M(9,2,4)

M(9,2,4) - [math]k(S_3 \times S_3)[/math]

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Representative:	[math]k(S_3 \times S_3)[/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]	4
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{cccc} 4 & 2 & 1 & 2 \\ 2 & 4 & 2 & 1 \\ 1 & 2 & 4 & 2 \\ 2 & 1 & 2 & 4 \\ \end{array} \right)[/math]
Defect group Morita invariant?
Inertial quotient Morita invariant?
[math]\mathcal{O}[/math]-Morita classes known?	Yes
[math]\mathcal{O}[/math]-Morita classes:	[math]\mathcal{O} ((S_3 \times S_3)[/math]
Decomposition matrices:	[math]\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 0 & 0 & 0 1 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 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 \wr 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:

Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,4>, d:<4,1>, e:<1,4>, f:<4,3>, g:<3,2>, h:<2,1>

Relations w.r.t. [math]k[/math]: [math]ab=ef[/math], [math]bc=he[/math], [math]cd=gh[/math], [math]da=fg[/math], [math]aha=ede=0[/math], [math]bgb=hah=0[/math], [math]cfc=gbg=0[/math], [math]ded=fcf=0[/math]

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]1,2,3,4[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{cccc} \begin{array}{ccccc} & & 1 & & \\ & 2 & & 4 & \\ 1 & & 3 & & 1 \\ & 4 & & 2 & \\ & & 1 & & \\ \end{array}, & \begin{array}{ccccc} & & 2 & & \\ & 1 & & 3 & \\ 2 & & 4 & & 2 \\ & 3 & & 1 & \\ & & 2 & & \\ \end{array}, & \begin{array}{ccccc} & & 3 & & \\ & 2 & & 4 & \\ 3 & & 1 & & 3 \\ & 4 & & 2 & \\ & & 3 & & \\ \end{array}, & \begin{array}{ccccc} & & 4 & & \\ & 1 & & 3 & \\ 4 & & 2 & & 4 \\ & 3 & & 1 & \\ & & 4 & & \\ \end{array} \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

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