M(8,4,1)

M(8,4,1) - [math]kQ_8[/math]

Representative:	[math]kQ_8[/math]
Defect groups:	[math]Q_8[/math]
Inertial quotients:	[math]1[/math]
[math]k(B)=[/math]	5
[math]l(B)=[/math]	1
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{c} 8 \\ \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}Q_8[/math]
Decomposition matrices:	[math]\left( \begin{array}{c} 2 \\ 1 \\ 1 \\ 1 \\ 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:	Forms a derived equivalence class
[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}}}

These are nilpotent blocks.

Basic algebra

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

Relations w.r.t. [math]k[/math]: [math]a^2=bab[/math], [math]b^2=aba[/math], [math](ab)^2=(ba)^2[/math], [math]ababa=0[/math]

Projective indecomposable modules

Labelling the unique simple [math]B[/math]-module by [math]1[/math], the unique projective indecomposable module has Loewy structure as follows:

[math]\begin{array}{c} 1 \\ 1 \ 1 \\ 1 \ 1 \\ 1 \ 1 \\ 1 \\ \end{array} [/math]

Irreducible characters

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