Quantum Grassmannians and their Associated Quantum Schubert Varieties at roots of unity

Abstract

We study the PI degree of various quantum algebras at roots of unity, including quantum Grassmannians, quantum Schubert varieties, partition subalgebras, and their associated quantum affine spaces. By a theorem of De Concini and Procesi, the PI degree of partition subalgebras and their associated quantum affine spaces is controlled by skew-symmetric integral matrices associated to (Cauchon-Le) diagrams. We prove that the invariant factors of these matrices are always powers of 2. This allows us to compute explicitly the PI degree of partition subalgebras. Our results also apply to certain completely prime (homogeneous) quotients of partition subalgebras. In particular, our results allow us to extend results of Jakobsen and Jondrup regarding the PI degree of quantum determinantal rings at roots of unity [JJ01] and we present a method to construct an irreducible representation of maximal dimension for quantum determinantal ideals. Building on these results, we use the strong connection between partition subalgebras and quantum Schubert varieties through noncommutative dehomogenisation [LR08] to obtain expressions for the PI degree of quantum Schubert varieties. In particular, we compute the PI degree of quantum Grassmannians.