http://wiki.manchester.ac.uk/blocks/index.php?action=history&feed=atom&title=M%2832%2C2%2C2%29M(32,2,2) - Revision history2026-09-14T20:06:32ZRevision history for this page on the wikiMediaWiki 1.30.1http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,2,2)&diff=1258&oldid=prevCharles Eaton: Created page with "<!-- Draft entry. Blank fields indicate data not verified here, not a claim that the corresponding question is open. The decomposition matrix, quiver relations and L..."2026-09-14T12:35:57Z<p>Created page with "<!-- Draft entry. Blank fields indicate data not verified here, not a claim that the corresponding question is open. The decomposition matrix, quiver relations and L..."</p>
<p><b>New page</b></p><div><!-- Draft entry. Blank fields indicate data not verified here, not a claim<br />
that the corresponding question is open. The decomposition matrix,<br />
quiver relations and Loewy layers are direct calculations for the<br />
representative, rather than quotations from [EKS12]. --><br />
<br />
{{blockbox<br />
|title = M(32,2,2) - <math>k(\mathrm{MNA}(2,2):C_3)</math><br />
|image = M(4,2,3)quiver.png<br />
|representative = <math>k(\mathrm{MNA}(2,2):C_3)</math><br />
|defect = [[MNA(2,2)|<math>\mathrm{MNA}(2,2)</math>]]<br />
|inertialquotients = <math>C_3</math><br />
|k(B) = 12<br />
|l(B) = 3<br />
|k-morita-frob = 1<br />
|Pic-k = <!-- Not verified. --><br />
|cartan = <math>\left( \begin{array}{ccc}<br />
12 & 10 & 10 \\<br />
10 & 12 & 10 \\<br />
10 & 10 & 12 \\<br />
\end{array} \right)</math><br />
|defect-morita-inv? = <!-- Not verified as an unrestricted Morita invariant. --><br />
|inertial-morita-inv? = <!-- Not verified as an unrestricted Morita invariant. --><br />
|O-morita? = Yes<br />
|O-morita = <math>\mathcal{O}(\mathrm{MNA}(2,2):C_3)</math><ref name="EKS12">Theorem 1 of [[References#E|[EKS12]]], with <math>r=2</math>.</ref><br />
|decomp = <math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 \\<br />
0 & 1 & 0 \\<br />
0 & 0 & 1 \\<br />
1 & 1 & 1 \\<br />
1 & 1 & 1 \\<br />
1 & 1 & 1 \\<br />
1 & 1 & 1 \\<br />
1 & 1 & 1 \\<br />
0 & 1 & 1 \\<br />
1 & 0 & 1 \\<br />
1 & 1 & 0 \\<br />
2 & 2 & 2 \\<br />
\end{array} \right)</math><br />
|O-morita-frob = 1<br />
|Pic-O = <!-- Not verified. --><br />
|PIgroup = <!-- Not verified. --><br />
|source? = <!-- Classification of source algebras not verified. --><br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = Forms its own derived equivalence class among blocks with defect group <math>\mathrm{MNA}(2,2)</math><br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks = [[M(32,2,1)]]<br />
|pcoveringblocks =<br />
}}<br />
<br />
Page adapted from output generated by ChatGPT 6.<br />
<br />
Here the defect group is<br />
<br />
<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><br />
<br />
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<br />
<br />
<math>txt^{-1}=y,\qquad tyt^{-1}=x^{-1}y^{-1}.</math><br />
<br />
<math>kG</math> is a single block and is already basic.<br />
<br />
<br />
<br />
== Basic algebra ==<br />
<br />
<br />
<br />
'''Quiver:''' a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f:<1,3><br />
<br />
Paths are composed from left to right.<br />
<br />
'''Relations w.r.t. <math>k</math>:'''<br />
<br />
<math>\begin{gathered}<br />
abca=bcab=cabc=0, \\<br />
dfed=fedf=edfe=0, \\<br />
abe=fca,\qquad bcf=dab,\qquad cad=ebc, \\<br />
adf=feb,\qquad bed=dfc,\qquad cfe=eda, \\<br />
adad=fcfc,\qquad bebe=dada,\qquad cfcf=ebeb.<br />
\end{gathered}</math><br />
<br />
Equivalently, put <math>U=a+b+c</math> and <math>V=d+e+f</math>. The relations are<br />
<br />
<math>U^4=V^4=0,\qquad U^2V=VU^2,\qquad UV^2=V^2U,\qquad (UV)^2=(VU)^2.</math><br />
<br />
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<br />
<br />
<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><br />
<br />
<br />
== Other notable representatives ==<br />
<br />
== Projective indecomposable modules ==<br />
<br />
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:<br />
<br />
<math>\begin{array}{ccc}<br />
\begin{array}{c}<br />
1 \\<br />
2\ 3 \\<br />
1\ 1\ 2\ 3 \\<br />
1\ 1\ 2\ 2\ 3\ 3 \\<br />
1\ 1\ 2\ 2\ 3\ 3 \\<br />
1\ 1\ 2\ 2\ 3\ 3 \\<br />
1\ 1\ 2\ 3 \\<br />
2\ 3 \\<br />
1<br />
\end{array},<br />
&<br />
\begin{array}{c}<br />
2 \\<br />
1\ 3 \\<br />
1\ 2\ 2\ 3 \\<br />
1\ 1\ 2\ 2\ 3\ 3 \\<br />
1\ 1\ 2\ 2\ 3\ 3 \\<br />
1\ 1\ 2\ 2\ 3\ 3 \\<br />
1\ 2\ 2\ 3 \\<br />
1\ 3 \\<br />
2<br />
\end{array},<br />
&<br />
\begin{array}{c}<br />
3 \\<br />
1\ 2 \\<br />
1\ 2\ 3\ 3 \\<br />
1\ 1\ 2\ 2\ 3\ 3 \\<br />
1\ 1\ 2\ 2\ 3\ 3 \\<br />
1\ 1\ 2\ 2\ 3\ 3 \\<br />
1\ 2\ 3\ 3 \\<br />
1\ 2 \\<br />
3<br />
\end{array}<br />
\end{array}</math><br />
<br />
<br />
<br />
== Irreducible characters ==<br />
<br />
There are eight irreducible ordinary characters of height zero and four of height one:<br />
<br />
<math>k_0(B)=8,\qquad k_1(B)=4,\qquad k_i(B)=0\quad(i\geq 2).</math><ref name="EKS12" /><br />
<br />
In the displayed decomposition matrix, the first eight rows correspond to height-zero characters and the last four to height-one characters.<br />
<br />
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>.<br />
<br />
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>.<br />
<br />
[[MNA(2,2)|Back to <math>\mathrm{MNA}(2,2)</math>]]<br />
<br />
== Notes ==<br />
<references /><br />
<br />
[[Category: Morita equivalence classes|32,2,2]]<br />
[[Category: Blocks with defect group MNA(2,2)]]</div>Charles Eaton