Large-dimensional homomorphism spaces between Weyl modules and Specht modules

We give a family of pairs of Weyl modules for the q-Schur algebras for which the corresponding homomorphism space is at least 2-dimensional. Using this result we show that for any field F and any q∈F× such that 1+q+⋯+qf−1=0 for some integer 2≤f<∞ there exist arbitrarily large homomorphism spaces between pairs of Weyl modules.