M(8,5,3) - Revision history

Charles Eaton at 21:07, 5 December 2018

2018-12-05T21:07:17Z

Charles Eaton at 20:32, 5 December 2018

2018-12-05T20:32:37Z

Charles Eaton at 16:16, 21 November 2018

2018-11-21T16:16:26Z

Charles Eaton at 19:57, 20 September 2018

2018-09-20T19:57:06Z

Charles Eaton: Added PIMs

2018-09-09T08:39:36Z

Added PIMs

Charles Eaton: Decomposition and Cartan corrected

2018-09-08T15:57:42Z

Decomposition and Cartan corrected

Charles Eaton: Created page with "{{blockbox |title = M(8,5,3) - $k(A_4 \times C_2)$ |image = |representative = $k(A_4 \times C_2)$ |defect = C2xC2xC2| $C_2 \times C_2 \times C_2..."$

2018-09-08T15:53:36Z

Created page with "{{blockbox |title = M(8,5,3) - <math>k(A_4 \times C_2)</math> |image = |representative = <math>k(A_4 \times C_2)</math> |defect = C2xC2xC2|<math>C_2 \times C_2 \times C_2..."

New page

{{blockbox
|title = M(8,5,3) - <math>k(A_4 \times C_2)</math>
|image =
|representative = <math>k(A_4 \times C_2)</math>
|defect = [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]]
|inertialquotients = <math>1</math>
|k(B) = 8
|l(B) = 3
|k-morita-frob = 1
|Pic-k=  
|cartan = <math>\left( \begin{array}{ccc}
2 & 1 & 1 \\
1 & 2 & 1 \\
1 & 1 & 2 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>\mathcal{O} (A_4 \times C_2)</math>
|decomp = <math>\left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 1 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O =  
|source? = No
|sourcereps =  
|k-derived-known? = Yes
|k-derived = [[M(8,5,2)]]
|O-derived-known? = Yes
}}

== Basic algebra ==

'''Quiver:''' a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f: <1,3>, g:<1,1>, h:<2,2>, i:<3,3>

'''Relations w.r.t. <math>k</math>:''' ab=bc=ca=0, df=fe=ed=0, ad=fc, be=da, cf=eb, g^2=h^2=i^2=0, ah=ga, bi=hb, cg=ic, dg=hd, eh=ie, fi=gf

== Other notatable representatives ==

== Covering blocks and covered blocks ==

== Projective indecomposable modules ==

== Irreducible characters ==

All irreducible characters have height zero.

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

@@ Line 30: / Line 30: @@
 |O-morita-frob = 1
 |Pic-O =  <math>S_3 \times C_2</math><ref>See [[References|[EL18c]]]</ref>
-|PIgroup =
+|PIgroup = <math>S_4 \times C_2 \times C_2</math><ref>By Theorem 3.7 of [[References|[EL18c]]].</ref>
 |source? = No
 |sourcereps = &nbsp;
@@ Line 36: / Line 36: @@
 |k-derived = [[M(8,5,2)]]
 |O-derived-known? = Yes
-|coveringblocks =
+|coveringblocks = [[M(16,10,1)]]<ref>For example consider the block of <math>PSL_3(7) \times C_2</math> covering the block number 2 of <math>PSL_3(7) \triangleleft PGL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/]. The covering block of <math>PGL_3(7) \times C_2</math> is nilpotent.</ref>, M(8,5,3) (complete)
 |coveredblocks =
 |pcoveringblocks = [[M(16,10,3)]], [[M(16,14,3)]]

@@ Line 29: / Line 29: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = &nbsp;
+|Pic-O =  <math>S_3 \times C_2</math><ref>See [[References|[EL18c]]]</ref>
 |source? = No
 |sourcereps = &nbsp;
@@ Line 83: / Line 83: @@
 [[C2xC2xC2|Back to <math>C_2 \times C_2 \times C_2</math>]]

@@ Line 4: / Line 4: @@
 |representative =  <math>k(A_4 \times C_2)</math>
 |defect = [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]]
-|inertialquotients = <math>1</math>
+|inertialquotients = <math>C_3</math>
 |k(B) = 8
 |l(B) = 3

@@ Line 1: / Line 1: @@
 {{blockbox
 |title = M(8,5,3) - <math>k(A_4 \times C_2)</math>
-|image =
+|image = M(8,5,3)quiver.png
 |representative =  <math>k(A_4 \times C_2)</math>
 |defect = [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]]
@@ Line 52: / Line 52: @@
 == Projective indecomposable modules ==
 == Irreducible characters ==

@@ Line 10: / Line 10: @@
 |Pic-k= &nbsp;
 |cartan = <math>\left( \begin{array}{ccc}
-& 1 & 1 \\
+& 2 & 2 \\
-& 2 & 1 \\
+& 4 & 2 \\
-& 1 & 2 \\
+& 2 & 4 \\
 \end{array} \right)</math>
 |defect-morita-inv? = Yes
@@ Line 19: / Line 19: @@
 |O-morita = <math>\mathcal{O} (A_4 \times C_2)</math>
 |decomp = <math>\left( \begin{array}{ccc}
 & 0 & 0 \\
 & 1 & 0 \\
 & 0 & 1 \\
 & 1 & 1 \\
 \end{array}\right)</math>