Let S(n,r) be a classical or quantised Schur algebra with n ≥ r. Totaro determined the global dimension of classical S(n,r) and Donkin extended this result to quantum S(n,r). Ming Fang and I have determined the dominant dimension (which in some sense measures the strength of Schur-Weyl duality) in both cases.
Schur algebras are, however, not indecomposable as algebras, and thus we should ask about the homological invariants of their blocks. This has been answered in recent joint work with Ming Fang and Wei Hu. We have been proving that - unlike in the general case - global dimension is invariant under derived equivalences between finite dimensional algebras with simple preserving dualities, and under an additional assumption satisfied by Schur algebras, dominant dimension is invariant, too. Using the derived equivalences constructed by Chuang and Rouquier, we then get explicit formulae for global and dominant dimensions of blocks.