M(8,5,2)

M(8,5,2) - [math]B_0(k(A_5 \times C_2))[/math]

Representative:	[math]B_0(k(A_5 \times C_2))[/math]
Defect groups:	[math]C_2 \times C_2 \times C_2[/math]
Inertial quotients:	[math]C_3[/math]
[math]k(B)=[/math]	8
[math]l(B)=[/math]	3
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{ccc} 8 & 4 & 4 \\ 4 & 4 & 2 \\ 4 & 2 & 4 \\ \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} (A_5 \times C_2))[/math]
Decomposition matrices:	[math]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 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:	M(8,5,2)
[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: a:<1,2>, b:<2,3>, c:<3,2>, d:<2,1>, e:<1,1>, f: <2,2>, g:<3,3>

Relations w.r.t. [math]k[/math]: ad=cb=0, bcda=dabc, e^2=f^2=g^2=0, af=ea, bg=fb, cf=gc, de=fd

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, S_2, S_3[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccc} \begin{array}{c} S_1 \\ S_1 S_2 S_3 \\ S_2 S_3 S_1 S_1 \\ S_1 S_1 S_3 S_2 \\ S_3 S_2 S_1 \\ S_1 \\ \end{array}, & \begin{array}{c} S_2 \\ S_2 S_1 \\ S_1 S_3 \\ S_3 S_1 \\ S_1 S_2 \\ S_2 \\ \end{array}, & \begin{array}{c} S_3 \\ S_2 S_1 \\ S_1 S_2 \\ S_3 S_1 \\ S_1 S_3 \\ S_3 \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

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