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M(5,1,3) - [math]B_0(kA_5)[/math]
Representative: [math]B_0(kA_5)[/math]
Defect groups: [math]C_5[/math]
Inertial quotients: [math]C_2[/math]
[math]k(B)=[/math] 4
[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} 2 & 1 \\ 1 & 3 \\ \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}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 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_4[/math]
[math]PI(B)=[/math] {{{PIgroup}}}
Source algebras known? Yes
Source algebra reps:
[math]k[/math]-derived equiv. classes known? Yes
[math]k[/math]-derived equivalent to: M(5,1,2)
[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:<2,1>, c:<2,2>

Relations w.r.t. [math]k[/math]: ac=cb=ba-c^2=0

Other notatable representatives

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].

If [math]b[/math] lies in M(5,1,3), then [math]B[/math] must lie in M(5,1,3) or M(5,1,5). Examples needed.

Projective indecomposable modules

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

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

Irreducible characters

All irreducible characters have height zero.