Difference between revisions of "Notation"

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<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.
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<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>.
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{|  
 
{|  
 
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|<math>k(B)</math> || Number of irreducible characters in <math>B</math> ||
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|<math>k(B)</math> || Number of irreducible characters in <math>B</math>, equal to <math>\dim_k(Z(kB))</math> ||
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|-
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|<math>k_i(B)</math> || Number of irreducible characters in <math>B</math> of height <math>i</math> ||
 
|-
 
|-
 
|<math>l(B)</math> || Number of isomorphism classes of simple <math>B</math>-modules ||
 
|<math>l(B)</math> || Number of isomorphism classes of simple <math>B</math>-modules ||
 
|-
 
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| <math>{\rm mf_k(B)}</math> || The Morita-Frobenius number of <math>B</math>
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| <math>{\rm mf_k(B)}</math> || The Morita-Frobenius number of <math>kB</math> || [[References|[Ke04] ]]
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|-
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|<math>{\rm mf_\mathcal{O}(B)}</math> || The <math>\mathcal{O}</math>-Morita Frobenius number ||
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|-
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|<math>{\rm Pic}_\mathcal{O}(B)</math> || The Picard group of <math>B</math> ||
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|-
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|<math>{\rm Pic}_k(B)</math> || The Picard group of <math>kB</math> ||
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|-
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|<math>\mathcal{T}(B)</math> || The subgroup of <math>{\rm Pic}_\mathcal{O}(B)</math> consisting of trivial source bimodules || [[References|[BKL18] ]]
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|-
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|<math>\mathcal{L}(B)</math> || The subgroup of <math>{\rm Pic}_\mathcal{O}(B)</math> consisting of linear source bimodules || [[References|[BKL18] ]]
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|-
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|<math>\mathcal{E}(B)</math> || The subgroup of <math>{\rm Pic}_\mathcal{O}(B)</math> consisting of endopermutation source bimodules || [[References|[BKL18] ]]
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|-
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|<math>M(x,y,z)</math> || A <math>k</math>-Morita equivalence class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class.
 
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Latest revision as of 17:45, 9 December 2019

[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].

[math]k(B)[/math] Number of irreducible characters in [math]B[/math], equal to [math]\dim_k(Z(kB))[/math]
[math]k_i(B)[/math] Number of irreducible characters in [math]B[/math] of height [math]i[/math]
[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_\mathcal{O}(B)}[/math] The [math]\mathcal{O}[/math]-Morita Frobenius number
[math]{\rm Pic}_\mathcal{O}(B)[/math] The Picard group of [math]B[/math]
[math]{\rm Pic}_k(B)[/math] The Picard group of [math]kB[/math]
[math]\mathcal{T}(B)[/math] The subgroup of [math]{\rm Pic}_\mathcal{O}(B)[/math] consisting of trivial source bimodules [BKL18]
[math]\mathcal{L}(B)[/math] The subgroup of [math]{\rm Pic}_\mathcal{O}(B)[/math] consisting of linear source bimodules [BKL18]
[math]\mathcal{E}(B)[/math] The subgroup of [math]{\rm Pic}_\mathcal{O}(B)[/math] consisting of endopermutation source bimodules [BKL18]
[math]M(x,y,z)[/math] A [math]k[/math]-Morita equivalence class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class.