Difference between revisions of "M(9,1,1)"

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|O-morita-frob = 1
|O-morita-frob = 1
|Pic-O = <math>\mathcal{L}(B)=C_9:C_6</math>
|Pic-O = <math>\mathcal{L}(B)=C_9:C_6</math>
|PIgroup =
|source? = Yes
|source? = Yes
|sourcereps = <math>kC_9</math>
|sourcereps = <math>kC_9</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 =
|coveredblocks =
|pcoveringblocks =
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== Other notatable representatives ==
== Other notatable representatives ==
== 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(9,1,1), then <math>B</math> must lie in M(9,1,1) or [[M(9,1,2)]]. For example consider the principal blocks of <math>C_9 \triangleleft D_{18}</math>.
If <math>B</math> lies in M(9,1,1), then <math>b</math> must lie in M(9,1,1) or [[M(9,1,2)]]. <span style="color: red">Examples needed.</span>
== Projective indecomposable modules ==
== Projective indecomposable modules ==

Latest revision as of 17:56, 1 February 2019

M(9,1,1) - [math]kC_9[/math]
Representative: [math]kC_9[/math]
Defect groups: [math]C_9[/math]
Inertial quotients: [math]1[/math]
[math]k(B)=[/math] 9
[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} 9 \\ \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_9[/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_9:C_6[/math]
Source algebras known? Yes
Source algebra reps: [math]kC_9[/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]p'[/math]-index covered blocks:
Index [math]p[/math] covering blocks:

These are nilpotent blocks.

Basic algebra

Quiver: a:<1,1>

Relations w.r.t. [math]k[/math]: a^9=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_9[/math]