Difference between revisions of "M(8,5,8)"

From Block library
Jump to: navigation, search
(Created page with "{{blockbox |title = M(8,5,8) - <math>B_0(k(\rm Aut (SL_2(8)))</math> |image = M(8,5,8)quiver.png |representative = <math>B_0(k(\rm Aut (SL_2(8)))</math> |defect = C2xC2xC2...")
(No difference)

Revision as of 10:31, 5 October 2018

M(8,5,8) - [math]B_0(k(\rm Aut (SL_2(8)))[/math]
M(8,5,8)quiver.png
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}}}

Basic algebra

Quiver:

Relations w.r.t. [math]k[/math]:

Other notatable representatives

Projective indecomposable modules

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_2 \times C_2 \times C_2[/math]