Difference between revisions of "M(8,5,7)"
(Created page with "{{blockbox |title = M(8,5,7) - <math>B_0(kJ_1)</math> |image = M(8,5,7)quiver.png |representative = <math>B_0(kJ_1)</math> |defect = C2xC2xC2|<math>C_2 \times C_2 \times C...") |
(Added PIMs) |
||
Line 80: | Line 80: | ||
<math>a_5b_6c_1=b_5c_7a_4=c_5a_2b_3</math> | <math>a_5b_6c_1=b_5c_7a_4=c_5a_2b_3</math> | ||
<math>a_6b_7c_2=b_6c_1a_5=c_6a_3b_4</math> | <math>a_6b_7c_2=b_6c_1a_5=c_6a_3b_4</math> | ||
− | <math>a_7b_1c_3=b_7c_2a_6=c_7a_4b_5</math> | + | <math>a_7b_1c_3=b_7c_2a_6=c_7a_4b_5</math> --> |
== Other notatable representatives == | == Other notatable representatives == | ||
Line 86: | Line 86: | ||
== Projective indecomposable modules == | == Projective indecomposable modules == | ||
− | Labelling the simple <math>B</math>-modules by <math> | + | Labelling the simple <math>B</math>-modules by <math>1,2,3,4,5</math>, the projective indecomposable modules have Loewy structure as follows: |
− | <math>\begin{array}{ | + | <math>\begin{array}{ccccc} |
\begin{array}{c} | \begin{array}{c} | ||
− | + | 1 \\ | |
− | + | 3 \ 4 \ 5 \\ | |
− | + | 1 \ 2 \ 1 \ 2 \ 1 \\ | |
− | + | 3 \ 3 \ 4 \ 4 \ 5 \ 5 \\ | |
+ | 1 \ 2 \ 1 \ 2 \ 1 \\ | ||
+ | 3 \ 4 \ 5 \\ | ||
+ | 1 \\ | ||
\end{array}, | \end{array}, | ||
& | & | ||
− | \begin{array}{ | + | \begin{array}{c} |
− | + | 2 \\ | |
− | + | 3 \ 4 \\ | |
− | + | 1 \ 2 \ 1 \\ | |
− | + | 3 \ 4 \ 5 \\ | |
+ | 1 \ 2 \ 1 \\ | ||
+ | 3 \ 4 \\ | ||
+ | 2 \\ | ||
\end{array}, | \end{array}, | ||
& | & | ||
− | \begin{array}{ | + | \begin{array}{c} |
− | + | 3 \\ | |
− | + | 1 \ 2 \\ | |
− | + | 3 \ 4 \ 5 \\ | |
− | + | 1 \ 2 \ 1 \\ | |
+ | 3 \ 4 \ 5 \\ | ||
+ | 1 \ 2 \\ | ||
+ | 3 \\ | ||
\end{array} | \end{array} | ||
, | , | ||
& | & | ||
− | \begin{array}{ | + | \begin{array}{c} |
− | + | 4 \\ | |
− | + | 1 \ 2 \\ | |
− | + | 3 \ 4 \ 5 \\ | |
− | + | 1 \ 2 \ 1 \\ | |
− | + | 3 \ 4 \ 5 \\ | |
− | + | 1 \ 2 \\ | |
− | + | 4 \\ | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
\end{array}, | \end{array}, | ||
& | & | ||
\begin{array}{ccc} | \begin{array}{ccc} | ||
− | + | 5 \\ | |
− | + | 1 \ 5 \\ | |
− | + | 3 \ 4 \\ | |
− | + | 1 \ 2 \ 1 \\ | |
− | \end{array} | + | 3 \ 4 \ 5 \\ |
+ | 1 \\ | ||
+ | 5 \\ | ||
+ | \end{array} | ||
\end{array} | \end{array} | ||
− | </math | + | </math> |
== Irreducible characters == | == Irreducible characters == |
Revision as of 17:38, 24 September 2018
Representative: | [math]B_0(kJ_1)[/math] |
---|---|
Defect groups: | [math]C_2 \times C_2 \times C_2[/math] |
Inertial quotients: | [math]C_7:C_3[/math] |
[math]k(B)=[/math] | 8 |
[math]l(B)=[/math] | 5 |
[math]{\rm mf}_k(B)=[/math] | 1 |
[math]{\rm Pic}_k(B)=[/math] | |
Cartan matrix: | [math]\left( \begin{array}{ccccc} 8 & 4 & 4 & 4 & 4 \\ 4 & 4 & 3 & 3 & 1 \\ 4 & 3 & 4 & 2 & 2 \\ 4 & 3 & 2 & 4 & 2 \\ 4 & 1 & 2 & 2 & 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]B_0(\mathcal{O}J_1)[/math] |
Decomposition matrices: | [math]\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 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? | No |
Source algebra reps: | |
[math]k[/math]-derived equiv. classes known? | Yes |
[math]k[/math]-derived equivalent to: | M(8,5,6), M(8,5,8) |
[math]\mathcal{O}[/math]-derived equiv. classes known? | Yes |
[math]p'[/math]-index covering blocks: | |
[math]p'[/math]-index covered blocks: | |
Index [math]p[/math] covering blocks: | {{{pcoveringblocks}}} |
Contents
Basic algebra
Other notatable representatives
Projective indecomposable modules
Labelling the simple [math]B[/math]-modules by [math]1,2,3,4,5[/math], the projective indecomposable modules have Loewy structure as follows:
[math]\begin{array}{ccccc} \begin{array}{c} 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \ 2 \ 1 \\ 3 \ 3 \ 4 \ 4 \ 5 \ 5 \\ 1 \ 2 \ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 3 \ 4 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \\ 2 \\ \end{array}, & \begin{array}{c} 3 \\ 1 \ 2 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \\ 3 \\ \end{array} , & \begin{array}{c} 4 \\ 1 \ 2 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \\ 4 \\ \end{array}, & \begin{array}{ccc} 5 \\ 1 \ 5 \\ 3 \ 4 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \\ 5 \\ \end{array} \end{array} [/math]
Irreducible characters
All irreducible characters have height zero.