Glossary - Revision history

Charles Eaton: Resistant p-groups

2020-11-18T22:02:15Z

Resistant p-groups

Charles Eaton: /* Fusion system */

2019-01-04T11:12:31Z

‎Fusion system

Charles Eaton: Height

2019-01-04T10:23:59Z

Height

Charles Eaton: Isotypy

2019-01-02T22:23:08Z

Isotypy

Charles Eaton at 09:49, 19 December 2018

2018-12-19T09:49:51Z

Charles Eaton: MNA(r,s)

2018-11-04T21:07:57Z

MNA(r,s)

Charles Eaton at 12:28, 27 October 2018

2018-10-27T12:28:50Z

Charles Eaton at 12:27, 27 October 2018

2018-10-27T12:27:32Z

Charles Eaton: covering/covered block entries

2018-10-27T12:21:10Z

covering/covered block entries

Charles Eaton: Trivial intersection

2018-10-14T09:08:23Z

Trivial intersection

← Older revision		Revision as of 22:02, 18 November 2020
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	The possibile Brauer trees (for <math>P</math> and <math>e</math> a divisor of <math>p-1</math>) are the Brauer trees whose vertex multiplicities add to <math>e+\frac{\|P\|-1}{e}</math> where the exceptional vertex multiplicity is <math>\frac{\|P\|-1}{e}</math> and non-exceptional vertices are regarded as having multiplicity <math>1</math>.		The possibile Brauer trees (for <math>P</math> and <math>e</math> a divisor of <math>p-1</math>) are the Brauer trees whose vertex multiplicities add to <math>e+\frac{\|P\|-1}{e}</math> where the exceptional vertex multiplicity is <math>\frac{\|P\|-1}{e}</math> and non-exceptional vertices are regarded as having multiplicity <math>1</math>.
		+
		+	=== Resistant p-group ===
		+
		+	A p-group <math>P</math> is resistant if whenever <math>\mathcal{F}</math> is a saturated fusion system on <math>P</math>, we have <math>\mathcal{F}=N_{\mathcal{F}}(P)</math>, or equivalently <math>\mathcal{F}=\mathcal{F}_P(G)</math> for some finite group <math>G</math> with <math>P</math> as a normal Sylow p-subgroup. Resistant p-groups were introduced in [[References#S\|[St02]]] in terms of fusion systems for groups, and for arbitrary saturated fusion systems in [[References#S\|[St06]]].

	=== Source algebra ===		=== Source algebra ===

@@ Line 10: / Line 10: @@
 === Fusion system ===
-Fusion system on a finite <math>p</math>-group. See [[References|[Cr11] or [AKO11]]]. The fusion system given by a finite group <math>G</math> on a Sylow <math>p</math>-subgroup <math>P</math> is written <math>\mathcal{F}_P(G)</math>.
+Fusion system on a finite <math>p</math>-group. See [[References|[Cr11] or [AKO11]]]. The fusion system given by a finite group <math>G</math> on a Sylow <math>p</math>-subgroup <math>P</math> is written <math>\mathcal{F}_P(G)</math>. The Fusion system for a block <math>B</math> (sometimes calle the Brauer category) is defined with respect to a maximal subpair <math>(D,b_D)</math>, and is written <math>\mathcal{F}_{(D,b_D)}(G,B)</math>.
 === Height of an irreducible character ===

← Older revision		Revision as of 10:23, 4 January 2019
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	Fusion system on a finite <math>p</math>-group. See [[References\|[Cr11] or [AKO11]]]. The fusion system given by a finite group <math>G</math> on a Sylow <math>p</math>-subgroup <math>P</math> is written <math>\mathcal{F}_P(G)</math>.		Fusion system on a finite <math>p</math>-group. See [[References\|[Cr11] or [AKO11]]]. The fusion system given by a finite group <math>G</math> on a Sylow <math>p</math>-subgroup <math>P</math> is written <math>\mathcal{F}_P(G)</math>.
		+
		+	=== Height of an irreducible character ===
		+
		+	An irreducible character <math>\chi</math> in a block <math>B</math> of <math>\mathcal{O} G</math> with defect group <math>D</math> has height <math>h</math> if <math>\chi(1)_p=p^h[G:D]_p</math>.

	=== Index p covering blocks ===		=== Index p covering blocks ===

← Older revision		Revision as of 22:23, 2 January 2019
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	Fix a Morita equivalence class <math>M</math>. This lists Morita equivalence classes containing a block <math>B</math> of <math>kG</math> for some finite group <math>G</math> such that <math>B</math> covers a block <math>b</math> in <math>M</math> of <math>kN</math> for some normal subgroup <math>N</math> of <math>G</math> of index <math>p</math>.		Fix a Morita equivalence class <math>M</math>. This lists Morita equivalence classes containing a block <math>B</math> of <math>kG</math> for some finite group <math>G</math> such that <math>B</math> covers a block <math>b</math> in <math>M</math> of <math>kN</math> for some normal subgroup <math>N</math> of <math>G</math> of index <math>p</math>.
		+
		+	=== Isotypy ===
		+
		+	A family of perfect isometries resulting from the existence of a splendid derived equivalence. Introduced by Broué. See [[References#L\|[Li18d,9.5]]].

	=== # lifts / <math>\mathcal{O}</math> ===		=== # lifts / <math>\mathcal{O}</math> ===

← Older revision		Revision as of 09:49, 19 December 2018
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	The possibile Brauer trees (for <math>P</math> and <math>e</math> a divisor of <math>p-1</math>) are the Brauer trees whose vertex multiplicities add to <math>e+\frac{\|P\|-1}{e}</math> where the exceptional vertex multiplicity is <math>\frac{\|P\|-1}{e}</math> and non-exceptional vertices are regarded as having multiplicity <math>1</math>.		The possibile Brauer trees (for <math>P</math> and <math>e</math> a divisor of <math>p-1</math>) are the Brauer trees whose vertex multiplicities add to <math>e+\frac{\|P\|-1}{e}</math> where the exceptional vertex multiplicity is <math>\frac{\|P\|-1}{e}</math> and non-exceptional vertices are regarded as having multiplicity <math>1</math>.
		+
		+	=== Source algebra ===
		+
		+	Introduced by Puig, these are subalgebras of a block <math>B</math> of a finite group (and more generally for a <math>G</math>-algebra) not only Morita equivalent to <math>B</math> but also determining the fusion system. See [[References#L\|[Li18c,5.6.12]]].

	=== Splendid equivalence ===		=== Splendid equivalence ===

← Older revision		Revision as of 21:07, 4 November 2018
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	The number of <math>\mathcal{O}</math>-Morita equivalence classes of blocks reducing to a representative of the given <math>k</math>-class.		The number of <math>\mathcal{O}</math>-Morita equivalence classes of blocks reducing to a representative of the given <math>k</math>-class.
		+
		+	=== MNA(r,s) ===
		+
		+	A class of minimal nonabelian <math>2</math>-groups, that is nonabelian <math>2</math>-groups such that every proper subgroup is abelian. For <math>r \geq s \geq 1</math>
		+
		+	\[ MNA(r,s) = \langle x,y\|x^{2^r}=y^{2^s}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle . \]

	=== p'-index covered blocks ===		=== p'-index covered blocks ===

← Older revision		Revision as of 12:28, 27 October 2018
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−	~~This page will contain an alphabetical glossary of terms used.~~

	=== Basic Morita/stable equivalence ===		=== Basic Morita/stable equivalence ===

@@ Line 13: / Line 13: @@
 Fusion system on a finite <math>p</math>-group. See [[References|[Cr11] or [AKO11]]]. The fusion system given by a finite group <math>G</math> on a Sylow <math>p</math>-subgroup <math>P</math> is written <math>\mathcal{F}_P(G)</math>.
-=== Index p covering blocks (table entry) ===
+=== Index p covering blocks ===
 Fix a Morita equivalence class <math>M</math>. This lists Morita equivalence classes containing a block <math>B</math> of <math>kG</math> for some finite group <math>G</math> such that <math>B</math> covers a block <math>b</math> in <math>M</math> of <math>kN</math> for some normal subgroup <math>N</math> of <math>G</math> of index <math>p</math>.
-=== # lifts / <math>\mathcal{O}</math> (table entry) ===
+=== # lifts / <math>\mathcal{O}</math> ===
 The number of <math>\mathcal{O}</math>-Morita equivalence classes of blocks reducing to a representative of the given <math>k</math>-class.
-=== p'-index covered blocks (table entry) ===
+=== p'-index covered blocks ===
 Fix a Morita equivalence class <math>M</math>. This lists Morita equivalence classes containing a block <math>b</math> of <math>kN</math> for some finite group <math>N</math> such that <math>b</math> is covered by a block <math>B</math> in <math>M</math> of <math>kG</math> for some finite group <math>G</math> containing <math>N</math> as a normal subgroup of prime index different to <math>p</math>.
-=== p'-index covering blocks (table entry) ===
+=== p'-index covering blocks ===
 Fix a Morita equivalence class <math>M</math>. This lists Morita equivalence classes containing a block <math>B</math> of <math>kG</math> for some finite group <math>G</math> such that <math>B</math> covers a block <math>b</math> in <math>M</math> of <math>kN</math> for some normal subgroup <math>N</math> of <math>G</math> of prime index different to <math>p</math>.

← Older revision		Revision as of 09:08, 14 October 2018
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	May apply to Morita equivalences, stable equivalences and derived equivalences. See 9.7 an 9.8 of [[References\|[Li18d]]]. It means roughly equivalences given by (complexes of) trivial source bimodules.		May apply to Morita equivalences, stable equivalences and derived equivalences. See 9.7 an 9.8 of [[References\|[Li18d]]]. It means roughly equivalences given by (complexes of) trivial source bimodules.
		+
		+	=== Trivial intersection subgroup ===
		+
		+	A subgroup <math>H \leq G</math> such that <math>\forall g \in G \setminus N_G(H)</math> we have <math>H^g\cap H=1</math>.