M(4,2,3)

M(4,2,3) - [math]kA_4[/math]

Representative:	[math]kA_4[/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^* \times k^* \times C_3):C_2[/math]
Cartan matrix:	[math]\left( \begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 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}A_4[/math]
Decomposition matrices:	[math]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 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)=S_3[/math]
[math]PI(B)=[/math]	{{{PIgroup}}}
Source algebras known?	Yes
Source algebra reps:	[math]\mathcal{O}A_4[/math]
[math]k[/math]-derived equiv. classes known?	Yes
[math]k[/math]-derived equivalent to:	M(4,2,2)
[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: <2,3> , c: <3,1>, d: <2,1>, e: <3,2>, f: <1,3>

Relations w.r.t. [math]k[/math]: ab=bc=ca=0, df=fe=ed=0, ad=fc, be=da, cf=eb

Other notatable representatives

Block number 2 of [math]k PSL_3(7)[/math] in the labelling used in [1]

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(4,2,3), then [math]B[/math] must lie in M(4,2,1) or M(4,2,3). For example consider blocks of [math]PSL_3(7) \triangleleft PGL_3(7)[/math].

If [math]B[/math] lies in M(4,2,3), then [math]b[/math] must lie in M(4,2,1) or M(4,2,3). For example consider the principal blocks of [math]O_2(A_4) \triangleleft A_4[/math].

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

Irreducible characters

All irreducible characters have height zero.