# Morita invariants

From Block library

## 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 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 listed above
- Isomorphism type of a defect group
^{[4]}
- Fusion system of the block
^{[5]}

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

## Notes

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