Difference between revisions of "Tame blocks"
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− | + | A finite dimensional <math>k</math>-algebra <math>A</math> is said to have ''finite representation type'' if there are only finitely many isomorphism classes of indecomposable modules. Algebras of infinite representation type are split into two cases: ''tame'' and ''wild''. For definitions see Section I.4 of [[References#E|[Er90]]], but tame essentially means that almost all modules of a given dimension fit into finitely many one-parameter families and wild means that the module category is comparable to that for <math>k\langle X,Y \rangle</math>. The properties of having finite or infinite representation type and of being tame or wild are all Morita invariants. | |
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+ | Let <math>B</math> be a block of <math>kG</math> for a finite group <math>G</math> with defect group <math>D</math>. Then | ||
+ | *<math>B</math> has finite representation type if and only if <math>D</math> is cyclic. | ||
+ | *<math>B</math> is tame if and only if <math>D</math> contains no noncyclic abelian <math>p</math>-subgroup of order greater than four. Equivalently, <math>D</math> is generalized quaternion, dihedral or semidihedral. | ||
+ | *Otherwise, <math>B</math> is wild. | ||
+ | |||
+ | In a series of papers and her book [[References#E|[Er90]]], Erdmann describes the basic algebra of tame type (see page vi of [[References#E|[Er90]]] for a definition), and in most cases describes which of these occur as basic algebras for blocks of finite groups. | ||
+ | |||
+ | <!--== Notes == | ||
+ | <references /> | ||
+ | --> |
Revision as of 15:18, 30 September 2019
A finite dimensional [math]k[/math]-algebra [math]A[/math] is said to have finite representation type if there are only finitely many isomorphism classes of indecomposable modules. Algebras of infinite representation type are split into two cases: tame and wild. For definitions see Section I.4 of [Er90], but tame essentially means that almost all modules of a given dimension fit into finitely many one-parameter families and wild means that the module category is comparable to that for [math]k\langle X,Y \rangle[/math]. The properties of having finite or infinite representation type and of being tame or wild are all Morita invariants.
Let [math]B[/math] be a block of [math]kG[/math] for a finite group [math]G[/math] with defect group [math]D[/math]. Then
- [math]B[/math] has finite representation type if and only if [math]D[/math] is cyclic.
- [math]B[/math] is tame if and only if [math]D[/math] contains no noncyclic abelian [math]p[/math]-subgroup of order greater than four. Equivalently, [math]D[/math] is generalized quaternion, dihedral or semidihedral.
- Otherwise, [math]B[/math] is wild.
In a series of papers and her book [Er90], Erdmann describes the basic algebra of tame type (see page vi of [Er90] for a definition), and in most cases describes which of these occur as basic algebras for blocks of finite groups.