M(4,2,1)

M(4,2,1) - [math]k(C_2 \times C_2)[/math]

Representative:	[math]k(C_2 \times C_2)[/math]
Defect groups:	[math]C_2 \times C_2[/math]
Inertial quotients:	[math]1[/math]
[math]k(B)=[/math]	4
[math]l(B)=[/math]	1
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]	[math](k \times k):GL_2(k)[/math]
Cartan matrix:	[math]\left( \begin{array}{c} 4 \\ \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} (C_2 \times C_2)[/math]
Decomposition matrices:	[math]\left( \begin{array}{c} 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{L}(B)=S_4[/math]
[math]PI(B)=[/math]
Source algebras known?	Yes
Source algebra reps:	[math]k(C_2 \times C_2)[/math]
[math]k[/math]-derived equiv. classes known?	Yes
[math]k[/math]-derived equivalent to:	Forms a derived equivalence class
[math]\mathcal{O}[/math]-derived equiv. classes known?	Yes
[math]p'[/math]-index covering blocks:	M(4,2,3) (complete)
[math]p'[/math]-index covered blocks:	M(4,2,3) (complete)
Index [math]p[/math] covering blocks:	M(8,2,1), M(8,5,1)

These are nilpotent blocks.

Basic algebra

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

Relations w.r.t. [math]k[/math]: a^2=b^2=ab+ba=0

Other notatable representatives

Block number 4 of [math]k PGL_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(4,2,1), 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].

If [math]B[/math] lies in M(4,2,1), 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].

Projective indecomposable modules

Labelling the unique simple [math]B[/math]-module by [math]S_1[/math], the unique projective indecomposable module has Loewy structure as follows:

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

Irreducible characters

All irreducible characters have height zero.