M(3,1,2)

M(3,1,2) - [math]kS_3[/math]

Representative:	[math]kS_3[/math]
Defect groups:	[math]C_3[/math]
Inertial quotients:	[math]C_2[/math]
[math]k(B)=[/math]	3
[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 & 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]\mathcal{O} S_3[/math]
Decomposition matrices:	[math]\left( \begin{array}{cc} 1 & 0 \\ 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_2[/math]
[math]PI(B)=[/math]	{{{PIgroup}}}
Source algebras known?	Yes
Source algebra reps:	[math]kS_3[/math]
[math]k[/math]-derived equiv. classes known?	Yes
[math]k[/math]-derived equivalent to:	M(3,1,1)
[math]\mathcal{O}[/math]-derived equiv. classes known?	Yes
[math]p'[/math]-index covering blocks:	M(3,1,1)
[math]p'[/math]-index covered blocks:	M(3,1,1)
Index [math]p[/math] covering blocks:	{{{pcoveringblocks}}}

These are very frequently occuring blocks with cyclic defect groups, so are described in work culminating in [Li96] .

Basic algebra

Quiver: a: <1,2>, b: <2,1>

Relations w.r.t. [math]k[/math]: aba=bab=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].

If [math]b[/math] lies in M(3,1,2), then [math]B[/math] must lie in M(3,1,1) or M(3,1,2). Example needed.

If [math]B[/math] lies in M(3,1,2), then [math]b[/math] must lie in M(3,1,1) or M(3,1,2). For example consider the principal blocks of [math]C_3 \triangleleft S_3[/math].

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}{c} S_2 \\ S_1 \\ S_2 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.