M(9,2,2) - Revision history

Charles Eaton at 07:45, 23 October 2018

2018-10-23T07:45:30Z

Charles Eaton at 19:02, 22 October 2018

2018-10-22T19:02:01Z

Charles Eaton at 17:41, 21 October 2018

2018-10-21T17:41:34Z

Charles Eaton: Created page with "{{blockbox |title = M(9,2,2) - $k(S_3 \times C_3)$ |image = M(8,3,3)quiver.png |representative = $k(S_3 \times C_3)$ |defect = C3xC3| $C_3 \times..."$

2018-10-21T17:37:19Z

Created page with "{{blockbox |title = M(9,2,2) - <math>k(S_3 \times C_3)</math> |image = M(8,3,3)quiver.png |representative = <math>k(S_3 \times C_3)</math> |defect = C3xC3|<math>C_3 \times..."

New page

{{blockbox
|title = M(9,2,2) - <math>k(S_3 \times C_3)</math>
|image = M(8,3,3)quiver.png
|representative = <math>k(S_3 \times C_3)</math>
|defect = [[C3xC3|<math>C_3 \times C_3</math>]]
|inertialquotients = <math>C_2</math>
|k(B) = 9
|l(B) = 2
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{cc}
6 & 3 \\
3 & 6 \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>\mathcal{O} (C_3 \times C_3)</math>
|decomp = <math>\left( \begin{array}{cc}
1 & 0 \\
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O =
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = No
|k-derived = none known
|O-derived-known? = No
|coveringblocks =
|coveredblocks =
|pcoveringblocks =
}}

These are [[Nilpotent blocks|nilpotent blocks]].

== Basic algebra ==

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

'''Relations w.r.t. <math>k</math>:''' <math>a^3=d^3=0</math>, <math>ab=bd</math>, <math>ca=dc</math>, <math>bcb=cbc=0</math>

== Other notatable representatives ==

== Covering blocks and covered blocks ==

== Projective indecomposable modules ==

Labelling the unique simple <math>B</math>-module by <math>1,2</math>, the unique projective indecomposable module has Loewy structure as follows:

<math>\begin{array}{cc}
\begin{array}{ccccc}
& & 1 & & \\
& 1 & & 2 & \\
1 & & 2 & & 1 \\
& 2 & & 1 & \\
& & 1 & & \\
\end{array}, &
\begin{array}{ccccc}
& & 2 & & \\
& 2 & & 1 & \\
2 & & 1 & & 2 \\
& 1 & & 2 & \\
& & 2 & & \\
\end{array} \\
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

[[C3xC3|Back to <math>C_3 \times C_3</math>]]

@@ Line 15: / Line 15: @@
 |defect-morita-inv? = Yes
 |inertial-morita-inv? = Yes
-|O-morita? = Yes
+|O-morita? = No
 |O-morita = <math>\mathcal{O} (C_3 \times C_3)</math>
 |decomp = <math>\left( \begin{array}{cc}
@@ Line 40: / Line 40: @@
 |pcoveringblocks =
 }}
 == Basic algebra ==
@@ Line 50: / Line 48: @@
 == Other notatable representatives ==
 == Projective indecomposable modules ==

@@ Line 41: / Line 41: @@
 }}
-These are [[Nilpotent blocks|nilpotent blocks]].
 == Basic algebra ==

@@ Line 55: / Line 55: @@
 == Projective indecomposable modules ==
-Labelling the unique simple <math>B</math>-module by <math>1,2</math>, the unique projective indecomposable module has Loewy structure as follows:
+Labelling the unique simple <math>B</math>-module by <math>1,2</math>, the projective indecomposable modules have Loewy structure as follows:
 <math>\begin{array}{cc}

M(9,2,2) - Revision history

Charles Eaton at 07:45, 23 October 2018

Charles Eaton at 19:02, 22 October 2018

Charles Eaton at 17:41, 21 October 2018

Charles Eaton: Created page with "{{blockbox |title = M(9,2,2) - k(S_3 \times C_3) |image = M(8,3,3)quiver.png |representative = k(S_3 \times C_3) |defect = C3xC3|C_3 \times..."

Charles Eaton: Created page with "{{blockbox |title = M(9,2,2) - $k(S_3 \times C_3)$ |image = M(8,3,3)quiver.png |representative = $k(S_3 \times C_3)$ |defect = C3xC3| $C_3 \times..."$