M(4,2,2)

M(4,2,2) - [math]B_0(kA_5)[/math]

Representative:	[math]B_0(kA_5)[/math]
Defect groups:	[math]C_2 \times C_2[/math]
Inertial quotients:	[math]C_3[/math]
[math]k(B)=[/math]	4
[math]l(B)=[/math]	3
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]	[math]k^* \wr C_2[/math]
Cartan matrix:	[math]\left( \begin{array}{ccc} 2 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 2 \\ \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}A_5)[/math]
Decomposition matrices:	[math]\left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 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]\mathcal{T}(B)=C_2[/math]
[math]PI(B)=[/math]	{{{PIgroup}}}
Source algebras known?	Yes
Source algebra reps:	[math]B_0(\mathcal{O}A_5)[/math]
[math]k[/math]-derived equiv. classes known?	Yes
[math]k[/math]-derived equivalent to:	M(4,2,3)
[math]\mathcal{O}[/math]-derived equiv. classes known?	Yes
[math]p'[/math]-index covering blocks:	{{{coveringblocks}}}
[math]p'[/math]-index covered blocks:	{{{coveredblocks}}}
Index [math]p[/math] covering blocks:	{{{pcoveringblocks}}}

Basic algebra

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

Relations w.r.t. [math]k[/math]: ad=cb=bcda+dabc=0

Covering blocks and covered blocks

Let [math]N \triangleleft G[/math] with [math]p'[/math]-index and let [math]B[/math] be a block of [math]\mathcal{O} G[/math] covering a block [math]b[/math] of [math]\mathcal{O} N[/math].

[math]B[/math] lies in M(4,2,2) iff [math]b[/math] does.

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, S_2, S_3[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccc} \begin{array}{c} S_1 \\ S_2 \\ S_3 \\ S_2 \\ S_1 \\ \end{array}, & \begin{array}{ccc} & S_2 & \\ S_1 & & S_3 \\ S_2 & & S_2 \\ S_3 & & S_1 \\ & S_2 & \\ \end{array}, & \begin{array}{c} S_3 \\ S_2 \\ S_1 \\ S_2 \\ S_3 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_2 \times C_2[/math]