8 S_9 S_3 S_{15} S_{12} S_{11} \\
S_{14} S_{11} S_{13} S_2 S_4 S_7 S_{15} S_3 S_9 S_6 \\
S_{13} S_{14} S_6 S_5 S_2 \\
S_5 \\
\end{array}
\end{array}
</math>

<br>  <br>

<math>\begin{array}{ccccc}
\begin{array}{c}
S_6 \\
S_1 S_5 S_{11} S_6 S_{13} \\
S_{12} S_{11} S_{10} S_1 S_5 S_2 S_8 S_9 S_4 S_{13} \\
S_9 S_3 S_{10} S_{14} S_{15} S_8 S_7 S_2 S_4 S_{12} \\
S_7 S_{14} S_{15} S_3 S_6 \\
S_6 \\
\end{array}
&
\begin{array}{c}
S_7 \\
S_4 S_6 S_{14} S_1 S_7 \\
S_5 S_9 S_{13} S_{14} S_4 S_6 S_{11} S_{15} S_1 S_{10} \\
S_3 S_{13} S_8 S_2 S_9 S_{12} S_{15} S_5 S_{11} S_{10} \\
S_{12} S_2 S_7 S_3 S_8 \\
S_7 \\
\end{array}
&
\begin{array}{c}
S_8 \\
S_8 S_1 S_3 S_7 S_9 \\
S_9 S_{14} S_1 S_4 S_{10} S_6 S_2 S_{11} S_7 S_3 \\
S_{14} S_{15} S_6 S_2 S_4 S_{11} S_{10} S_{13} S_{12} S_5 \\
S_{15} S_5 S_{13} S_8 S_{12} \\
S_8 \\
\end{array}
&
\begin{array}{c}
S_9 \\
S_3 S_{14} S_{10} S_2 S_9 \\
S_{10} S_{15} S_2 S_7 S_6 S_{14} S_{12} S_{11} S_5 S_3 \\
S_4 S_6 S_5 S_{12} S_{13} S_{11} S_7 S_1 S_{15} S_8 \\
S_1 S_{13} S_8 S_4 S_9 \\
S_9 \\
\end{array}
&
\begin{array}{c}
S_{10} \\
S_{15} S_{12} S_3 S_{10} S_{11} \\
S_4 S_2 S_{13} S_8 S_7 S_6 S_3 S_{15} S_{11} S_{12} \\
S_9 S_5 S_7 S_{14} S_6 S_{13} S_2 S_8 S_4 S_1 \\
S_{14} S_5 S_9 S_1 S_{10} \\
S_{10} \\
\end{array}
\end{array}
</math>

<br>  <br>

<math>\begin{array}{ccccc}
\begin{array}{c}
S_{11} \\
S_{13} S_{11} S_2 S_{12} S_4 \\
S_8 S_4 S_9 S_7 S_{12} S_{15} S_2 S_{13} S_{14} S_5 \\
S_1 S_{14} S_8 S_3 S_5 S_9 S_6 S_{15} S_{10} S_7 \\
S_1 S_{10} S_6 S_3 S_{11} \\
S_{11} \\
\end{array}
&
\begin{array}{c}
S_{12} \\
S_{15} S_4 S_{12} S_7 S_8 \\
S_6 S_3 S_4 S_9 S_1 S_7 S_{14} S_{13} S_{15} S_8 \\
S_2 S_{13} S_3 S_1 S_9 S_6 S_{10} S_{11} S_{14} S_5 \\
S_{11} S_5 S_2 S_{10} S_{12} \\
S_{12} \\
\end{array}
&
\begin{array}{c}
S_{13} \\
S_8 S_5 S_{13} S_9 S_2 \\
S_3 S_{12} S_7 S_1 S_5 S_{10} S_9 S_8 S_2 S_{14} \\
S_{12} S_7 S_3 S_{10} S_{14} S_1 S_{11} S_{15} S_6 S_4 \\
S_{11} S_4 S_{15} S_6 S_{13} \\
S_{13} \\
\end{array}
&
\begin{array}{c}
S_{14} \\
S_6 S_{10} S_{14} S_{15} S_5 \\
S_{15} S_5 S_3 S_{12} S_{13} S_{11} S_1 S_{10} S_8 S_6 \\
S_2 S_{12} S_8 S_{13} S_3 S_1 S_{11} S_4 S_9 S_7 \\
S_7 S_4 S_9 S_2 S_{14} \\
S_{14} \\
\end{array}
&
\begin{array}{c}
S_{15} \\
S_{15} S_6 S_8 S_{13} S_3 \\
S_2 S_8 S_{13} S_3 S_5 S_1 S_6 S_9 S_{11} S_7 \\
S_2 S_{11} S_7 S_5 S_1 S_9 S_{10} S_4 S_{12} S_{14} \\
S_{14} S_{12} S_{10} S_4 S_{15} \\
S_{15} \\
\end{array}
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

== Cartan matrix ==
<math>\left( \begin{array}{ccc}
4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 \\
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 \\
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 \\
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 \\
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 \\
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 \\
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4
\end{array} \right)</math>

== Decomposition matrix ==

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

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

@@ Line 27: / Line 27: @@
 |pcoveringblocks =
 }}
-A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is Morita equivalent to M(32,51,11) or [[M(32,51,12)]], and the latter are derived equivalent.
+A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is in this Morita equivalence class or in [[M(32,51,12)]], which is derived equivalent to this class.
 It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).

@@ Line 4: / Line 4: @@
 |representative =  <math>k(((C_2)^4 : C_{15}) \times C_2)</math>
 |defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
-|inertialquotients = <math>C_{15}</math>
+|inertialquotients = <math>C_{15}</math>, ?
 |k(B) = 32
 |l(B) = 15

@@ Line 27: / Line 27: @@
 |pcoveringblocks =
 }}
-A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is Morita equivalent to M(32,51,11) or [[M(32,51,12)]], and the latter are derived equivalent.
+A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is Morita equivalent to M(32,51,11) or [[M(32,51,12)]], and the latter are derived equivalent.
 It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).

← Older revision		Revision as of 15:06, 9 December 2019
Line 27:		Line 27:
	\|pcoveringblocks =		\|pcoveringblocks =
	}}		}}
		+	A block with defect group [[(C2)%5E5\|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is Morita equivalent to M(32,51,11) or [[M(32,51,12)]], and the latter are derived equivalent.
		+
		+	It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).

@@ Line 1: / Line 1: @@
 {{blockbox
-|title = M(32,51,11) - <math>k((C_2)^4 : C_{15}) \times C_2)</math>
+|title = M(32,51,11) - <math>k(((C_2)^4 : C_{15}) \times C_2)</math>
 |image = &nbsp;
-|representative =  <math>k((C_2)^4 : C_{15}) \times C_2)</math>
+|representative =  <math>k(((C_2)^4 : C_{15}) \times C_2)</math>
 |defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
 |inertialquotients = <math>C_{15}</math>
@@ Line 13: / Line 13: @@
 |inertial-morita-inv? = Yes
 |O-morita? = Yes
-|O-morita = <math>\mathcal{O} ((C_2)^4 : C_{15}) \times C_2)</math>
+|O-morita = <math>\mathcal{O} (((C_2)^4 : C_{15}) \times C_2)</math>
 |decomp = See below.
 |O-morita-frob = 1

@@ Line 5: / Line 5: @@
 |defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
 |inertialquotients = <math>C_{15}</math>
-|k(B) = 16
+|k(B) = 32
-|l(B) = 5
+|l(B) = 15
 |k-morita-frob = 1
 |Pic-k= &nbsp;