M(9,2,1)

M(9,2,1) - [math]k(C_3 \times C_3)[/math]

Representative:	[math]k(C_3 \times C_3)[/math]
Defect groups:	[math]C_3 \times C_3[/math]
Inertial quotients:	[math]1[/math]
[math]k(B)=[/math]	9
[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} 9 \\ \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_3 \times C_3)[/math]
Decomposition matrices:	[math]\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 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_3 \times C_3):GL_2(3)[/math]
[math]PI(B)=[/math]
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:	M(9,2,2), M(9,2,3) (complete)
[math]p'[/math]-index covered blocks:
Index [math]p[/math] covering blocks:	M(27,2,1), M(27,5,1)

These are nilpotent blocks.

Basic algebra

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

Relations w.r.t. [math]k[/math]: a^3=b^3=ab-ba=0

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}{ccccc} & & 1 & & \\ & 1 & & 1 & \\ 1 & & 1 & & 1 \\ & 1 & & 1 & \\ & & 1 & & \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.