Difference between revisions of "M(16,3,3)"
(Was M(16,3,2).) |
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1 & 1 \\ | 1 & 1 \\ | ||
1 & 1 \\ | 1 & 1 \\ | ||
| − | \end{array}\right)</math><ref>This is the only possible decomposition matrix with the given Cartan matrix.</ref> | + | \end{array}\right)</math><ref>This is the only possible decomposition matrix with the given Cartan matrix and <math>k(B)=10</math>.</ref> |
|O-morita-frob = | |O-morita-frob = | ||
|Pic-O = | |Pic-O = | ||
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== Projective indecomposable modules == | == Projective indecomposable modules == | ||
| + | |||
| + | Labelling the simple <math>B</math>-modules by <math>1, 2</math>, the projective indecomposable modules have Loewy structure as follows: | ||
| + | |||
| + | <math>\begin{array}{cc} | ||
| + | \begin{array}{c} 1 \\ 1 \ 2 \\ 1 \ 1 \ 2 \\ 1 \ 1 \ 2 \\ 1 \ 2 \\ 1 \\ \end{array}, | ||
| + | & | ||
| + | \begin{array}{c} 2 \\ 1 \ 2 \ 2 \\ 1 \ 2 \\ 1 \ 2 \\ 1 \\ 2 \\ \end{array} \\ | ||
| + | \end{array} | ||
| + | </math> | ||
== Irreducible characters == | == Irreducible characters == | ||
Latest revision as of 09:56, 15 August 2019
M(16,3,3) - [math]k(A_4:C_4)[/math][1]
[[File:|250px]]
| Representative: | [math]k(A_4:C_4)[/math] |
|---|---|
| Defect groups: | MNA(2,1) |
| Inertial quotients: | [math]1[/math] |
| [math]k(B)=[/math] | 10 |
| [math]l(B)=[/math] | 2 |
| [math]{\rm mf}_k(B)=[/math] | 1 |
| [math]{\rm Pic}_k(B)=[/math] | |
| Cartan matrix: | [math]\left( \begin{array}{cc} 8 & 4 \\ 4 & 6 \\ \end{array} \right)[/math] |
| Defect group Morita invariant? | |
| Inertial quotient Morita invariant? | |
| [math]\mathcal{O}[/math]-Morita classes known? | |
| [math]\mathcal{O}[/math]-Morita classes: | |
| Decomposition matrices: | [math]\left( \begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \\ \end{array}\right)[/math][2] |
| [math]{\rm mf}_\mathcal{O}(B)=[/math] | |
| [math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | |
| [math]PI(B)=[/math] | |
| Source algebras known? | |
| Source algebra reps: | |
| [math]k[/math]-derived equiv. classes known? | |
| [math]k[/math]-derived equivalent to: | |
| [math]\mathcal{O}[/math]-derived equiv. classes known? | |
| [math]p'[/math]-index covering blocks: | |
| [math]p'[/math]-index covered blocks: | |
| Index [math]p[/math] covering blocks: |
Contents
Basic algebra
Quiver: a: <1,2>, b:=<1,1>, c:=<2,1>, d:=<2,2>, e:=<2,2>
Relations w.r.t. [math]k[/math]:
Other notatable representatives
Projective indecomposable modules
Labelling the simple [math]B[/math]-modules by [math]1, 2[/math], the projective indecomposable modules have Loewy structure as follows:
[math]\begin{array}{cc} \begin{array}{c} 1 \\ 1 \ 2 \\ 1 \ 1 \ 2 \\ 1 \ 1 \ 2 \\ 1 \ 2 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 1 \ 2 \ 2 \\ 1 \ 2 \\ 1 \ 2 \\ 1 \\ 2 \\ \end{array} \\ \end{array} [/math]
Irreducible characters
[math]k_0(B)=8[/math], [math]k_1(B)=2[/math]
