Products of Closed k-Schur Katalan Functions

Following applications of Catalan functions to resolve conjectures about k-Schur functions,
which are Schubert homology representatives of the affine Grassmannian, the K-theoretic Catalan
functions (or Katalan functions) have offered a rich combinatorial structure with connections to
the K-homology K∗(Gr), Hopf isomorphic to Λ(k). We prove a multiplication rule for the closed
k-Schur Katalan functions conjectured to be equivalent to a cancellation-free product that matches
Lenart and Maeno’s Monk rule for quantum Grothendieck polynomials under a K-theoretic ana-
logue of the Peterson isomorphism. Our methods include root expansions of the k-Schur root ideal,
the combinatorics of covers, and the relationship between the two. We conclude with an involution
that proves the equivalence of certain Katalan functions; we offer progress alongside this result
which could establish the cancellation-free conjecture.