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. | ||
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| + | 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. As a consequence the <math>k</math>-Donovan conjecture holds for all tame blocks except when <math>D</math> is generalized quaternion of order greater than eight and <math>l(B)=2</math>. In this last case there is an infinite class of basic algebras where it is not determined which may occur as basic algbras for blocks of finite groups. In general it is difficult to extend these results to blocks defined over <math>\mathcal{O}</math>. For blocks with Klein four defect groups this was done by Linckelmann in [[References#L|[Li94]]], and for blocks with generalized quaternion defect groups and <math>l(B)=3</math> by Eisele in [[References#E|[Ei16]]]. Note that by Erdmann's classifications tame blocks with just one simple module are nilpotent, and so the classification extends to <math>\mathcal{O}</math> in this case. | ||
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| + | Linckelmann's results on Klein four defect groups were extended using the classification of finite simple groups to classify such blocks up to Puig equivalence in [[References#C|[CEKL11]]]. | ||
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| + | [[Statements of conjectures|Puig's conjecture]] holds for all principal tame blocks by [[References#K|[KoLa20]]], [[References#K|[KoLa20b]]], [[References#K|[KoLaSa22]]] . | ||
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| + | <!--== Notes == | ||
| + | <references /> | ||
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Latest revision as of 18:28, 9 November 2022
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. As a consequence the [math]k[/math]-Donovan conjecture holds for all tame blocks except when [math]D[/math] is generalized quaternion of order greater than eight and [math]l(B)=2[/math]. In this last case there is an infinite class of basic algebras where it is not determined which may occur as basic algbras for blocks of finite groups. In general it is difficult to extend these results to blocks defined over [math]\mathcal{O}[/math]. For blocks with Klein four defect groups this was done by Linckelmann in [Li94], and for blocks with generalized quaternion defect groups and [math]l(B)=3[/math] by Eisele in [Ei16]. Note that by Erdmann's classifications tame blocks with just one simple module are nilpotent, and so the classification extends to [math]\mathcal{O}[/math] in this case.
Linckelmann's results on Klein four defect groups were extended using the classification of finite simple groups to classify such blocks up to Puig equivalence in [CEKL11].
Puig's conjecture holds for all principal tame blocks by [KoLa20], [KoLa20b], [KoLaSa22] .
