Morita invariants - Revision history

Charles Eaton at 11:40, 2 May 2024

2024-05-02T11:40:11Z

Charles Eaton at 10:48, 4 January 2019

2019-01-04T10:48:22Z

Charles Eaton at 09:13, 4 January 2019

2019-01-04T09:13:34Z

Charles Eaton at 17:57, 3 January 2019

2019-01-03T17:57:55Z

Charles Eaton: Created page with "== Invariants preserved under Morita equivalence of blocks of $k$ -algebras == == Invariants preserved under $k$ -Morita equivalence of blocks of finite g..."

2019-01-03T17:38:05Z

Created page with "== Invariants preserved under Morita equivalence of blocks of <math>k</math>-algebras == == Invariants preserved under <math>k</math>-Morita equivalence of blocks of finite g..."

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== Invariants preserved under Morita equivalence of blocks of <math>k</math>-algebras ==

== Invariants preserved under <math>k</math>-Morita equivalence of blocks of finite groups ==

== Invariants preserved under <math>\mathcal{O}</math>-Morita equivalence of blocks of finite groups ==

== Invariants preserved under basic Morita equivalence of blocks of finite groups<ref>Note that by [[References#K|[KL18]]], if <math>k</math>-blocks are basic Morita equivalent, then the corresponding <math>\mathcal{O}</math>-blocks are Morita equivalent</ref> ==

== Invariants preserved under splendid Morita equivalence of blocks of finite groups ==

== Notes ==

<references />

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 * Number of isomorphism classes of simple modules
-* Isomorphism type of centre of algebra, and so dimension of centre
+* Isomorphism type of centre of algebra, and so dimension of centre (hence, in the case of <math>k</math>-blocks of finite groups, the number of irreducible characters associated to the corresponding <math>\mathcal{O}</math>-block)
 * Loewy structure and socle structure

← Older revision		Revision as of 09:13, 4 January 2019
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	== Invariants preserved under <math>k</math>-Morita equivalence of blocks of finite groups ==		== Invariants preserved under <math>k</math>-Morita equivalence of blocks of finite groups ==
		+
		+	* Cartan matrix, up to rearrangement of rows and columns
		+	* Order of defect group
		+	* Exponent of defect group<ref>See (78) of [[References#K\|[Ku91]]]</ref>
		+	* <math>p</math>-rank of defect group<ref>See Corollary 4 of [[References#A\|[AE81]]], attributed to Donovan</ref>

	== Invariants preserved under <math>\mathcal{O}</math>-Morita equivalence of blocks of finite groups ==		== Invariants preserved under <math>\mathcal{O}</math>-Morita equivalence of blocks of finite groups ==

← Older revision		Revision as of 17:57, 3 January 2019
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−	== Invariants preserved under Morita equivalence of blocks of <math>k</math>-algebras ==	+	[[Image:under-construction.png\|50px\|left]]
		+
		+	== Invariants preserved under Morita equivalence of blocks of f.d. <math>k</math>-algebras ==
		+
		+	* Number of isomorphism classes of simple modules
		+	* Isomorphism type of centre of algebra, and so dimension of centre
		+	* Loewy structure and socle structure

	== Invariants preserved under <math>k</math>-Morita equivalence of blocks of finite groups ==		== Invariants preserved under <math>k</math>-Morita equivalence of blocks of finite groups ==