Morita invariants

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
• [math]p[/math]-rank of defect group

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

• Invariants listed above
• Isomorphism type of a defect group
• Fusion system of the block

Invariants preserved under splendid Morita equivalence of blocks of finite groups

• Invariants listed above
• Source algebra