M(3^n,1,3)

M(3^n,1,3) - [math]B_0(kPSL_2(2^{(3^{n-1})}))[/math]

Representative:	[math]B_0(kPSL_2(2^{(3^{n-1})}))[/math]
Defect groups:	[math]C_{3^n}[/math]
Inertial quotients:	[math]C_2[/math]
[math]k(B)=[/math]	[math]\frac{3^n+3}{2}[/math]
[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 & \frac{3^n+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]B_0(\mathcal{O} PSL_2(2^{(3^{n-1})}))[/math]
Decomposition matrices:	[math]\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ \vdots & \vdots \\ 0 & 1 \\ 1 & 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math]	1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/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(3^n,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]: [math]ac=cb=ba-c^{(3^n-1)/2}=0[/math]

Other notatable representatives

[math]B_0(kPSL_2(q_n))[/math] for any [math]q_n[/math] a prime power such that [math](q_n+1)_3=3^n[/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}{ccc} & S_2 & \\ S_1 & & \begin{array}{c} S_2 \\ S_2 \\ \vdots \\ S_2 \\ \end{array} \\ & S_2 & \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_{3^n}[/math]