We construct graded left dual exceptional collections to the exceptional collections generating the blocks of Kuznetsov and Polishchuk on Lagrangian Grassmannians. As an application, we find explicit resolutions for some natural irreducible equivariant vector bundles.of the braid group on n strands on the set of all semiorthonormal bases. Moreover, the left dual basis to a semiorthonormal basis (v 1 , v 2 , . . . , v n ) is given by (3) L