M(9,2,4) - Revision history

Charles Eaton at 22:26, 2 January 2019

2019-01-02T22:26:33Z

Charles Eaton at 07:40, 23 October 2018

2018-10-23T07:40:48Z

Charles Eaton: Created page with "{{blockbox |title = M(9,2,4) - $k(S_3 \times S_3)$ |image = M(9,2,4)quiver.png |representative = $k(S_3 \times S_3)$ |defect = C3xC3| $C_3 \times..."$

2018-10-23T07:38:32Z

Created page with "{{blockbox |title = M(9,2,4) - <math>k(S_3 \times S_3)</math> |image = M(9,2,4)quiver.png |representative = <math>k(S_3 \times S_3)</math> |defect = C3xC3|<math>C_3 \times..."

New page

{{blockbox
|title = M(9,2,4) - <math>k(S_3 \times S_3)</math>
|image = M(9,2,4)quiver.png
|representative = <math>k(S_3 \times S_3)</math>
|defect = [[C3xC3|<math>C_3 \times C_3</math>]]
|inertialquotients = <math>C_2</math>
|k(B) = 9
|l(B) = 4
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{cccc}
4 & 2 & 1 & 2 \\
2 & 4 & 2 & 1 \\
1 & 2 & 4 & 2 \\
2 & 1 & 2 & 4 \\
\end{array} \right)</math>
|defect-morita-inv? =
|inertial-morita-inv? =
|O-morita? = Yes
|O-morita = <math>\mathcal{O} ((S_3 \times S_3)</math>
|decomp = <math>\left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 0 & 0 & 0 1 \\
0 & 1 & 1 & 0 \\
1 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 \\
1 & 1 & 1 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O = <math>C_2 \wr C_2</math><ref>Proposition 4.3 of [[References|[BKL18]]]</ref>
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = No
|k-derived = none known
|O-derived-known? = No
|coveringblocks =
|coveredblocks =
|pcoveringblocks =
}}

== Basic algebra ==

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

'''Relations w.r.t. <math>k</math>:''' <math>ab=ef</math>, <math>bc=he</math>, <math>cd=gh</math>, <math>da=fg</math>, <math>aha=ede=0</math>, <math>bgb=hah=0</math>, <math>cfc=gbg=0</math>, <math>ded=fcf=0</math>

== Other notatable representatives ==

== Projective indecomposable modules ==

Labelling the simple <math>B</math>-modules by <math>1,2,3,4</math>, the projective indecomposable modules have Loewy structure as follows:

<math>\begin{array}{cccc}
\begin{array}{ccccc}
& & 1 & & \\
& 2 & & 4 & \\
1 & & 3 & & 1 \\
& 4 & & 2 & \\
& & 1 & & \\
\end{array}, &
\begin{array}{ccccc}
& & 2 & & \\
& 1 & & 3 & \\
2 & & 4 & & 2 \\
& 3 & & 1 & \\
& & 2 & & \\
\end{array}, &
\begin{array}{ccccc}
& & 3 & & \\
& 2 & & 4 & \\
3 & & 1 & & 3 \\
& 4 & & 2 & \\
& & 3 & & \\
\end{array}, &
\begin{array}{ccccc}
& & 4 & & \\
& 1 & & 3 & \\
4 & & 2 & & 4 \\
& 3 & & 1 & \\
& & 4 & & \\
\end{array} \\
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

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

== Notes ==
<references />

@@ Line 31: / Line 31: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>C_2 \wr C_2</math><ref>Proposition 4.3 of [[References|[BKL18]]]</ref>
+|Pic-O = <math>C_2 \wr C_2</math><ref>Proposition 4.3 of [[References#B|[BKL18]]]</ref>
 |PIgroup =
 |source? = No

@@ Line 4: / Line 4: @@
 |representative =  <math>k(S_3 \times S_3)</math>
 |defect = [[C3xC3|<math>C_3 \times C_3</math>]]
-|inertialquotients = <math>C_2</math>
+|inertialquotients = <math>C_2 \times C_2</math>
 |k(B) = 9
 |l(B) = 4
@@ Line 18: / Line 18: @@
 |inertial-morita-inv? =
 |O-morita? = Yes
-|O-morita = <math>\mathcal{O} ((S_3 \times S_3)</math>
+|O-morita = <math>\mathcal{O} (S_3 \times S_3)</math>
 |decomp = <math>\left( \begin{array}{cccc}
 & 0 & 0 & 0 \\
@@ Line 24: / Line 24: @@
 & 0 & 1 & 0 \\
 & 0 & 0 & 1 \\
-0 & 0 & 0 1 \\
+& 0 & 0 & 1 \\
 & 1 & 1 & 0 \\
 & 1 & 0 & 0 \\
@@ Line 36: / Line 36: @@
 |sourcereps =
 |k-derived-known? = No
-|k-derived = none known
+|k-derived =
 |O-derived-known? = No
 |coveringblocks =

M(9,2,4) - Revision history

Charles Eaton at 22:26, 2 January 2019

Charles Eaton at 07:40, 23 October 2018

Charles Eaton: Created page with "{{blockbox |title = M(9,2,4) - k(S_3 \times S_3) |image = M(9,2,4)quiver.png |representative = k(S_3 \times S_3) |defect = C3xC3|C_3 \times..."

Charles Eaton: Created page with "{{blockbox |title = M(9,2,4) - $k(S_3 \times S_3)$ |image = M(9,2,4)quiver.png |representative = $k(S_3 \times S_3)$ |defect = C3xC3| $C_3 \times..."$