Perspectives on Catalanimals part I

Over two days, Mark Haiman and I gave 2 lectures on Catalanimals, describing the framework we use with collaborators to approach various “Shuffle theorems.” I gave the first lecture (broken up over two session).

Full abstract: Catalanimals have been a central tool to a family of proofs of various “shuffle identities” such as the shuffle theorem, the extended Delta theorem, and the Loehr-Warrington theorem. In this series of talks, we will attempt to give a self-contained treatment of the theory of Catalanimals and their various properties that enable these proofs. The first half of the series will focus on the theory of nonsymmetric Hall-Littlewood polynomials and LLT series, culminating in a statement of a Cauchy identity that is useful for extracting combinatorial formulae from certain Catalanimals. The second half of the series will focus on the realization of certain Catalanimals as elements of Negut’s Shuffle Algebra and how they correspond to elements in Schiffmann’s elliptic Hall algebra

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Date published: Monday, June 9, 2025