Difference between revisions of "M(3^n,1,1)"
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{{blockbox | {{blockbox | ||
|title = M(3^n,1,1) - <math>kC_{3^n}</math> | |title = M(3^n,1,1) - <math>kC_{3^n}</math> | ||
| − | |image = | + | |image = M(2,1,1)quiver.png |
|representative = <math>kC_{3^n}</math> | |representative = <math>kC_{3^n}</math> | ||
|defect = <math>C_{3^n}</math> | |defect = <math>C_{3^n}</math> | ||
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|O-morita-frob = 1 | |O-morita-frob = 1 | ||
|Pic-O = <math>\mathcal{L}(B)=C_{3^n}:C_{2.3^{n-1}}</math> | |Pic-O = <math>\mathcal{L}(B)=C_{3^n}:C_{2.3^{n-1}}</math> | ||
| + | |PIgroup = | ||
|source? = Yes | |source? = Yes | ||
|sourcereps = <math>kC_{3^n}</math> | |sourcereps = <math>kC_{3^n}</math> | ||
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|k-derived = Forms a derived equivalence class | |k-derived = Forms a derived equivalence class | ||
|O-derived-known? = Yes | |O-derived-known? = Yes | ||
| + | |coveringblocks = <math>M(3^n,1,2)</math>, [[M(3^n,1,2)|<math>M(3^n,1,2)</math>]]<ref>For example consider the principal blocks of <math>C_{3^n} \triangleleft D_{2.3^{n-1}}</math></ref> | ||
| + | |coveredblocks = <math>M(3^n,1,2)</math><ref>Could cover blocks in [[M(3^n,1,2)|<math>M(3^n,1,2)</math>]], but examples needed.</ref> | ||
| + | |pcoveringblocks = | ||
}} | }} | ||
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== Other notatable representatives == | == Other notatable representatives == | ||
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== Projective indecomposable modules == | == Projective indecomposable modules == | ||
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[[C(3^n)|Back to <math>C_{3^n}</math>]] | [[C(3^n)|Back to <math>C_{3^n}</math>]] | ||
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| + | == Notes == | ||
| + | |||
| + | <references /> | ||
Latest revision as of 07:44, 3 December 2018
M(3^n,1,1) - [math]kC_{3^n}[/math]
quiver.png)
| 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.
