M(16,14,2) - Revision history

CesareGArdito at 13:48, 27 November 2019

2019-11-27T13:48:14Z

CesareGArdito: /* Decomposition matrix */

2019-11-27T13:47:25Z

‎Decomposition matrix

CesareGArdito: Created page with "{{blockbox |title = M(16,14,2) - $k(C_2 \times C_2 \times A_5)$ |image = |representative = $k(C_2 \times C_2 \times A_5)$ |defect = (C2)%5E4|..."

2019-11-27T13:42:10Z

Created page with "{{blockbox |title = M(16,14,2) - <math>k(C_2 \times C_2 \times A_5)</math> |image = |representative = <math>k(C_2 \times C_2 \times A_5)</math> |defect = (C2)%5E4|..."

New page

{{blockbox
|title = M(16,14,2) - <math>k(C_2 \times C_2 \times A_5)</math>
|image =  
|representative = <math>k(C_2 \times C_2 \times A_5)</math>
|defect = [[(C2)%5E4|<math>(C_2)^4</math>]]
|inertialquotients = <math>C_3</math>
|k(B) = 16
|l(B) = 3
|k-morita-frob = 1
|Pic-k=  
|cartan = <math>\left( \begin{array}{ccc}
16 & 8 & 8\\
8 & 8 & 4 \\
8 & 4 & 8
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O} (C_2 \times C_2 \times A_5)</math>
|decomp = See below
|O-morita-frob = 1
|Pic-O = <math>(C_2 \times C_2):S_3 \times C_2</math>
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = Forms a derived equivalence class
|O-derived-known? = Yes
|coveringblocks =
|coveredblocks =
|pcoveringblocks =
}}

== Basic algebra ==

== Other notatable representatives ==

== Covering blocks and covered blocks ==

Let <math>N \triangleleft G</math> with <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.

If <math>b</math> is in M(16,14,2), then <math>B</math> is in M(16,14,2) or [[M(16,14,9)]].

== Projective indecomposable modules ==

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

<math>\begin{array}{ccc}
\begin{array}{c}
S_1 \\
S_1 S_1 S_2 S_3 \\
S_1 S_1 S_1 S_2 S_3 S_2 S_3 \\
S_1 S_1 S_1 S_1 S_2 S_3 S_2 S_3 \\
S_1 S_1 S_1 S_2 S_3 S_2 S_3 \\
S_1 S_1 S_2 S_3 \\
S_1 \\

\end{array}
&
\begin{array}{c}
S_2 \\
S_1 S_2 S_2 \\
S_1 S_1 S_3 S_2 \\
S_1 S_1 S_3 S_3 \\
S_1 S_1 S_3 S_2 \\
S_1 S_2 S_2 \\
S_2 \\

\end{array}
&
\begin{array}{c}
S_3 \\
S_1 S_3 S_3 \\
S_1 S_1 S_2 S_3 \\
S_1 S_1 S_2 S_2 \\
S_1 S_1 S_2 S_3 \\
S_1 S_3 S_3 \\
S_3 \\

\end{array}
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

== Decomposition matrix ==
<math>\left( \begin{array}{ccc}
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 \\
1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0
\end{array}\right)</math>

[[(C2)%5E4|Back to <math>(C_2)^4</math>]]

@@ Line 25: / Line 25: @@
 |sourcereps =
 |k-derived-known? = Yes
-|k-derived = Forms a derived equivalence class
+|k-derived = [[M(16,14,3)]]
 |O-derived-known? = Yes
 |coveringblocks =

@@ Line 88: / Line 88: @@
 == Decomposition matrix ==
 <math>\left( \begin{array}{ccc}
-& 0 & 0 & 0 \\
+& 0 & 0  \\
-& 0 & 0 & 0 \\
+& 0 & 0  \\
-& 0 & 0 & 0 \\
+& 0 & 0  \\
-& 0 & 0 & 0 \\
+& 0 & 0  \\
-& 0 & 1 & 0 \\
+& 0 & 1  \\
-& 1 & 0 & 0 \\
+& 1 & 0  \\
-& 0 & 1 & 0 \\
+& 0 & 1  \\
-& 1 & 0 & 0 \\
+& 1 & 0  \\
-& 0 & 1 & 0 \\
+& 0 & 1  \\
-& 1 & 0 & 0 \\
+& 1 & 0  \\
-& 1 & 0 & 0 \\
+& 1 & 0  \\
-& 0 & 1 & 0 \\
+& 0 & 1  \\
-& 0 & 0 & 1 \\
+& 1 & 1  \\
-& 0 & 0 & 1 \\
+& 1 & 1  \\
-& 0 & 0 & 1 \\
+& 1 & 1  \\
-& 0 & 0 & 1 \\
+& 1 & 1
 \end{array}\right)</math>
 [[(C2)%5E4|Back to <math>(C_2)^4</math>]]

M(16,14,2) - Revision history

CesareGArdito at 13:48, 27 November 2019

CesareGArdito: /* Decomposition matrix */

CesareGArdito: Created page with "{{blockbox |title = M(16,14,2) - k(C_2 \times C_2 \times A_5) |image = |representative = k(C_2 \times C_2 \times A_5) |defect = (C2)%5E4|..."

CesareGArdito: Created page with "{{blockbox |title = M(16,14,2) - $k(C_2 \times C_2 \times A_5)$ |image = |representative = $k(C_2 \times C_2 \times A_5)$ |defect = (C2)%5E4|..."