M(8,5,7) - Revision history

Charles Eaton at 09:10, 5 June 2019

2019-06-05T09:10:59Z

Charles Eaton: [Ne02] references

2018-12-06T12:18:20Z

[Ne02] references

Charles Eaton at 17:21, 5 December 2018

2018-12-05T17:21:01Z

Charles Eaton at 16:26, 21 November 2018

2018-11-21T16:26:30Z

Charles Eaton: quiver

2018-09-24T21:34:57Z

quiver

Charles Eaton: Added PIMs

2018-09-24T17:38:40Z

Added PIMs

Charles Eaton: Created page with "{{blockbox |title = M(8,5,7) - $B_0(kJ_1)$ |image = M(8,5,7)quiver.png |representative = $B_0(kJ_1)$ |defect = C2xC2xC2| $C_2 \times C_2 \times C..."$

2018-09-24T15:54:29Z

Created page with "{{blockbox |title = M(8,5,7) - <math>B_0(kJ_1)</math> |image = M(8,5,7)quiver.png |representative = <math>B_0(kJ_1)</math> |defect = C2xC2xC2|<math>C_2 \times C_2 \times C..."

New page

{{blockbox
|title = M(8,5,7) - <math>B_0(kJ_1)</math>
|image = M(8,5,7)quiver.png
|representative = <math>B_0(kJ_1)</math>
|defect = [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]]
|inertialquotients = <math>C_7:C_3</math>
|k(B) = 8
|l(B) = 5
|k-morita-frob = 1
|Pic-k=  
|cartan = <math>\left( \begin{array}{ccccc}
8 & 4 & 4 & 4 & 4 \\
4 & 4 & 3 & 3 & 1 \\
4 & 3 & 4 & 2 & 2 \\
4 & 3 & 2 & 4 & 2 \\
4 & 1 & 2 & 2 & 4 \\

\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O}J_1)</math>
|decomp = <math>\left( \begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 \\
1 & 1 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 \\
1 & 1 & 1 & 1 & 0 \\
1 & 0 & 0 & 1 & 1 \\
1 & 0 & 1 & 0 & 1 \\
1 & 1 & 1 & 1 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O =
|source? = No
|sourcereps =  
|k-derived-known? = Yes
|k-derived = [[M(8,5,6)]], [[M(8,5,8)]]
|O-derived-known? = Yes
|coveringblocks =
|coveredblocks =
}}

== Basic algebra ==



== Irreducible characters ==

All irreducible characters have height zero.

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

@@ Line 32: / Line 32: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>1</math><ref>See [[References|[EL18c]]]. Also shown by Eisele by a different method, using [[References|[Ne02]]].</ref>
+|Pic-O = <math>1</math><ref>Shown by Eisele, using [[References|[Ne02]]].</ref>
 |PIgroup = <math>(C_2 \wr S_4) \times C_2</math><ref>See 8.7 of [[References|[Ru11]]], which uses GAP.</ref>
 |source? = No

@@ Line 33: / Line 33: @@
 |O-morita-frob = 1
 |Pic-O = <math>1</math><ref>See [[References|[EL18c]]]. Also shown by Eisele by a different method.</ref>
 |source? = No
 |sourcereps = &nbsp;
@@ Line 38: / Line 39: @@
 |k-derived = [[M(8,5,6)]], [[M(8,5,8)]]
 |O-derived-known? = Yes
-|coveringblocks =
+|coveringblocks = M(8,5,7) (complete)
-|coveredblocks =
+|coveredblocks = M(8,5,7) (complete)
 }}
 == Basic algebra ==

@@ Line 46: / Line 46: @@
 == 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:''' a:<1,4>, b:<4,2>, c:<2,3>, d:<3,1>, e:<1,5>, f:<5,5>, g:<5,1>, h:<1,3>, i:<3,2>, j:<2,4>, k:<4,1>
-'''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>:'''
 == Other notatable representatives ==

@@ Line 80: / Line 80: @@
 <math>a_5b_6c_1=b_5c_7a_4=c_5a_2b_3</math>
 <math>a_6b_7c_2=b_6c_1a_5=c_6a_3b_4</math>
-<math>a_7b_1c_3=b_7c_2a_6=c_7a_4b_5</math>
+<math>a_7b_1c_3=b_7c_2a_6=c_7a_4b_5</math> -->
 == Other notatable representatives ==
@@ Line 86: / Line 86: @@
 == 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:
-<math>\begin{array}{ccccccc}
+<math>\begin{array}{ccccc}
     \begin{array}{c}
-      S_1 \\
+\\
-     S_2 S_3 S_5 \\
+\ 4 \ 5 \\
-     S_4 S_6 S_7 \\
+\ 2 \ 1 \ 2 \ 1 \\
-      S_1 \\
+\ 3 \ 4 \ 4 \ 5 \ 5 \\
     \end{array},
+&
-    \begin{array}{ccc}
+    \begin{array}{c}
-      S_2 \\
+\\
-     S_3 S_4 S_6 \\
+\ 4 \\
-     S_1 S_5 S_7 \\
+\ 2 \ 1 \\
-      S_2 \\
+\ 4 \ 5 \\
     \end{array},
+&
-   \begin{array}{ccc}
+   \begin{array}{c}
-      S_3 \\
+\\
-     S_4 S_5 S_7 \\
+\ 2 \\
-     S_1 S_2 S_6 \\
+\ 4 \ 5 \\
-      S_3 \\
+\ 2 \ 1 \\
     \end{array}
+,
+&
-   \begin{array}{ccc}
+   \begin{array}{c}
-      S_4 \\
+\\
-     S_5 S_6 S_1 \\
+\ 2 \\
-     S_2 S_3 S_7 \\
+\ 4 \ 5 \\
-     S_4 \\
+\ 2 \ 1 \\
-   \end{array},
+\ 4 \ 5 \\
+\ 2 \\
-  \begin{array}{ccc}
+\\
     \end{array},
+&
    \begin{array}{ccc}
-      S_7 \\
+\\
-     S_1 S_2 S_4 \\
+\ 5 \\
-     S_3 S_5 S_6 \\
+\ 4 \\
-      S_7 \\
+\ 2 \ 1 \\
-    \end{array}
+\ 4 \ 5 \\
 \end{array}
-</math> -->
+</math>
 == Irreducible characters ==