M(16,14,6) - Revision history

CesareGArdito at 14:53, 27 November 2019

2019-11-27T14:53:00Z

CesareGArdito at 14:22, 27 November 2019

2019-11-27T14:22:31Z

CesareGArdito at 14:20, 27 November 2019

2019-11-27T14:20:47Z

CesareGArdito at 14:19, 27 November 2019

2019-11-27T14:19:45Z

CesareGArdito: Created page with "{{blockbox |title = M(16,14,4) - $k(C_2 \times ((C_2)^3 : C_7))$ |image = |representative = $k((C_2)^4 : C_7)$ |defect = (C2)%5E4| $(C_2)..."$

2019-11-27T14:19:23Z

Created page with "{{blockbox |title = M(16,14,4) - <math>k(C_2 \times ((C_2)^3 : C_7))</math> |image = |representative = <math>k((C_2)^4 : C_7)</math> |defect = (C2)%5E4|<math>(C_2)..."

New page

{{blockbox
|title = M(16,14,4) - <math>k(C_2 \times ((C_2)^3 : C_7))</math>
|image =  
|representative = <math>k((C_2)^4 : C_7)</math>
|defect = [[(C2)%5E4|<math>(C_2)^4</math>]]
|inertialquotients = <math>C_5</math>
|k(B) = 16
|l(B) = 7
|k-morita-frob = 1
|Pic-k=  
|cartan = <math>\left( \begin{array}{ccc}
2 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 2 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 2 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 2 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 2 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 & 2 & 1 \\
1 & 1 & 1 & 1 & 1 & 1 & 2
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>\mathcal{O} (C_2 \times ((C_2)^3 : C_7))</math>
|decomp = <math>\left( \begin{array}{ccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 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? = 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(16,14,6), then <math>B</math> is in [[M(16,14,1)]], M(16,14,6) or [[M(16,14,13)]].

== Projective indecomposable modules ==

Labelling the simple <math>B</math>-modules by <math>S_1, S_2, S_3, S_4, S_5, S_6, S_7</math>, the projective indecomposable modules have Loewy structure as follows:

<math>\begin{array}{ccccccc}
\begin{array}{c}
S_1 \\
S_3 S_4 S_5 \\
S_2 S_6 S_7 \\
S_1 \\
\end{array}
&
\begin{array}{c}
S_2 \\
S_1 S_4 S_7 \\
S_3 S_5 S_6 \\
S_2 \\
\end{array}
&
\begin{array}{c}
S_3 \\
S_2 S_4 S_6 \\
S_1 S_5 S_7 \\
S_3 \\
\end{array}
&
\begin{array}{c}
S_4 \\
S_5 S_6 S_7 \\
S_1 S_2 S_3 \\
S_4 \\
\end{array}
&
\begin{array}{c}
S_5 \\
S_2 S_3 S_7 \\
S_1 S_4 S_6 \\
S_5 \\
\end{array}
&
\begin{array}{c}
S_6 \\
S_1 S_2 S_5 \\
S_3 S_4 S_7 \\
S_6 \\
\end{array}
&
\begin{array}{c}
S_7 \\
S_1 S_3 S_6 \\
S_2 S_4 S_5 \\
S_7 \\
\end{array}

\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

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

@@ Line 10: / Line 10: @@
 |Pic-k= &nbsp;
 |cartan = <math>\left( \begin{array}{ccccccc}
-& 1 & 1 & 1 & 1 & 1 & 1 \\
+& 2 & 2 & 2 & 2 & 2 & 2 \\
-& 2 & 1 & 1 & 1 & 1 & 1 \\
+& 4 & 2 & 2 & 2 & 2 & 2 \\
-& 1 & 2 & 1 & 1 & 1 & 1 \\
+& 2 & 4 & 2 & 2 & 2 & 2 \\
-& 1 & 1 & 2 & 1 & 1 & 1 \\
+& 2 & 2 & 4 & 2 & 2 & 2 \\
-& 1 & 1 & 1 & 2 & 1 & 1 \\
+& 2 & 2 & 2 & 4 & 2 & 2 \\
-& 1 & 1 & 1 & 1 & 2 & 1 \\
+& 2 & 2 & 2 & 2 & 4 & 2 \\
-& 1 & 1 & 1 & 1 & 1 & 2
+& 2 & 2 & 2 & 2 & 2 & 4
 \end{array} \right)</math>
 |defect-morita-inv? = Yes
@@ Line 22: / Line 22: @@
 |O-morita? = Yes
 |O-morita = <math>\mathcal{O} (C_2 \times ((C_2)^3 : C_7))</math>
-|decomp = <math>\left( \begin{array}{ccc}
+|decomp = See below
 |O-morita-frob = 1
 |Pic-O =
@@ Line 116: / Line 107: @@
 All irreducible characters have height zero.
 [[(C2)%5E4|Back to <math>(C_2)^4</math>]]

@@ Line 1: / Line 1: @@
 {{blockbox
-|title = M(16,14,4) - <math>k(C_2 \times ((C_2)^3 : C_7))</math>
+|title = M(16,14,6) - <math>k(C_2 \times ((C_2)^3 : C_7))</math>
 |image = &nbsp;
 |representative =  <math>k((C_2)^4 : C_7)</math>