Difference between revisions of "M(8,2,1)"
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Revision as of 15:12, 8 September 2018
M(8,2,1) - [math]k(C_4 \times C_2)[/math]
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Representative: | [math]k(C_4 \times C_2)[/math] |
---|---|
Defect groups: | [math]C_4 \times C_2[/math] |
Inertial quotients: | [math]1[/math] |
[math]k(B)=[/math] | 8 |
[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} 8 \\ \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_4 \times C_2)[/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_4 \times C_2):(C_2 \times C_2 \times C_2)[/math] |
[math]PI(B)=[/math] | {{{PIgroup}}} |
Source algebras known? | Yes |
Source algebra reps: | [math]k(C_4 \times C_2)[/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.
Contents
Basic algebra
Quiver: a:<1,1>, b:<1,1>
Relations w.r.t. [math]k[/math]: a^4=b^2=ab+ba=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] or [math]b[/math] is in M(8,2,1), then [math]B[/math] and [math]b[/math] must be Morita equivalent.
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.