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2019-11-28T13:13:05Z

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{{blockbox
|title = M(16,14,9) - <math>B_0(k(A_4 \times A_5))</math>
|image =  
|representative = <math>B_0(k(A_4 \times A_5))</math>
|defect = [[(C2)%5E4|<math>(C_2)^4</math>]]
|inertialquotients = <math>C_3 \times C_3</math>
|k(B) = 16
|l(B) = 9
|k-morita-frob = 1
|Pic-k=  
|cartan = <math>\left( \begin{array}{ccccccccc}
8 & 4 & 4 & 4 & 4 & 2 & 2 & 2 & 2 \\
4 & 8 & 4 & 2 & 2 & 4 & 2 & 2 & 4 \\
4 & 4 & 8 & 2 & 2 & 2 & 4 & 4 & 2 \\
4 & 2 & 2 & 4 & 2 & 2 & 1 & 2 & 1 \\
4 & 2 & 2 & 2 & 4 & 1 & 2 & 1 & 2 \\
2 & 4 & 2 & 2 & 1 & 4 & 1 & 2 & 2 \\
2 & 2 & 4 & 1 & 2 & 1 & 4 & 2 & 2 \\
2 & 2 & 4 & 2 & 1 & 2 & 2 & 4 & 1 \\
2 & 4 & 2 & 1 & 2 & 2 & 2 & 1 & 4
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O} (A_4 \times A_5))</math>
|decomp = See below
|O-morita-frob = 1
|Pic-O = <math>S_3 \times C_2</math>
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(16,14,8)]], [[M(16,14,10)]]
|O-derived-known? = Yes
|coveringblocks =
|coveredblocks =
|pcoveringblocks =
}}

== Basic algebra ==

== Other notatable representatives ==

== 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(16,14,9), 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, \dots, S_9</math>, the projective indecomposable modules have Loewy structure as follows:

<math>\begin{array}{ccccccccc}
\begin{array}{c}
S_1 \\
S_2 S_3 S_4 S_5 \\
S_1 S_1 S_1 S_6 S_7 S_8 S_9\\
S_2 S_2 S_3 S_3 S_4 S_4 S_5 S_5 \\
S_1 S_1 S_1 S_6 S_7 S_8 S_9\\
S_2 S_3 S_4 S_5 \\
S_1 \\

\end{array}
&
\begin{array}{c}
S_2 \\
S_1 S_3 S_6 S_9 \\
S_2 S_2 S_2 S_4 S_5 S_7 S_8 \\
S_1 S_1 S_3 S_3 S_6 S_6 S_9 S_9 \\
S_2 S_2 S_2 S_4 S_5 S_7 S_8 \\
S_1 S_3 S_6 S_9 \\
S_2 \\

\end{array}
&
\begin{array}{c}
S_3 \\
S_1 S_2 S_7 S_8 \\
S_3 S_3 S_3 S_4 S_5 S_6 S_9 \\
S_1 S_1 S_2 S_2 S_7 S_7 S_8 S_8 \\
S_3 S_3 S_3 S_4 S_5 S_6 S_9 \\
S_1 S_2 S_7 S_8 \\
S_3 \\

\end{array}

&
\begin{array}{c}
S_4 \\
S_1 S_6 S_8 \\
S_2 S_3 S_4 S_5 \\
S_1 S_1 S_7 S_9 \\
S_2 S_3 S_4 S_5 \\
S_1 S_6 S_8 \\
S_4 \\

\end{array}
&
\begin{array}{c}
S_5 \\
S_1 S_7 S_9 \\
S_2 S_3 S_4 S_5 \\
S_1 S_1 S_6 S_8 \\
S_2 S_3 S_4 S_5 \\
S_1 S_7 S_9 \\
S_5 \\

\end{array}
&
\begin{array}{c}
S_6 \\
S_2 S_4 S_8 \\
S_1 S_3 S_6 S_9 \\
S_2 S_2 S_5 S_7 \\
S_1 S_3 S_6 S_9 \\
S_2 S_4 S_8 \\
S_6 \\

\end{array}
&
\begin{array}{c}
S_7 \\
S_3 S_5 S_9 \\
S_1 S_2 S_7 S_8 \\
S_3 S_3 S_4 S_6 \\
S_1 S_2 S_7 S_8 \\
S_3 S_5 S_9 \\
S_7 \\

\end{array}
&
\begin{array}{c}
S_8 \\
S_3 S_4 S_6 \\
S_1 S_2 S_7 S_8 \\
S_3 S_3 S_5 S_9 \\
S_1 S_2 S_7 S_8 \\
S_3 S_4 S_6 \\
S_8 \\

\end{array}
&
\begin{array}{c}
S_9 \\
S_2 S_5 S_7 \\
S_1 S_3 S_6 S_9 \\
S_2 S_2 S_4 S_8 \\
S_1 S_3 S_6 S_9 \\
S_2 S_5 S_7 \\
S_9 \\

\end{array}

\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

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

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

M(16,14,9) - Revision history

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