M(3^n,1,1)

M(3^n,1,1) - [math]kC_{3^n}[/math]

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Representative:	[math]kC_{3^n}[/math]
Defect groups:	[math]C_{3^n}[/math]
Inertial quotients:	[math]1[/math]
[math]k(B)=[/math]	[math]3^n[/math]
[math]l(B)=[/math]	1
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{c} 3^n \\ \end{array} \right)[/math]
Defect group Morita invariant?	Yes
Inertial quotient Morita invariant?	Yes
[math]\mathcal{O}[/math]-Morita classes known?	Yes
[math]\mathcal{O}[/math]-Morita classes:	[math]\mathcal{O} C_{3^n}[/math]
Decomposition matrices:	[math]\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math]	1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math]	[math]\mathcal{L}(B)=C_{3^n}:C_{2.3^{n-1}}[/math]
[math]PI(B)=[/math]	{{{PIgroup}}}
Source algebras known?	Yes
Source algebra reps:	[math]kC_{3^n}[/math]
[math]k[/math]-derived equiv. classes known?	Yes
[math]k[/math]-derived equivalent to:	Forms a derived equivalence class
[math]\mathcal{O}[/math]-derived equiv. classes known?	Yes
[math]p'[/math]-index covering blocks:	{{{coveringblocks}}}
[math]p'[/math]-index covered blocks:	{{{coveredblocks}}}
Index [math]p[/math] covering blocks:	{{{pcoveringblocks}}}

These are nilpotent blocks.

Basic algebra

Quiver: a:<1,1>

Relations w.r.t. [math]k[/math]: a^{3^n}=0

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] lies in M(3^n,1,1), then [math]B[/math] must lie in M(3^n,1,1) or M(3^n,1,2). For example consider the principal blocks of [math]C_{3^n} \triangleleft D_{2.3^{n-1}}[/math].

If [math]B[/math] lies in M(3^n,1,1), then [math]b[/math] must lie in M(3^n,1,1) or M(3^n,1,2). Examples needed.

Projective indecomposable modules

Labelling the unique simple [math]B[/math]-module by [math]S_1[/math], the unique projective indecomposable module has Loewy structure as follows:

[math]\begin{array}{c} S_1 \\ S_1 \\ \vdots \\ S_1 \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_{3^n}[/math]