Steenrod Operators, the Coulomb Branch and the Frobenius Twist, I

TL;DR

This paper explores the use of Steenrod operators to establish Frobenius-constant quantizations of the quantum Coulomb branch and extends these ideas to categorical structures in representation theory.

Contribution

It introduces a novel application of Steenrod's construction to quantum Coulomb branches and constructs a functor extending the Frobenius twist in categorical settings.

Findings

Quantum Coulomb branch is a Frobenius-constant quantization.

K-theoretic quantum Coulomb branch also exhibits Frobenius-constant properties.

Constructs a functor of categorical p-center extending Frobenius twist.

Abstract

In Part I, we use Steenrod's construction to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We will also demonstrate the corresponding result for the $K$-theoretic version of the quantum Coulomb branch. In Part II, we use the same method to construct a functor of categorical $p$-center between the derived Satake categories with and without loop-rotation, which extends the Frobenius twist functor for representations of the dual group.