Difference between revisions of "M(8,1,1)"
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{{blockbox | {{blockbox | ||
|title = M(8,1,1) - <math>kC_8</math> | |title = M(8,1,1) - <math>kC_8</math> | ||
| − | |image = | + | |image = M(2,1,1)quiver.png |
|representative = <math>kC_8</math> | |representative = <math>kC_8</math> | ||
|defect = [[C8|<math>C_8</math>]] | |defect = [[C8|<math>C_8</math>]] | ||
| Line 8: | Line 8: | ||
|l(B) = 1 | |l(B) = 1 | ||
|k-morita-frob = 1 | |k-morita-frob = 1 | ||
| − | |Pic-k= | + | |Pic-k= <math>k^6:k^*</math> |
|cartan = <math>\left( \begin{array}{c} | |cartan = <math>\left( \begin{array}{c} | ||
8 \\ | 8 \\ | ||
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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 = M(8,1,1) | ||
| + | |coveredblocks = M(8,1,1) | ||
}} | }} | ||
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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>. | 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>. | ||
| − | <math>B</math> and <math>b</math> must be Morita equivalent. | + | If <math>B</math> or <math>b</math> is in M(8,1,1), then <math>B</math> and <math>b</math> must be Morita equivalent. |
== Projective indecomposable modules == | == Projective indecomposable modules == | ||
Latest revision as of 14:16, 7 October 2018
M(8,1,1) - [math]kC_8[/math]
quiver.png)
| Representative: | [math]kC_8[/math] |
|---|---|
| Defect groups: | [math]C_8[/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] | [math]k^6:k^*[/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_8[/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_8:C_4[/math] |
| [math]PI(B)=[/math] | {{{PIgroup}}} |
| Source algebras known? | Yes |
| Source algebra reps: | [math]kC_8[/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: | M(8,1,1) |
| [math]p'[/math]-index covered blocks: | M(8,1,1) |
| Index [math]p[/math] covering blocks: | {{{pcoveringblocks}}} |
These are nilpotent blocks.
Contents
Basic algebra
Quiver: a:<1,1>
Relations w.r.t. [math]k[/math]: a^8=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(8,1,1), 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.
