Difference between revisions of "M(16,14,10)"
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Let <math>N \triangleleft G</math> with prime <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 prime <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> is in M(16,14, | + | If <math>b</math> is in M(16,14,10), then <math>B</math> is also in M(16,14,10). |
== Projective indecomposable modules == | == Projective indecomposable modules == | ||
Latest revision as of 13:19, 28 November 2019
| Representative: | [math]B_0(k(A_5 \times A_5))[/math] |
|---|---|
| Defect groups: | [math](C_2)^4[/math] |
| Inertial quotients: | [math]C_3 \times C_3[/math] |
| [math]k(B)=[/math] | 16 |
| [math]l(B)=[/math] | 9 |
| [math]{\rm mf}_k(B)=[/math] | 1 |
| [math]{\rm Pic}_k(B)=[/math] | |
| Cartan matrix: | [math]\left( \begin{array}{ccccccccc} 16 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 \\ 8 & 8 & 4 & 4 & 4 & 4 & 2 & 2 & 4 \\ 8 & 4 & 8 & 4 & 4 & 2 & 4 & 4 & 2 \\ 8 & 4 & 4 & 8 & 4 & 4 & 2 & 4 & 2 \\ 8 & 4 & 4 & 4 & 8 & 2 & 4 & 2 & 4 \\ 4 & 4 & 2 & 4 & 2 & 4 & 1 & 2 & 2 \\ 4 & 2 & 4 & 2 & 4 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 4 & 2 & 2 & 2 & 4 & 1 \\ 4 & 4 & 2 & 2 & 4 & 2 & 2 & 1 & 4 \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]B_0(\mathcal{O} (A_5 \times A_5))[/math] |
| Decomposition matrices: | See below |
| [math]{\rm mf}_\mathcal{O}(B)=[/math] | 1 |
| [math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | |
| [math]PI(B)=[/math] | |
| Source algebras known? | No |
| Source algebra reps: | |
| [math]k[/math]-derived equiv. classes known? | Yes |
| [math]k[/math]-derived equivalent to: | M(16,14,8), M(16,14,9) |
| [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: |
Contents
Basic algebra
Other notatable representatives
Covering blocks and covered blocks
Let [math]N \triangleleft G[/math] with prime [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] is in M(16,14,10), then [math]B[/math] is also in M(16,14,10).
Projective indecomposable modules
Irreducible characters
All irreducible characters have height zero.
Decomposition matrix
[math]\left( \begin{array}{ccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array}\right)[/math]
