Highest weight $$\mathfrak{sl }_2$$ -categorifications I: crystals

Abstract

We define highest weight categorical actions of \(\mathfrak{sl }_2\) on highest weight categories and show that basically all known examples of categorical \(\mathfrak{sl }_2\)-actions on highest weight categories (including rational and polynomial representations of general linear groups, parabolic categories \(\mathcal O \) of type \(A\), categories \(\mathcal O \) for cyclotomic Rational Cherednik algebras) are highest weight in our sense. Our main result is an explicit combinatorial description of (the labels of) the crystal on the set of simple objects. A new application of this is to determining the supports of simple modules over the cyclotomic Rational Cherednik algebras starting from their labels.