Difference between revisions of "M(3^n,1,1)"
(Image added) |
|||
Line 24: | Line 24: | ||
|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> | ||
Line 29: | Line 30: | ||
|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 = | ||
}} | }} | ||
Line 40: | Line 44: | ||
== Other notatable representatives == | == Other notatable representatives == | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
== Projective indecomposable modules == | == Projective indecomposable modules == | ||
Line 66: | Line 62: | ||
[[C(3^n)|Back to <math>C_{3^n}</math>]] | [[C(3^n)|Back to <math>C_{3^n}</math>]] | ||
+ | |||
+ | == Notes == | ||
+ | |||
+ | <references /> |
Latest revision as of 07:44, 3 December 2018
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.