Charles Eaton at 09:20, 4 September 2018

2018-09-04T09:20:11Z

Charles Eaton: Created page with "{{blockbox |title = M(5,1,5) - $B_0(kA_7)$ |image = M(5,1,5)tree.png |representative = $B_0(kA_7)$ |defect = $C_5$ |inertialquotients..."

2018-09-04T09:05:00Z

Created page with "{{blockbox |title = M(5,1,5) - <math>B_0(kA_7)</math> |image = M(5,1,5)tree.png |representative = <math>B_0(kA_7)</math> |defect = <math>C_5</math> |inertialquotients..."

New page

{{blockbox
|title = M(5,1,5) - <math>B_0(kA_7)</math>
|image = M(5,1,5)tree.png
|representative = <math>B_0(kA_7)</math>
|defect = [[C5|<math>C_5</math>]]
|inertialquotients = <math>C_4</math>
|k(B) = 5
|l(B) = 4
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{cccc}
2 & 1 & 0 & 0 \\
1 & 2 & 1 & 0 \\
0 & 1 & 2 & 1 \\
0 & 0 & 1 & 2 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O}A_7)</math>
|decomp = <math>\left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O = 
|source? = Yes
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(5,1,4)]], [[M(5,1,6)]]
|O-derived-known? = Yes
}}

== Basic algebra ==

'''Quiver:''' a:<1,2>, b:<2,3>, c:<3,4>, d:<4,3>, e:<3,2>, f:<2,1>

'''Relations w.r.t. <math>k</math>:''' ab=bc=de=ef=0, fa=be, eb=cd

== Other notatable representatives ==

== Covering blocks and covered blocks ==



== Projective indecomposable modules ==

Labelling the simple <math>B</math>-modules by <math>S_1, S_2, S_3, S_4</math>, the projective indecomposable modules have Loewy structure as follows:

<math>\begin{array}{cccc}
\begin{array}{c}
S_1 \\
S_2 \\
S_1 \\
\end{array}, &
\begin{array}{ccc}
& S_2 & \\
S_1 & & S_3 \\
& S_2 & \\
\end{array}, &
\begin{array}{ccc}
& S_3 & \\
S_2 & & S_4 \\
& S_3 & \\
\end{array}, &
\begin{array}{c}
S_4 \\
S_3 \\
S_4 \\
\end{array}
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

[[C5|Back to <math>C_5</math>]]

@@ Line 1: / Line 1: @@
 {{blockbox
 |title = M(5,1,5) - <math>B_0(kA_7)</math>
-|image = M(5,1,5)tree.png
+|image = M(5,1,5)quiver.png
 |representative =  <math>B_0(kA_7)</math>
 |defect = [[C5|<math>C_5</math>]]

M(5,1,5) - Revision history

Charles Eaton at 09:20, 4 September 2018

Charles Eaton: Created page with "{{blockbox |title = M(5,1,5) - B_0(kA_7) |image = M(5,1,5)tree.png |representative = B_0(kA_7) |defect = C_5 |inertialquotients..."

Charles Eaton: Created page with "{{blockbox |title = M(5,1,5) - $B_0(kA_7)$ |image = M(5,1,5)tree.png |representative = $B_0(kA_7)$ |defect = $C_5$ |inertialquotients..."