M(8,4,2)

M(8,4,2) - [math]B_0(kSL_2(5))[/math]

Representative:	[math]B_0(kSL_2(5))[/math]
Defect groups:	[math]Q_8[/math]
Inertial quotients:	[math]C_3[/math]
[math]k(B)=[/math]	7
[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 & 4 & 2 \\ 4 & 8 & 4 \\ 2 & 4 & 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]B_0(\mathcal{O}SL_2(5))[/math]
Decomposition matrices:	[math]\left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ 1 & 2 & 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(8,4,3)
[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:

These are tame blocks, and appear in the family [math]D(3 {\cal A})_2[/math] in Erdmann's classification (see [Er88a], [Er88b]). The class lifts to a unique [math]\mathcal{O}[/math]-Morita equivalence class by [Ei16]. A derived equivalence with M(8,4,3) over [math]k[/math] was established in [Ho97].

Basic algebra

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

Relations w.r.t. [math]k[/math]: ada=abcdabc, dad=bcdabcd, cbc=cdabcda, bcb=dabcdab, adab=cbcd

Projective indecomposable modules

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

[math]\begin{array}{ccc} \begin{array}{ccc} & 1 & \\ & 2 & \\ \begin{array}{c} 1 \\ \end{array} & \oplus & \begin{array}{c} 3 \\ 2 \\ 1 \\ 2 \\ 3 \\ \end{array} \\ & 2 & \\ & 1 & \\ \end{array}, & \begin{array}{c} 2 \\ 1 \ 3 \\ 2 \ 2 \\ 3 \ 1 \\ 2 \ 2 \\ 1 \ 3 \\ 2 \ 2 \\ 3 \ 1 \\ 2 \\ \end{array}, & \begin{array}{ccc} & 3 & \\ & 2 & \\ \begin{array}{c} 3 \\ \end{array} & \oplus & \begin{array}{c} 1 \\ 2 \\ 3 \\ 2 \\ 1 \\ \end{array} \\ & 2 & \\ & 3 & \\ \end{array} \end{array} [/math]

Irreducible characters

[math]k_0(B)=4, k_1(B)=3[/math]

Back to [math]Q_8[/math]