M(16,3,3) - Revision history

Charles Eaton: Loewy layers

2019-08-15T09:56:39Z

Loewy layers

Charles Eaton at 09:47, 15 August 2019

2019-08-15T09:47:05Z

Charles Eaton: Was M(16,3,2).

2019-08-15T09:29:25Z

Was M(16,3,2).

New page

{{blockbox
|title = M(16,3,3) - <math>k(A_4:C_4)</math><ref><math>A_4:C_4</math> is SmallGroup(48,30).</ref>
|image =
|representative = <math>k(A_4:C_4)</math>
|defect = [[MNA(2,1)]]
|inertialquotients = <math>1</math>
|k(B) = 10
|l(B) = 2
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{cc}
8 & 4 \\
4 & 6 \\
\end{array} \right)</math>
|defect-morita-inv? =
|inertial-morita-inv? =
|O-morita? =
|O-morita =
|decomp = <math>\left( \begin{array}{cc}
1 & 0 \\
1 & 0 \\
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
\end{array}\right)</math><ref>This is the only possible decomposition matrix with the given Cartan matrix.</ref>
|O-morita-frob =
|Pic-O =
|PIgroup =
|source? =
|sourcereps =
|k-derived-known? =
|k-derived =
|O-derived-known? =
|coveringblocks =
|coveredblocks =
|pcoveringblocks =
}}

== Basic algebra ==

'''Quiver:''' a: <1,2>, b:=<1,1>, c:=<2,1>, d:=<2,2>, e:=<2,2>

'''Relations w.r.t. <math>k</math>:'''

== Other notatable representatives ==

== Projective indecomposable modules ==

== Irreducible characters ==

<math>k_0(B)=8</math>, <math>k_1(B)=2</math>

[[MNA(2,1)|Back to <math>MNA(2,1)</math>]]

== Notes ==
<references />

← Older revision		Revision as of 09:56, 15 August 2019
Line 51:		Line 51:

	== Projective indecomposable modules ==		== Projective indecomposable modules ==
		+
		+	Labelling the simple <math>B</math>-modules by <math>1, 2</math>, the projective indecomposable modules have Loewy structure as follows:
		+
		+	<math>\begin{array}{cc}
		+	\begin{array}{c} 1 \\ 1 \ 2 \\ 1 \ 1 \ 2 \\ 1 \ 1 \ 2 \\ 1 \ 2 \\ 1 \\ \end{array},
		+	&
		+	\begin{array}{c} 2 \\ 1 \ 2 \ 2 \\ 1 \ 2 \\ 1 \ 2 \\ 1 \\ 2 \\ \end{array} \\
		+	\end{array}
		+	</math>

	== Irreducible characters ==		== Irreducible characters ==

@@ Line 28: / Line 28: @@
 & 1 \\
 & 1 \\
-\end{array}\right)</math><ref>This is the only possible decomposition matrix with the given Cartan matrix.</ref>
+\end{array}\right)</math><ref>This is the only possible decomposition matrix with the given Cartan matrix and <math>k(B)=10</math>.</ref>
 |O-morita-frob =
 |Pic-O =