Motivated by the intersection theory of moduli spaces of curves, we introduce psi classes in matroid Chow rings and prove a number of properties that naturally generalize properties of psi classes in Chow rings of Losev-Manin spaces. We use these properties of matroid psi classes to give new proofs of (1) a Chow-theoretic interpretation for the coefficients of the reduced characteristic polynomials of matroids, (2) explicit formulas for the volume polynomials of matroids, and (3) Poincaré duality for matroid Chow rings.

For any simply-laced type simple Lie algebra and any height function adapted to an orientation of the Dynkin diagram of , Hernandez-Leclerc introduced a certain category of representations of the quantum affine algebra , as well as a subcategory of whose complexified Grothendieck ring is isomorphic to the coordinate ring of a maximal unipotent subgroup. In this paper, we define an algebraic morphism on a torus containing the image of under the truncated -character morphism. We prove that the restriction of to coincides with the morphism recently introduced by Baumann-Kamnitzer-Knutson in their study of equivariant multiplicities of Mirković-Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov-Reshetikhin modules in , as well as certain results by Brundan-Kleshchev-McNamara on the representation theory of quiver Hecke algebras. This alternative description of allows us to prove a conjecture by the first author on the distinguished values of on the flag minors of . We also provide applications of our results from the perspective of Kang-Kashiwara-Kim-Oh's generalized Schur-Weyl duality. Finally, we use Kashiwara-Kim-Oh-Park's recent constructions to define a cluster algebra as a subquotient of naturally containing , and suggest the existence of an analogue of the Mirković-Vilonen basis in on which the values of may be interpreted as certain equivariant multiplicities.