M(9,2,3)

M(9,2,3) - [math]k((C_3 \times C_3):C_2)[/math]

Representative:	[math]k((C_3 \times C_3):C_2)[/math]
Defect groups:	[math]C_3 \times C_3[/math]
Inertial quotients:	[math]C_2[/math]
[math]k(B)=[/math]	9
[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} 5 & 4 \\ 4 & 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_3 \times C_3):C_2)[/math]
Decomposition matrices:	[math]\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 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]C_2[/math]
[math]PI(B)=[/math]
Source algebras known?	No
Source algebra reps:
[math]k[/math]-derived equiv. classes known?	No
[math]k[/math]-derived equivalent to:	none known
[math]\mathcal{O}[/math]-derived equiv. classes known?	No
[math]p'[/math]-index covering blocks:
[math]p'[/math]-index covered blocks:
Index [math]p[/math] covering blocks:

The representative [math]k((C_3 \times C_3):C_2)[/math] is given by [math]C_3 \times C_3[/math] acted on by an element inverting those of [math]C_3 \times C_3[/math], i.e., it is the group SmallGroup(18,4).

Basic algebra

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

Relations w.r.t. [math]k[/math]: [math]ad=bc[/math], [math]cb=da[/math], [math]aca=bdb=0[/math], [math]cac=dbd=0[/math]

Projective indecomposable modules

Labelling the unique simple [math]B[/math]-module by [math]1,2[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{cc} \begin{array}{ccccc} & & 1 & & \\ & 2 & & 2 & \\ 1 & & 1 & & 1 \\ & 2 & & 2 & \\ & & 1 & & \\ \end{array}, & \begin{array}{ccccc} & & 2 & & \\ & 1 & & 1 & \\ 2 & & 2 & & 2 \\ & 1 & & 1 & \\ & & 2 & & \\ \end{array} \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_3 \times C_3[/math]