Charles Eaton at 17:16, 21 October 2018

2018-10-21T17:16:47Z

Charles Eaton: Created page with "{{blockbox |title = M(9,2,1) - $k(C_3 \times C_3)$ |image = M(4,2,1)quiver.png |representative = $k(C_3 \times C_3)$ |defect = $C_3 \times C_3$
2018-10-21T17:11:01Z

Created page with "{{blockbox |title = M(9,2,1) - <math>k(C_3 \times C_3)</math> |image = M(4,2,1)quiver.png |representative = <math>k(C_3 \times C_3)</math> |defect = <math>C_3 \times C_3</ma..."

New page
{{blockbox
|title = M(9,2,1) - <math>k(C_3 \times C_3)</math>
|image = M(4,2,1)quiver.png
|representative = <math>k(C_3 \times C_3)</math>
|defect = <math>C_3 \times C_3</math>
|inertialquotients = <math>1</math>
|k(B) = 9
|l(B) = 1
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{c}
9 \\
\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}{c}
1 \\
1 \\
\vdots \\
1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O = <math>\mathcal{L}(B)=(C_3 \times C_3):GL_2(3)</math>
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = Forms a derived equivalence class
|O-derived-known? = Yes
|coveringblocks = [[M(9,2,2)]], [[M(9,2,3)]] (complete)
|coveredblocks =
|pcoveringblocks = [[M(27,2,1)]], [[M(27,5,1)]]
}}

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

== Basic algebra ==

'''Quiver:''' a:<1,1>, b:<1,1>

'''Relations w.r.t. <math>k</math>:''' a^3=b^3=ab-ba=0

== Other notatable representatives ==

== Covering blocks and covered blocks ==

== Projective indecomposable modules ==

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

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

== Irreducible characters ==

All irreducible characters have height zero.

@@ Line 3: / Line 3: @@
 |image = M(4,2,1)quiver.png
 |representative =  <math>k(C_3 \times C_3)</math>
-|defect = <math>C_3 \times C_3</math>
+|defect = [[C3xC3|<math>C_3 \times C_3</math>]]
 |inertialquotients = <math>1</math>
 |k(B) = 9
@@ Line 63: / Line 63: @@
 All irreducible characters have height zero.