Charles Eaton: Corrected rep

2018-09-28T07:56:32Z

Corrected rep

Charles Eaton: Created page with "{{blockbox |title = M(5,1,6) - $k(2.A_7)$ |image = M(5,1,6)quiver.png |representative = faithful block of $k(2.A_7)$ |defect = $C_5$ ..."

2018-09-04T09:19:14Z

Created page with "{{blockbox |title = M(5,1,6) - <math>k(2.A_7)</math> |image = M(5,1,6)quiver.png |representative = faithful block of <math>k(2.A_7)</math> |defect = <math>C_5</math>..."

New page

{{blockbox
|title = M(5,1,6) - <math>k(2.A_7)</math>
|image = M(5,1,6)quiver.png
|representative = faithful block of <math>k(2.A_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 & 1 \\
0 & 1 & 2 & 1 \\
0 & 1 & 1 & 2 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = faithful block of <math>\mathcal{O}(2.A_7)</math>
|decomp = <math>\left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 \\
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(5,1,5)]]
|O-derived-known? = Yes
}}

== Basic algebra ==

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

'''Relations w.r.t. <math>k</math>:''' ea=bcd, ab=de=cdbc=0

== Other notatable representatives ==

Block 17 in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/ON/]

== 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 & & \begin{array}{c} S_3 \\ S_4 \\ \end{array} \\
& S_2 & \\
\end{array}, &
\begin{array}{c}
S_3 \\
S_4 \\
S_2 \\
S_3 \\
\end{array}, &
\begin{array}{c}
S_4 \\
S_2 \\
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(5,1,6) - <math>k(2.A_7)</math>
+|title = M(5,1,6) - <math>B_{15}(k(6.A_7))</math>
 |image = M(5,1,6)quiver.png
-|representative =  faithful block of <math>k(2.A_7)</math>
+|representative = <math>B_{15}(k(6.A_7))</math>
 |defect = [[C5|<math>C_5</math>]]
 |inertialquotients = <math>C_4</math>
@@ Line 18: / Line 18: @@
 |inertial-morita-inv? = Yes
 |O-morita? = Yes
-|O-morita = faithful block of <math>\mathcal{O}(2.A_7)</math>
+|O-morita = <math>B_{15}(\mathcal{O}(6.A_7))</math>
 |decomp = <math>\left( \begin{array}{cccc}
 & 0 & 0 & 0 \\
@@ Line 33: / Line 33: @@
 |k-derived = [[M(5,1,4)]], [[M(5,1,5)]]
 |O-derived-known? = Yes
 }}
@@ Line 42: / Line 44: @@
 == Other notatable representatives ==
 == Projective indecomposable modules ==

M(5,1,6) - Revision history

Charles Eaton: Corrected rep

Charles Eaton: Created page with "{{blockbox |title = M(5,1,6) - k(2.A_7) |image = M(5,1,6)quiver.png |representative = faithful block of k(2.A_7) |defect = C_5..."

Charles Eaton: Created page with "{{blockbox |title = M(5,1,6) - $k(2.A_7)$ |image = M(5,1,6)quiver.png |representative = faithful block of $k(2.A_7)$ |defect = $C_5$ ..."