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M(3^n,1,1) - [math]kC_{3^n}[/math]
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]
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: [math]M(3^n,1,2)[/math], [math]M(3^n,1,2)[/math][1]
[math]p'[/math]-index covered blocks: [math]M(3^n,1,2)[/math][2]
Index [math]p[/math] covering blocks:

These are nilpotent blocks.

Basic algebra

Quiver: a:<1,1>

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

Other notatable representatives

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]


  1. For example consider the principal blocks of [math]C_{3^n} \triangleleft D_{2.3^{n-1}}[/math]
  2. Could cover blocks in [math]M(3^n,1,2)[/math], but examples needed.