PI Degree and Irreducible Representations of Quantum Determinantal Rings and their Associated Quantum Schubert Varieties

We study quantum determinantal rings at roots of unity and calculate the PI degree using results of Lenagan-Rigal and Haynal to reduce the problem to finding properties of their associated matrices. These matrices, in turn, correspond to Cauchon-Le diagrams from which we can calculate the required matrix properties. In particular, we show that any matrix corresponding to an $m\times n$ diagram has invariant factors which are powers of 2. Our calculations allow us to state an explicit expression for the PI degree of quantum determinantal rings when the deformation parameter $q$ is a primitive $\ell^{\text{th}}$ root of unity with $\ell$ odd. Using this newly calculated PI degree we present a method to construct an irreducible representation of maximal dimension. Building on these results, we use the strong connection between quantum determinantal rings and certain quantum Schubert varieties through noncommutative dehomogenisation to obtain expressions for the PI degree of such quantum Schubert varieties under the same conditions on $q$.
Comment: 36 pages