M(8,4,3)

M(8,4,3) - [math]kSL_2(3)[/math]

Representative:	[math]kSL_2(3)[/math]
Defect groups:	[math]Q_8[/math]
Inertial quotients:	[math]C_3[/math]
[math]k(B)=[/math]	5
[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} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 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]\mathcal{O}SL_2(3)[/math]
Decomposition matrices:	[math]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 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(8,4,2)
[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:	{{{pcoveringblocks}}}

These are tame blocks, and appear in the family [math]D(3 {\cal K})[/math] in Erdmann's classification (see [Er88a], [Er88b]). The class lifts to a unique [math]\mathcal{O}[/math]-Morita equivalence class by [HKL07]. A derived equivalence with M(8,4,2) over [math]k[/math] was established in [Ho97].

Basic algebra

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

Relations w.r.t. [math]k[/math]: ab=fcf, bc=dad, ca=ebe, fe=ada, df=beb, ed=cfc, dab=0=bed=cfe

Irreducible characters

[math]k_0(B)=4, k_1(B)=1[/math]

Back to [math]Q_8[/math]