M(32,2,2)

M(32,2,2) - [math]k(\mathrm{MNA}(2,2):C_3)[/math]

Representative:	[math]k(\mathrm{MNA}(2,2):C_3)[/math]
Defect groups:	[math]\mathrm{MNA}(2,2)[/math]
Inertial quotients:	[math]C_3[/math]
[math]k(B)=[/math]	12
[math]l(B)=[/math]	3
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{ccc} 12 & 10 & 10 \\ 10 & 12 & 10 \\ 10 & 10 & 12 \\ \end{array} \right)[/math]
Defect group Morita invariant?
Inertial quotient Morita invariant?
[math]\mathcal{O}[/math]-Morita classes known?	Yes
[math]\mathcal{O}[/math]-Morita classes:	[math]\mathcal{O}(\mathrm{MNA}(2,2):C_3)[/math]
Decomposition matrices:	[math]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ 2 & 2 & 2 \\ \end{array} \right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math]	1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math]
[math]PI(B)=[/math]
Source algebras known?
Source algebra reps:
[math]k[/math]-derived equiv. classes known?	Yes
[math]k[/math]-derived equivalent to:	Forms its own derived equivalence class among blocks with defect group [math]\mathrm{MNA}(2,2)[/math]
[math]\mathcal{O}[/math]-derived equiv. classes known?	Yes
[math]p'[/math]-index covering blocks:
[math]p'[/math]-index covered blocks:	M(32,2,1)
Index [math]p[/math] covering blocks:

Page adapted from output generated by ChatGPT 6.

Here the defect group is

[math]D=\mathrm{MNA}(2,2)=\langle x,y\mid x^4=y^4=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1\rangle.[/math]

The representative is [math]kG[/math], where [math]G=D:\langle t\rangle[/math], [math]t^3=1[/math], and the action may be chosen as

[math]txt^{-1}=y,\qquad tyt^{-1}=x^{-1}y^{-1}.[/math]

[math]kG[/math] is a single block and is already basic.

Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f:<1,3>

Paths are composed from left to right.

Relations w.r.t. [math]k[/math]:

[math]\begin{gathered} abca=bcab=cabc=0, \\ dfed=fedf=edfe=0, \\ abe=fca,\qquad bcf=dab,\qquad cad=ebc, \\ adf=feb,\qquad bed=dfc,\qquad cfe=eda, \\ adad=fcfc,\qquad bebe=dada,\qquad cfcf=ebeb. \end{gathered}[/math]

Equivalently, put [math]U=a+b+c[/math] and [math]V=d+e+f[/math]. The relations are

[math]U^4=V^4=0,\qquad U^2V=VU^2,\qquad UV^2=V^2U,\qquad (UV)^2=(VU)^2.[/math]

Writing [math]W=UV+VU[/math], these imply that [math]W[/math] is central and [math]W^2=0[/math]. If [math]e_1,e_2,e_3[/math] are the vertex idempotents, a basis is

[math]\{U^iV^jW^\epsilon e_s\mid 0\leq i,j\leq 3,\ 0\leq\epsilon\leq 1,\ 1\leq s\leq 3\}.[/math]

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]1,2,3[/math], the projective indecomposable modules have the following radical layers, listed from top to socle:

[math]\begin{array}{ccc} \begin{array}{c} 1 \\ 2\ 3 \\ 1\ 1\ 2\ 3 \\ 1\ 1\ 2\ 2\ 3\ 3 \\ 1\ 1\ 2\ 2\ 3\ 3 \\ 1\ 1\ 2\ 2\ 3\ 3 \\ 1\ 1\ 2\ 3 \\ 2\ 3 \\ 1 \end{array}, & \begin{array}{c} 2 \\ 1\ 3 \\ 1\ 2\ 2\ 3 \\ 1\ 1\ 2\ 2\ 3\ 3 \\ 1\ 1\ 2\ 2\ 3\ 3 \\ 1\ 1\ 2\ 2\ 3\ 3 \\ 1\ 2\ 2\ 3 \\ 1\ 3 \\ 2 \end{array}, & \begin{array}{c} 3 \\ 1\ 2 \\ 1\ 2\ 3\ 3 \\ 1\ 1\ 2\ 2\ 3\ 3 \\ 1\ 1\ 2\ 2\ 3\ 3 \\ 1\ 1\ 2\ 2\ 3\ 3 \\ 1\ 2\ 3\ 3 \\ 1\ 2 \\ 3 \end{array} \end{array}[/math]

Irreducible characters

There are eight irreducible ordinary characters of height zero and four of height one:

[math]k_0(B)=8,\qquad k_1(B)=4,\qquad k_i(B)=0\quad(i\geq 2).[/math]

In the displayed decomposition matrix, the first eight rows correspond to height-zero characters and the last four to height-one characters.

For the representative [math]G=D:C_3[/math], the ordinary character degrees are three of degree [math]1[/math], three of degree [math]2[/math], five of degree [math]3[/math], and one of degree [math]6[/math].

The first eight rows of the decomposition matrix come from the characters inflated from [math]G/D'\cong(C_4\times C_4):C_3[/math]. The remaining rows arise from the four degree-two irreducible characters of [math]D[/math]: one is invariant under [math]C_3[/math] and has three extensions to [math]G[/math], while the other three form a single orbit and induce one irreducible character of degree [math]6[/math].

Back to [math]\mathrm{MNA}(2,2)[/math]