M(8,5,4) - Revision history

Charles Eaton at 09:09, 5 June 2019

2019-06-05T09:09:30Z

Charles Eaton at 22:31, 2 January 2019

2019-01-02T22:31:36Z

Charles Eaton at 16:21, 21 November 2018

2018-11-21T16:21:49Z

Charles Eaton at 09:50, 5 October 2018

2018-10-05T09:50:53Z

Charles Eaton at 19:25, 23 September 2018

2018-09-23T19:25:24Z

Charles Eaton: Created page with "{{blockbox |title = M(8,5,4) - $k((C_2 \times C_2 \times C_2):C_7)$ |image = M(8,5,4)quiver.png |representative = $k((C_2 \times C_2 \times C_2):C_7)$ ..."

2018-09-23T17:12:29Z

Created page with "{{blockbox |title = M(8,5,4) - <math>k((C_2 \times C_2 \times C_2):C_7)</math> |image = M(8,5,4)quiver.png |representative = <math>k((C_2 \times C_2 \times C_2):C_7)</math>..."

New page

{{blockbox
|title = M(8,5,4) - <math>k((C_2 \times C_2 \times C_2):C_7)</math>
|image = M(8,5,4)quiver.png
|representative = <math>k((C_2 \times C_2 \times C_2):C_7)</math>
|defect = [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]]
|inertialquotients = <math>C_7</math>
|k(B) = 8
|l(B) = 7
|k-morita-frob = 1
|Pic-k=  
|cartan = <math>\left( \begin{array}{ccccccc}
2 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 2 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 2 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 2 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 2 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 & 2 & 1 \\
1 & 1 & 1 & 1 & 1 & 1 & 2 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>\mathcal{O} ((C_2 \times C_2 \times C_2):C_7)</math>
|decomp = <math>\left( \begin{array}{ccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O = <math>C_7</math>;
|source? = No
|sourcereps =  
|k-derived-known? = Yes
|k-derived = [[M(8,5,5)]]
|O-derived-known? = Yes
|coveringblocks =
|coveredblocks =
}}

== 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>,
<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>,
<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>c_1c_5=c_2c_6=c_3c_7=c_4c_1=c_5c_2=c_6c_3=c_7c_4=0</math>,
<math>a_1b_2=b_1a_3</math>,
<math>a_2b_3=b_2a_4</math>,
<math>a_3b_4=b_3a_5</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>

== Other notatable representatives ==

== 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:

<math>\begin{array}{ccccccc}
\begin{array}{c}
S_1 \\
S_2 S_3 S_5 \\
S_4 S_6 S_7 \\
S_1 \\
\end{array},
&
\begin{array}{ccc}
S_2 \\
S_3 S_4 S_6 \\
S_1 S_5 S_7 \\
S_2 \\
\end{array},
&
\begin{array}{ccc}
S_3 \\
S_4 S_5 S_7 \\
S_1 S_2 S_6 \\
S_3 \\
\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}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

[[C2xC2xC2|Back to <math>C_2 \times C_2 \times C_2</math>]]

@@ Line 33: / Line 33: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>C_7</math><ref>See [[References#B|[BKL18]]]</ref>
+|Pic-O = <math>C_7 : C_3</math><ref>See [[References#B|[BKL18]]]</ref>
 |PIgroup = <math>S_7 \times C_2</math><ref>Using GAP programme from [[References#R|[Ru11]]].</ref>
 |source? = No

@@ Line 33: / Line 33: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>C_7</math><ref>See [[References|[BKL18]]]</ref>
+|Pic-O = <math>C_7</math><ref>See [[References#B|[BKL18]]]</ref>
 |source? = No
 |sourcereps = &nbsp;
@@ Line 41: / Line 42: @@
 |coveringblocks =
 |coveredblocks =
 }}
@@ Line 47: / Line 49: @@
 '''Quiver:''' <math>a_1:<1,2>, a_2:<2,3></math>, <math>a_3:<3,4>, a_4:<4,5></math>, <math>a_5:<5,6>, a_6:<6,7>, a_7:<7,1></math>,
 <math>b_1:<1,3>, b_2:<2,4></math>, <math>b_3:<3,5>, b_4:<4,6></math>, <math>b_5:<5,7>, b_6:<6,1>, b_7:<7,2></math>,
-<math>c_1:<1,5>, c_2:<2,6></math>, <math>c_3:<3,7>, c_4:<4,1></math>, <math>c_5:<5,2>, c_6:<6,3>, c_7:<7,4></math><ref>Thanks to Malka Schaps for showing us this way of drawing the quiver, from [[References|[[SSS98]]].</ref>
+<math>c_1:<1,5>, c_2:<2,6></math>, <math>c_3:<3,7>, c_4:<4,1></math>, <math>c_5:<5,2>, c_6:<6,3>, c_7:<7,4></math><ref>Thanks to Malka Schaps for showing us this way of drawing the quiver, from [[References#S|[SSS98]]].</ref>
 '''Relations w.r.t. <math>k</math>:'''

@@ Line 33: / Line 33: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>C_7</math>;
+|Pic-O = <math>C_7</math><ref>See [[References|[BKL18]]]</ref>
 |source? = No
 |sourcereps = &nbsp;
@@ Line 47: / Line 47: @@
 '''Quiver:''' <math>a_1:<1,2>, a_2:<2,3></math>, <math>a_3:<3,4>, a_4:<4,5></math>, <math>a_5:<5,6>, a_6:<6,7>, a_7:<7,1></math>,
 <math>b_1:<1,3>, b_2:<2,4></math>, <math>b_3:<3,5>, b_4:<4,6></math>, <math>b_5:<5,7>, b_6:<6,1>, b_7:<7,2></math>,
-<math>c_1:<1,5>, c_2:<2,6></math>, <math>c_3:<3,7>, c_4:<4,1></math>, <math>c_5:<5,2>, c_6:<6,3>, c_7:<7,4></math>
+<math>c_1:<1,5>, c_2:<2,6></math>, <math>c_3:<3,7>, c_4:<4,1></math>, <math>c_5:<5,2>, c_6:<6,3>, c_7:<7,4></math><ref>Thanks to Malka Schaps for showing us this way of drawing the quiver, from [[References|[[SSS98]]].</ref>
 '''Relations w.r.t. <math>k</math>:'''
@@ Line 147: / Line 147: @@
 [[C2xC2xC2|Back to <math>C_2 \times C_2 \times C_2</math>]]

@@ Line 45: / Line 45: @@
 == 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:''' <math>a_1:<1,2>, a_2:<2,3></math>, <math>a_3:<3,4>, a_4:<4,5></math>, <math>a_5:<5,6>, a_6:<6,7>, a_7:<7,1></math>,
-<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>b_1:<1,3>, b_2:<2,4></math>, <math>b_3:<3,5>, b_4:<4,6></math>, <math>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>
+<math>c_1:<1,5>, c_2:<2,6></math>, <math>c_3:<3,7>, c_4:<4,1></math>, <math>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>b_1b_3=b_2b_4=b_3b_5=b_4b_6=b_5b_7=b_6b_1=b_7b_2=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>,

@@ Line 73: / Line 73: @@
 <math>b_6c_1=c_6b_3</math>,
 <math>b_7c_2=c_7b_4</math>
 == Other notatable representatives ==