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

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|title = M(9,1,1) - <math>kC_9</math>  
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|image = M(2,1,1)quiver.png
|representative =  <math>kC_9</math>
|representative =  <math>kC_9</math>
|defect = <math>C_9</math>
|defect = <math>C_9</math>

Revision as of 13:29, 9 September 2018

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]
[math]PI(B)=[/math] {{{PIgroup}}}
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: {{{coveringblocks}}}
[math]p'[/math]-index covered blocks: {{{coveredblocks}}}
Index [math]p[/math] covering blocks: {{{pcoveringblocks}}}

These are nilpotent blocks.

Basic algebra

Quiver: a:<1,1>

Relations w.r.t. [math]k[/math]: a^9=0

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). Examples needed.

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]