# Morita invariants

## Invariants preserved under Morita equivalence of blocks of f.d. $k$-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 $k$-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]
• $p$-rank of defect group[2]

## Invariants preserved under $\mathcal{O}$-Morita equivalence of blocks of finite groups

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

## Notes

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