University of Michigan: Dens, nests, and Catalanimals: a walk through the zoo of shuffle theorems

Abstract: In the past few years, significant progress has been made in proving “shuffle theorems”: equations that express an algebraically defined operator on a symmetric functions as a \(q,t\)-weighted sum over some kind of combinatorial object. In 2005, Haglund-Haiman-Loehr-Remmel-Ulya posed The Shuffle Conjecture, expressing the image of the \(\nabla\) operator on an elementary symmetric function as a \(q,t\)-weighted sum over parking functions, that is, tableaux on Dyck paths. A decade later, Carlsson-Mellit proved the shuffle conjecture, but not before much work had been done discovering various conjectural generalizations of the shuffle conjecture. One such conjecture is the Loehr-Warrington conjecture, giving a combinatorial description of the image of $\nabla$ on any Schur function. In joint work with Jonah Blasiak, Mark Haiman, Jennifer Morse, and Anna Pun, we give a “Catalanimal”, or an infinite series of \(GL_n-characters\), whose polynomial part gives a scalar multiple of \(\nabla s_\lambda\). Furthermore, this realization lends itself to the combinatorics of what we call “nests” which are controlled by “dens.” This approach allows us to establish and generalize the Loehr-Warrington conjecture.

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Date published: Friday, March 17, 2023