Difference between revisions of "M(32,51,12)"
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|representative = <math>B_0(k(SL_2(16) \times C_2))</math> | |representative = <math>B_0(k(SL_2(16) \times C_2))</math> | ||
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]] | |defect = [[(C2)%5E5|<math>(C_2)^5</math>]] | ||
| − | |inertialquotients = <math>C_{15}</math> | + | |inertialquotients = <math>C_{15}</math>, ? |
|k(B) = 32 | |k(B) = 32 | ||
|l(B) = 15 | |l(B) = 15 | ||
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}} | }} | ||
| + | A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is in this Morita equivalence class or in [[M(32,51,11)]], which is derived equivalent to this class. | ||
| + | |||
| + | It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]). | ||
== Basic algebra == | == Basic algebra == | ||
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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(32,51, | + | If <math>b</math> is in M(32,51,12), then <math>B</math> is also in M(32,51,12). |
== Projective indecomposable modules == | == Projective indecomposable modules == | ||
Latest revision as of 15:55, 9 December 2019
| Representative: | [math]B_0(k(SL_2(16) \times C_2))[/math] |
|---|---|
| Defect groups: | [math](C_2)^5[/math] |
| Inertial quotients: | [math]C_{15}[/math], ? |
| [math]k(B)=[/math] | 32 |
| [math]l(B)=[/math] | 15 |
| [math]{\rm mf}_k(B)=[/math] | 1 |
| [math]{\rm Pic}_k(B)=[/math] | |
| Cartan matrix: | See below. |
| 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}(SL_2(16) \times C_2))[/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(32,51,11) |
| [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: |
A block with defect group [math](C_2)^5[/math] and inertial quotient [math]C_{15}[/math] is in this Morita equivalence class or in M(32,51,11), which is derived equivalent to this class.
It is unknown whether this Morita equivalence class contains blocks with inertial quotient [math]C_7:C_3 \times C_3[/math] (with action as in M(32,51,24)).
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(32,51,12), then [math]B[/math] is also in M(32,51,12).
Projective indecomposable modules
Irreducible characters
All irreducible characters have height zero.
Cartan matrix
[math]\left( \begin{array}{ccc} 32 & 16 & 16 & 16 & 16 & 8 & 8 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 \\ 16 & 16 & 8 & 8 & 8 & 4 & 8 & 4 & 4 & 8 & 0 & 0 & 2 & 4 & 0 \\ 16 & 8 & 16 & 8 & 8 & 8 & 4 & 0 & 4 & 4 & 8 & 0 & 0 & 2 & 4 \\ 16 & 8 & 8 & 16 & 8 & 4 & 8 & 8 & 0 & 4 & 4 & 4 & 0 & 0 & 2 \\ 16 & 8 & 8 & 8 & 16 & 8 & 4 & 4 & 8 & 0 & 4 & 2 & 4 & 0 & 0 \\ 8 & 4 & 8 & 4 & 8 & 8 & 2 & 0 & 4 & 0 & 4 & 0 & 0 & 0 & 0 \\ 8 & 8 & 4 & 8 & 4 & 2 & 8 & 4 & 0 & 4 & 0 & 0 & 0 & 0 & 0 \\ 8 & 4 & 0 & 8 & 4 & 0 & 4 & 8 & 0 & 2 & 0 & 4 & 0 & 0 & 0 \\ 8 & 4 & 4 & 0 & 8 & 4 & 0 & 0 & 8 & 0 & 2 & 0 & 4 & 0 & 0 \\ 8 & 8 & 4 & 4 & 0 & 0 & 4 & 2 & 0 & 8 & 0 & 0 & 0 & 4 & 0 \\ 8 & 0 & 8 & 4 & 4 & 4 & 0 & 0 & 2 & 0 & 8 & 0 & 0 & 0 & 4 \\ 4 & 0 & 0 & 4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 \\ 4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 \\ 4 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 \\ 4 & 0 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 \\ \end{array} \right)[/math]
Decomposition matrix
[math]\left( \begin{array}{ccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \end{array}\right)[/math]
