We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to "Howe duality" in the exterior algebra.
For the case of positive characteristic we use the classification of finite irreducible groups generated by pseudoreflections due to Kantor, Wagner, Zalesski? and Sere?kin.
In the interesting case when the group is of Coxeter typeDn (n≥4) we use higher polarization operators introduced by Wallach.
We also use known results about canonical bases forUq2 to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmannians), due to Lascoux-Schützenberger and Zelevinsky.
We use invariant theory to compute the exact number of nonzero idempotents of an arbitrary 2-dimensional real division algebra.