Morita invariants

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Under-construction.png

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 (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

Invariants preserved under [math]k[/math]-Morita equivalence of blocks of finite groups

  • Invariants listed above
  • Cartan matrix, up to rearrangement of rows and columns
  • Order of defect group
  • Exponent of defect group[1]
  • [math]p[/math]-rank of defect group[2]

Invariants preserved under [math]\mathcal{O}[/math]-Morita equivalence of blocks of finite groups

  • Invariants listed above
  • Number [math]k_h(B)[/math] of irreducible characters of a given height [math]h[/math]
  • Decomposition matrix, up to rearrangement of rows and columns

Invariants preserved under basic Morita equivalence of blocks of finite groups[3]

Invariants preserved under splendid Morita equivalence of blocks of finite groups

Notes

  1. See (78) of [Ku91]
  2. See Corollary 4 of [AE81], attributed to Donovan
  3. Note that by [KL18], if [math]k[/math]-blocks are basic Morita equivalent, then the corresponding [math]\mathcal{O}[/math]-blocks are Morita equivalent
  4. See Corollary 3.6 of [Pu99]
  5. See Corollary 3.6 of [Pu99]