CesareGArdito: /* Other notatable representatives */

2019-12-08T18:49:20Z

‎Other notatable representatives

CesareGArdito: /* Other notatable representatives */

2019-11-28T14:34:19Z

‎Other notatable representatives

CesareGArdito: Created page with "{{blockbox |title = M(16,14,15) - $B_0(k(C_2 \times \operatorname{Aut}(SL_2(8))))$ |image = |representative = $B_0(k(C_2 \times \operatorname{Aut}(SL..."$

2019-11-28T14:32:06Z

Created page with "{{blockbox |title = M(16,14,15) - <math>B_0(k(C_2 \times \operatorname{Aut}(SL_2(8))))</math> |image = |representative = <math>B_0(k(C_2 \times \operatorname{Aut}(SL..."

New page

{{blockbox
|title = M(16,14,15) - <math>B_0(k(C_2 \times \operatorname{Aut}(SL_2(8))))</math>
|image =  
|representative = <math>B_0(k(C_2 \times \operatorname{Aut}(SL_2(8))))</math>
|defect = [[(C2)%5E4|<math>(C_2)^4</math>]]
|inertialquotients = <math>C_7:C_3</math>
|k(B) = 16
|l(B) = 5
|k-morita-frob = 1
|Pic-k=  
|cartan = <math>\left( \begin{array}{ccccc}
8 & 4 & 4 & 8 & 4 \\
4 & 8 & 4 & 8 & 4 \\
4 & 4 & 8 & 8 & 4 \\
8 & 8 & 8 & 16 & 6 \\
4 & 4 & 4 & 6 & 4
\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 \operatorname{Aut}(SL_2(8))))</math>
|decomp = <math>\left( \begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 \\
1 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 & 0 \\
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 2 & 1 \\
1 & 1 & 1 & 2 & 1
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O =
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(16,14,13)]], [[M(16,14,14)]]
|O-derived-known? = Yes
|coveringblocks =
|coveredblocks =
|pcoveringblocks =
}}

== Basic algebra ==

== Other notatable representatives ==
<math>B_0(k(C_2 \times {}^2G_2(q)))</math>, the unique nonprincipal block with defect group $(C_2)^4$ of $C_2 \times \operatorname{Co}_3$.

== Covering blocks and covered blocks ==

Let <math>N \triangleleft G</math> with prime <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(16,14,15), then <math>B</math> is in [[M(16,14,7)]] or M(16,14,15).

== Projective indecomposable modules ==

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

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

\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

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

@@ Line 54: / Line 54: @@
 == Other notatable representatives ==
-<math>B_0(k(C_2 \times {}^2G_2(q)))</math>, and the unique nonprincipal block with defect group $(C_2)^4$ of $C_2 \times \operatorname{Co}_3$.
+<math>B_0(k(C_2 \times {}^2G_2(q)))</math>, and the unique nonprincipal block with defect group <math>(C_2)^4</math> of <math>C_2 \times \operatorname{Co}_3</math>.
 == Covering blocks and covered blocks ==

M(16,14,15) - Revision history

CesareGArdito: /* Other notatable representatives */

CesareGArdito: /* Other notatable representatives */

CesareGArdito: Created page with "{{blockbox |title = M(16,14,15) - B_0(k(C_2 \times \operatorname{Aut}(SL_2(8)))) |image = |representative = B_0(k(C_2 \times \operatorname{Aut}(SL..."

CesareGArdito: Created page with "{{blockbox |title = M(16,14,15) - $B_0(k(C_2 \times \operatorname{Aut}(SL_2(8))))$ |image = |representative = $B_0(k(C_2 \times \operatorname{Aut}(SL..."$