M(8,3,2)

M(8,3,2) - [math]B_0(kPGL_2(5))[/math]

Representative:	[math]B_0(kPGL_2(5))[/math]
Defect groups:	[math]D_8[/math]
Inertial quotients:	[math]1[/math]
[math]k(B)=[/math]	5
[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} 3 & 4 \\ 4 & 8 \\ \end{array} \right)[/math]
Defect group Morita invariant?	Yes
Inertial quotient Morita invariant?	Yes
[math]\mathcal{O}[/math]-Morita classes known?	No
[math]\mathcal{O}[/math]-Morita classes:
Decomposition matrices:	[math]\left( \begin{array}{c} 0 & 1 \\ 0 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 2 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math]
[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(8,3,3)
[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:	{{{pcoveringblocks}}}

These are tame blocks, and appear in the family [math]D(2 {\cal A})[/math] in Erdmann's classification (see [Er87] ). The classification of [math]\mathcal{O}[/math]-blocks is only known in the nilpotent case. Derived equivalences over [math]k[/math] are established in [Ho97] and [Li94b].

Basic algebra

Quiver: a:<1,2>, b:<2,1>, c:<2,2>

Relations w.r.t. [math]k[/math]: [math]ab=c^2=0[/math], [math](cba)^2=(bac)^2[/math]

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 \\ 2 \\ 2 \\ 1 \\ 2 \\ 2 \\ 1 \\ \end{array}, & \begin{array}{ccc} & 1 & \\ \begin{array}{c} 1 \\ 2 \\ 2 \\ 1 \\ 2 \\ \end{array} & \oplus & \begin{array}{c} 2 \\ 1 \\ 2 \\ 2 \\ 1 \\ \end{array} \\ & 2 & \\ \end{array} \\ \end{array} [/math]

Irreducible characters

[math]k_0(B)=4, k_1(B)=1[/math]