Notation - Revision history

CesareGArdito: fixed a typo

2019-12-09T17:45:39Z

fixed a typo

Charles Eaton at 09:52, 24 September 2019

2019-09-24T09:52:52Z

Charles Eaton at 08:46, 1 October 2018

2018-10-01T08:46:39Z

Charles Eaton at 13:51, 30 August 2018

2018-08-30T13:51:43Z

Charles Eaton at 18:26, 23 August 2018

2018-08-23T18:26:16Z

Charles Eaton at 17:14, 23 August 2018

2018-08-23T17:14:39Z

Charles Eaton: Created page with " $(K,\mathcal{O},k)$ is a $p$ -modular system, where $\mathcal{O}$ is a complete discrete valuation ring with algebraically closed residue field..."

2018-08-23T17:06:23Z

Created page with "<math>(K,\mathcal{O},k)</math> is a <math>p</math>-modular system, where <math>\mathcal{O}</math> is a complete discrete valuation ring with algebraically closed residue field..."

New page

<math>(K,\mathcal{O},k)</math> is a <math>p</math>-modular system, where <math>\mathcal{O}</math> is a complete discrete valuation ring with algebraically closed residue field <math>k=\mathcal{O}/J(\mathcal{O})</math> and <math>K</math> is the field of fractions of <math>\mathcal{O}</math>, of characteristic zero. In order to make a consistent choice of <math>(K,\mathcal{O},k)</math> we take <math>k</math> to be the algebraic closure of the field with <math>p</math> elements and <math>\mathcal{O}</math> to be the ring of Witt vectors for <math>k</math> This has the disadvantage that for <math>G</math> a finite group <math>KG</math> need not contain the primitive character idempotents, but this condition can usually be avoided.

In the below, <math>G</math> is a finite group and <math>B</math> is a block of <math>kG</math> or <math>\mathcal{O}G</math> depending on the context.

{|
|-
|<math>k(B)</math> || Number of irreducible characters in <math>B</math> ||
|-
|<math>l(B)</math> || Number of isomorphism classes of simple <math>B</math>-modules ||
|}

← Older revision		Revision as of 17:45, 9 December 2019
Line 1:		Line 1:
−	<math>(K,\mathcal{O},k)</math> is a <math>p</math>-modular system, where <math>\mathcal{O}</math> is a complete discrete valuation ring with algebraically closed residue field <math>k=\mathcal{O}/J(\mathcal{O})</math> and <math>K</math> is the field of fractions of <math>\mathcal{O}</math>, of characteristic zero. When we need to make a consistent choice of <math>(K,\mathcal{O},k)</math> we take <math>k</math> to be the algebraic closure of the field with <math>p</math> elements and <math>\mathcal{O}</math> to be the ring of Witt vectors for <math>k</math> This has the disadvantage that for <math>G</math> a finite group <math>KG</math> need not contain the primitive character idempotents, but this condition can usually be avoided. In general however the choice of <math>\mathcal{O}</math> is not consistent across the literature and some care has to be taken.	+	<math>(K,\mathcal{O},k)</math> is a <math>p</math>-modular system, where <math>\mathcal{O}</math> is a complete discrete valuation ring with algebraically closed residue field <math>k=\mathcal{O}/J(\mathcal{O})</math> and <math>K</math> is the field of fractions of <math>\mathcal{O}</math>, of characteristic zero. When we need to make a consistent choice of <math>(K,\mathcal{O},k)</math> we take <math>k</math> to be the algebraic closure of the field with <math>p</math> elements and <math>\mathcal{O}</math> to be the ring of Witt vectors for <math>k</math>. This has the disadvantage that for <math>G</math> a finite group <math>KG</math> need not contain the primitive character idempotents, but this condition can usually be avoided. In general however the choice of <math>\mathcal{O}</math> is not consistent across the literature and some care has to be taken.

	In the below, <math>G</math> is a finite group and <math>B</math> is a block of <math>\mathcal{O}G</math>. If it is clear from context, <math>B</math> may also mean the corresponding block of <math>kG</math>. When it is not otherwise clear from context <math>kB</math> will refer to the block of <math>kG</math>.		In the below, <math>G</math> is a finite group and <math>B</math> is a block of <math>\mathcal{O}G</math>. If it is clear from context, <math>B</math> may also mean the corresponding block of <math>kG</math>. When it is not otherwise clear from context <math>kB</math> will refer to the block of <math>kG</math>.

← Older revision		Revision as of 09:52, 24 September 2019
Line 1:		Line 1:
−	<math>(K,\mathcal{O},k)</math> is a <math>p</math>-modular system, where <math>\mathcal{O}</math> is a complete discrete valuation ring with algebraically closed residue field <math>k=\mathcal{O}/J(\mathcal{O})</math> and <math>K</math> is the field of fractions of <math>\mathcal{O}</math>, of characteristic zero. ~~In order~~ to make a consistent choice of <math>(K,\mathcal{O},k)</math> we take <math>k</math> to be the algebraic closure of the field with <math>p</math> elements and <math>\mathcal{O}</math> to be the ring of Witt vectors for <math>k</math> This has the disadvantage that for <math>G</math> a finite group <math>KG</math> need not contain the primitive character idempotents, but this condition can usually be avoided.	+	<math>(K,\mathcal{O},k)</math> is a <math>p</math>-modular system, where <math>\mathcal{O}</math> is a complete discrete valuation ring with algebraically closed residue field <math>k=\mathcal{O}/J(\mathcal{O})</math> and <math>K</math> is the field of fractions of <math>\mathcal{O}</math>, of characteristic zero. When we need to make a consistent choice of <math>(K,\mathcal{O},k)</math> we take <math>k</math> to be the algebraic closure of the field with <math>p</math> elements and <math>\mathcal{O}</math> to be the ring of Witt vectors for <math>k</math> This has the disadvantage that for <math>G</math> a finite group <math>KG</math> need not contain the primitive character idempotents, but this condition can usually be avoided. In general however the choice of <math>\mathcal{O}</math> is not consistent across the literature and some care has to be taken.

	In the below, <math>G</math> is a finite group and <math>B</math> is a block of <math>\mathcal{O}G</math>. If it is clear from context, <math>B</math> may also mean the corresponding block of <math>kG</math>. When it is not otherwise clear from context <math>kB</math> will refer to the block of <math>kG</math>.		In the below, <math>G</math> is a finite group and <math>B</math> is a block of <math>\mathcal{O}G</math>. If it is clear from context, <math>B</math> may also mean the corresponding block of <math>kG</math>. When it is not otherwise clear from context <math>kB</math> will refer to the block of <math>kG</math>.

@@ Line 9: / Line 9: @@
 |<math>l(B)</math> || Number of isomorphism classes of simple <math>B</math>-modules ||
 |-
-| <math>{\rm mf_k(B)}</math> || The Morita-Frobenius number of <math>kB</math> || [Ke04]
+| <math>{\rm mf_k(B)}</math> || The Morita-Frobenius number of <math>kB</math> || [[References|[Ke04] ]]
 |-
 |<math>{\rm mf_\mathcal{O}(B)}</math> || The <math>\mathcal{O}</math>-Morita Frobenius number ||
@@ Line 16: / Line 16: @@
 |-
 |<math>{\rm Pic}_k(B)</math> || The Picard group of <math>kB</math> ||
 |}

@@ Line 9: / Line 9: @@
 |<math>l(B)</math> || Number of isomorphism classes of simple <math>B</math>-modules ||
 |-
-| <math>{\rm mf_k(B)}</math> || The Morita-Frobenius number of <math>B</math>
+| <math>{\rm mf_k(B)}</math> || The Morita-Frobenius number of <math>kB</math> || [Ke04]
 |}

@@ Line 5: / Line 5: @@
 {|
 |-
-|<math>k(B)</math> || Number of irreducible characters in <math>B</math> ||
+|<math>k(B)</math> || Number of irreducible characters in <math>B</math>, equal to <math>\dim_k(Z(kB))</math> ||
 |-
 |<math>l(B)</math> || Number of isomorphism classes of simple <math>B</math>-modules ||

@@ Line 1: / Line 1: @@
 <math>(K,\mathcal{O},k)</math> is a <math>p</math>-modular system, where <math>\mathcal{O}</math> is a complete discrete valuation ring with algebraically closed residue field <math>k=\mathcal{O}/J(\mathcal{O})</math> and <math>K</math> is the field of fractions of <math>\mathcal{O}</math>, of characteristic zero. In order to make a consistent choice of <math>(K,\mathcal{O},k)</math> we take <math>k</math> to be the algebraic closure of the field with <math>p</math> elements and <math>\mathcal{O}</math> to be the ring of Witt vectors for <math>k</math> This has the disadvantage that for <math>G</math> a finite group <math>KG</math> need not contain the primitive character idempotents, but this condition can usually be avoided.
-In the below, <math>G</math> is a finite group and <math>B</math> is a block of <math>kG</math> or <math>\mathcal{O}G</math> depending on the context.
+In the below, <math>G</math> is a finite group and <math>B</math> is a block of <math>\mathcal{O}G</math>. If it is clear from context, <math>B</math> may also mean the corresponding block of <math>kG</math>. When it is not otherwise clear from context <math>kB</math> will refer to the block of <math>kG</math>.
 {|
@@ Line 8: / Line 8: @@
 |-
 |<math>l(B)</math> || Number of isomorphism classes of simple <math>B</math>-modules ||
 |}