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. | <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> | + | 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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|<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 || | ||
| + | |- | ||
| + | | <math>{\rm mf_k(B)}</math> || The Morita-Frobenius number of <math>B</math> | ||
|} | |} | ||
Revision as of 18:14, 23 August 2018
[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]\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] | |
| [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] |
