M(5,1,3)
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}}} |
Contents
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.