M(8,5,8)

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]B_0(\mathcal{O}(\rm Aut (SL_2(8))))[/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:	Potentially M(8,5,5)
[math]p'[/math]-index covered blocks:	M(8,5,5)
Index [math]p[/math] covering blocks:	{{{pcoveringblocks}}}

The principal 2-blocks of all Ree groups [math]{}^2G_2(3^{2m+1})[/math] belong to this Morita equivalence class.

Basic algebra

Quiver: a:<4,1>, b:<1,4>, c:<2,4>, d:<4,2>, e:<4,3>, f:<3,4>, g:<4,5>, h:<5,4>

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

Other notatable representatives

[math]{\rm Aut (SL_2(8))} \cong {}^2G_2(3)[/math], and the blocks [math]B_0(\mathcal{O}({}^2G_2(3^{2m+1})))[/math] are Morita equivalent for all [math]m[/math]. This follows from Example 3.3 of [Ok97], where, as noted in 6.2.2 of [CR13] the Morita equivalence is splendid and so lifts to [math]\mathcal{O}[/math]

By [Ea16] the principal block of each subgroup of [math]{\rm Aut}({}^2G_2(3^{2m+1}))[/math] containing [math]{}^2G_2(3^{2m+1})[/math] is in this Morita equivalence class.

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]1,2,3,4,5[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccccc} \begin{array}{c} 1 \\ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \\ 2 \\ \end{array}, & \begin{array}{c} 3 \\ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \\ 3 \\ \end{array}, & \begin{array}{c} 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \ 4 \ 4 \\ 1 \ 1 \ 2 \ 2 \ 3 \ 3 \ 5 \\ 4 \ 4 \ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \\ \end{array}, & \begin{array}{c} 5 \\ 4 \\ 1 \ 2 \ 3 \\ 4 \\ 1 \ 2 \ 3 \\ 4 \\ 5 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Notes

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