Dissertation: K-theoretic Catalan Functions

My dissertation shares a significant overlap with the paper by the same name, but includes some additional results and conjectures. Most notably, in Chapter 4, I provide a combinatorial proof of the so-called “k-rectangle rule” to the conjectured formula for closed \(K\)-\(k\)-Schur functions, using the techniques developed in the previous chapters. Ikeda, Iwao, and Naito subsequently proved the conjectured formula is true and thus give a different proof of the k-rectangle rule via work of Takigiku. In Chapter 5, I provide some conjectures related to Type C \(k\)-Schur functions, giving a conjectured combinatorial formula for them in terms of certain kinds of tableaux, reminiscient of the affine type A combinatorics, as well as a conjectured “raising operator” style formula.

You can access my dissertation at the University of Virginia library Libra archive here.

References

  • Blasiak, J., Morse, J., & Seelinger, G. H. (2022). K-theoretic Catalan functions. Advances in Mathematics, 404(),
    1. http://dx.doi.org/10.1016/j.aim.2022.108421

  • Ikeda, T., Iwao, S., & Naito, S. (2024). Closed k-Schur Katalan functions as K-homology Schubert representatives of the affine Grassmannian. Transactions of the American Mathematical Society, Series B, 11(20), 667–702. http://dx.doi.org/10.1090/btran/184

  • Takigiku, M. (2019). A Pieri formula and a factorization formula for sums of K-theoretic k-Schur functions. Algebraic Combinatorics, 2(4), 447–480. http://dx.doi.org/10.5802/alco.45
Date published: Thursday, May 13, 2021