C(3^n) - Revision history

Charles Eaton at 22:34, 2 January 2019

2019-01-02T22:34:16Z

2018-11-22T10:18:10Z

2018-09-09T20:26:11Z

2018-09-09T12:38:57Z

Images added

2018-09-08T14:05:12Z

Added example of q_n

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 These are [[blocks with cyclic defect groups]] and so they are described by [[Brauer trees]].
-For each <math>n>1</math> there are three <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the possible Brauer trees. For <math>n=1</math> there are just two Morita equivalence classes (see [[C3]]).
+For each <math>n>1</math> there are three <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the [[Glossary#Possible Brauer tree|possible Brauer trees]]. For <math>n=1</math> there are just two Morita equivalence classes (see [[C3|<math>C_3</math>]]).
 In the following <math>q_n</math> is a prime power such that <math>(q_n+1)_3=3^n</math>. For example take <math>q_n=2^{3^{n-1}}</math>.

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 == Blocks with defect group <math>C_{3^n}</math> ==
-These are blocks with [[cyclic defect groups]] and so they are described by [[Brauer trees]].
+These are [[blocks with cyclic defect groups]] and so they are described by [[Brauer trees]].
 For each <math>n>1</math> there are three <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the possible Brauer trees. For <math>n=1</math> there are just two Morita equivalence classes (see [[C3]]).

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 For each <math>n>1</math> there are three <math>\mathcal{O}</math>-Morita equivalence classes, accounting for all the possible Brauer trees. For <math>n=1</math> there are just two Morita equivalence classes (see [[C3]]).
-In the below <math>q_n</math> is a prime power such that <math>(q_n+1)_3=3^n</math>. For example take <math>q_n=2^{3^{n-1}}</math>.
+In the following <math>q_n</math> is a prime power such that <math>(q_n+1)_3=3^n</math>. For example take <math>q_n=2^{3^{n-1}}</math>.
 {| class="wikitable"

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 |-
-|[[M(3^n,1,1)|M(<math>3^n</math>,1,1)]] || <math>kC_{3^n}</math> || <math>3^n</math> ||1 ||<math>1</math> ||<math>C_{3^n} : C_{2.3^{n-1}}</math> || ||1 ||1 ||
+|[[M(3^n,1,1)|M(<math>3^n</math>,1,1)]] || <math>kC_{3^n}</math> || <math>3^n</math> ||1 ||<math>1</math> ||<math>C_{3^n} : C_{2.3^{n-1}}</math> || ||1 ||1 || [[Image:M(3^n,1,1)tree.png|45px]]
 |-
-|[[M(3^n,1,2)|M(<math>3^n</math>,1,2)]] || <math>kD_{2.3^n}</math> ||<math>\frac{(3^n+3)}{2}</math> ||2 ||<math>C_2</math> |||| ||1 ||1 ||
+|[[M(3^n,1,2)|M(<math>3^n</math>,1,2)]] || <math>kD_{2.3^n}</math> ||<math>\frac{(3^n+3)}{2}</math> ||2 ||<math>C_2</math> |||| ||1 ||1 || [[Image:M(3^n,1,2)tree.png|45px]]
 |-
-|[[M(3^n,1,3)|M(<math>3^n</math>,1,3)]] || <math>B_0(kPSL_2(q_n))</math> ||<math>\frac{(3^n+3)}{2}</math> ||2 ||<math>C_2</math> || || ||1 ||1 ||
+|[[M(3^n,1,3)|M(<math>3^n</math>,1,3)]] || <math>B_0(kPSL_2(q_n))</math> ||<math>\frac{(3^n+3)}{2}</math> ||2 ||<math>C_2</math> || || ||1 ||1 || [[Image:M(3^n,1,3)tree.png|45px]]
 |}
 Blocks in [[M(3^n,1,2)|M(<math>3^n</math>,1,2)]] are derived equivalent (over <math>\mathcal{O}</math>) to those in [[M(3^n,1,3)|M(<math>3^n</math>,1,3)]].