Charles Eaton: /* Other notatable representatives */

2018-09-08T14:40:16Z

‎Other notatable representatives

Charles Eaton: Created page with "{{blockbox |title = M(3^n,1,3) - $B_0(kPSL_2(2^{(3^{n-1})}))$ |image = M(5,1,3)quiver.png |representative = $B_0(kPSL_2(2^{(3^{n-1})}))$ |defect = C(..."

2018-09-08T14:37:02Z

Created page with "{{blockbox |title = M(3^n,1,3) - <math>B_0(kPSL_2(2^{(3^{n-1})}))</math> |image = M(5,1,3)quiver.png |representative = <math>B_0(kPSL_2(2^{(3^{n-1})}))</math> |defect = C(..."

New page

{{blockbox
|title = M(3^n,1,3) - <math>B_0(kPSL_2(2^{(3^{n-1})}))</math>
|image = M(5,1,3)quiver.png
|representative = <math>B_0(kPSL_2(2^{(3^{n-1})}))</math>
|defect = [[C(3^n)|<math>C_{3^n}</math>]]
|inertialquotients = <math>C_2</math>
|k(B) = <math>\frac{3^n+3}{2}</math>
|l(B) = 2
|k-morita-frob = 1
|Pic-k=
|cartan = <math>\left( \begin{array}{cc}
2 & 1 \\
1 & \frac{3^n+1}{2} \\
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>B_0(\mathcal{O} PSL_2(2^{(3^{n-1})}))</math>
|decomp = <math>\left( \begin{array}{cc}
1 & 0 \\
0 & 1 \\
0 & 1 \\
\vdots & \vdots \\
0 & 1 \\
1 & 1 \\
\end{array}\right)</math>
|O-morita-frob = 1
|Pic-O = |source? = Yes
|sourcereps =
|k-derived-known? = Yes
|k-derived = [[M(3^n,1,2)]]
|O-derived-known? = Yes
}}

== 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 ==

== Covering blocks and covered blocks ==



== 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.

[[C(3^n)|Back to <math>C_{3^n}</math>]]

← Older revision		Revision as of 14:40, 8 September 2018
Line 40:		Line 40:

	== Other notatable representatives ==		== 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>.

	== Covering blocks and covered blocks ==		== Covering blocks and covered blocks ==

M(3^n,1,3) - Revision history

Charles Eaton: /* Other notatable representatives */

Charles Eaton: Created page with "{{blockbox |title = M(3^n,1,3) - B_0(kPSL_2(2^{(3^{n-1})})) |image = M(5,1,3)quiver.png |representative = B_0(kPSL_2(2^{(3^{n-1})})) |defect = C(..."

Charles Eaton: Created page with "{{blockbox |title = M(3^n,1,3) - $B_0(kPSL_2(2^{(3^{n-1})}))$ |image = M(5,1,3)quiver.png |representative = $B_0(kPSL_2(2^{(3^{n-1})}))$ |defect = C(..."