Difference between revisions of "M(16,14,10)"
(Created page with "{{blockbox |title = M(16,14,9) - <math>B_0(k(A_5 \times A_5))</math> |image = |representative = <math>B_0(k(A_5 \times A_5))</math> |defect = (C2)%5E4|<math>(C_2)^...") |
|||
Line 1: | Line 1: | ||
{{blockbox | {{blockbox | ||
− | |title = M(16,14, | + | |title = M(16,14,10) - <math>B_0(k(A_5 \times A_5))</math> |
|image = | |image = | ||
|representative = <math>B_0(k(A_5 \times A_5))</math> | |representative = <math>B_0(k(A_5 \times A_5))</math> |
Revision as of 13:19, 28 November 2019
Representative: | [math]B_0(k(A_5 \times A_5))[/math] |
---|---|
Defect groups: | [math](C_2)^4[/math] |
Inertial quotients: | [math]C_3 \times C_3[/math] |
[math]k(B)=[/math] | 16 |
[math]l(B)=[/math] | 9 |
[math]{\rm mf}_k(B)=[/math] | 1 |
[math]{\rm Pic}_k(B)=[/math] | |
Cartan matrix: | [math]\left( \begin{array}{ccccccccc} 16 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 \\ 8 & 8 & 4 & 4 & 4 & 4 & 2 & 2 & 4 \\ 8 & 4 & 8 & 4 & 4 & 2 & 4 & 4 & 2 \\ 8 & 4 & 4 & 8 & 4 & 4 & 2 & 4 & 2 \\ 8 & 4 & 4 & 4 & 8 & 2 & 4 & 2 & 4 \\ 4 & 4 & 2 & 4 & 2 & 4 & 1 & 2 & 2 \\ 4 & 2 & 4 & 2 & 4 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 4 & 2 & 2 & 2 & 4 & 1 \\ 4 & 4 & 2 & 2 & 4 & 2 & 2 & 1 & 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} (A_5 \times A_5))[/math] |
Decomposition matrices: | See below |
[math]{\rm mf}_\mathcal{O}(B)=[/math] | 1 |
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | |
[math]PI(B)=[/math] | |
Source algebras known? | No |
Source algebra reps: | |
[math]k[/math]-derived equiv. classes known? | Yes |
[math]k[/math]-derived equivalent to: | M(16,14,8), M(16,14,9) |
[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: |
Contents
Basic algebra
Other notatable representatives
Covering blocks and covered blocks
Let [math]N \triangleleft G[/math] with prime [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] is in M(16,14,9), then [math]B[/math] is also in M(16,14,9).
Projective indecomposable modules
Irreducible characters
All irreducible characters have height zero.
Decomposition matrix
[math]\left( \begin{array}{ccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array}\right)[/math]