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