M(5,1,1)

M(5,1,1) - [math]kC_5[/math]

Representative:	[math]kC_5[/math]
Defect groups:	[math]C_5[/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} 5 \\ \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_5[/math]
Decomposition matrices:	[math]\left( \begin{array}{c} 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]\mathcal{L}(B)=C_5:C_4[/math]
[math]PI(B)=[/math]	[math]\mathcal{L}(B)=(C_5:C_4) \times C_2[/math]
Source algebras known?	Yes
Source algebra reps:	[math]kC_5[/math]
[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:	M(5,1,1), M(5,1,2) (complete)
[math]p'[/math]-index covered blocks:
Index [math]p[/math] covering blocks:

These are nilpotent blocks.

Basic algebra

Quiver: a:<1,1>

Relations w.r.t. [math]k[/math]: a^5=0

Projective indecomposable modules

Labelling the unique simple [math]B[/math]-module by [math]S_1[/math], the unique projective indecomposable module has Loewy structure as follows:

[math]\begin{array}{c} S_1 \\ S_1 \\ S_1 \\ S_1 \\ S_1 \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.