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

TL;DR

This paper investigates the PI degree of quantum algebras at roots of unity, including quantum Grassmannians and Schubert varieties, providing explicit calculations and extending previous results on quantum determinantal rings.

Contribution

It introduces a method to compute PI degrees of quantum Schubert varieties and Grassmannians at roots of unity, revealing that invariant factors are powers of 2 and extending prior work.

Findings

PI degree of partition subalgebras is explicitly computed.

Invariant factors of associated matrices are powers of 2.

PI degree of quantum Grassmannians is explicitly determined.

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.