Difference between revisions of "M(8,5,8)"
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Revision as of 10:31, 5 October 2018
M(8,5,8) - [math]B_0(k(\rm Aut (SL_2(8)))[/math]
Representative: | [math]B_0(k(\rm Aut (SL_2(8)))[/math] |
---|---|
Defect groups: | [math]C_2 \times C_2 \times C_2[/math] |
Inertial quotients: | [math]C_7:C_3[/math] |
[math]k(B)=[/math] | 8 |
[math]l(B)=[/math] | 5 |
[math]{\rm mf}_k(B)=[/math] | 1 |
[math]{\rm Pic}_k(B)=[/math] | |
Cartan matrix: | [math]\left( \begin{array}{ccccccc} 4 & 2 & 2 & 4 & 2 \\ 2 & 4 & 2 & 4 & 2 \\ 2 & 2 & 4 & 4 & 2 \\ 4 & 4 & 4 & 8 & 3 \\ 2 & 2 & 2 & 3 & 2 \\ \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_2 \times C_2 \times C_2):(C_7:C_3))[/math] |
Decomposition matrices: | [math]\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ \end{array}\right)[/math] |
[math]{\rm mf}_\mathcal{O}(B)=[/math] | 1 |
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | |
[math]PI(B)=[/math] | {{{PIgroup}}} |
Source algebras known? | No |
Source algebra reps: | |
[math]k[/math]-derived equiv. classes known? | Yes |
[math]k[/math]-derived equivalent to: | M(8,5,6), M(8,5,7) |
[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: | {{{pcoveringblocks}}} |
Contents
Basic algebra
Quiver:
Relations w.r.t. [math]k[/math]:
Other notatable representatives
Projective indecomposable modules
Irreducible characters
All irreducible characters have height zero.