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Browsing by Author "Gillespie, Maria, advisor"

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    A local characterization of domino evacuation-shuffling
    (Colorado State University. Libraries, 2024) McCann, Jacob, author; Gillespie, Maria, advisor; Peterson, Christopher, committee member; Huang, Dongzhou, committee member
    We consider linear intersection problems in the Grassmanian (the space of k-dimensional subspaces of Cn), where the dimension of the intersection is 2. These spaces are called Schubert surfaces. We build of the previous work of Speyer [1] and Gillespie and Levinson [2]. Speyer showed there is a combinatorial interpretation for what happens to fibers of Schubert intersections above a "wall crossing", where marked points corresponding to the coordinates of partitions coincide. Building off Speyer's work, Levinson showed there is a combinatorial operation associated with the monodromy operator on Schubert curves, involving rectification, promotion, and shuffling of Littlewood-Richardson Young Tableaux, which overall is christened evacuation-shuffling. Gillespie and Levinson [2] further developed a localization of the evacuation-shuffling algorithm for Schubert curves. We fully develop a local description of the monodromy operator on certain classes of curves embedded inside Schubert surfaces [3].
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    Generalized RSK for enumerating projective maps from n-pointed curves
    (Colorado State University. Libraries, 2022) Reimer-Berg, Andrew, author; Gillespie, Maria, advisor; Ghosh, Sudipto, committee member; Hulpke, Alexander, committee member; Shoemaker, Mark, committee member
    Schubert calculus has been studied since the 1800s, ever since the mathematician Hermann Schubert studied the intersections of lines and planes. Since then, it has grown to have a plethora of connections to enumerative geometry and algebraic combinatorics alike. These connections give us a way of using Schubert calculus to translate geometric problems into combinatorial ones, and vice versa. In this thesis, we define several combinatorial objects known as Young tableaux, as well as the well-known RSK correspondence between pairs of tableaux and sequences. We also define the Grassmannian space, as well as the Schubert cells that live inside it. Then, we describe how Schubert calculus and the Littlewood-Richardson rule allow us to turn problems of intersecting geometric spaces into ones of counting Young tableaux with particular characteristics. We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-d morphisms from a general genus g, n-marked curve C to Pr, sending the marked points on C to specified general points in Pr, is equal to (r + 1)g for sufficiently large d. This computation may be rephrased as an intersection problem on Grassmannians, which has a natural combinatorial interpretation in terms of Young tableaux by the classical Littlewood-Richardson rule. We give a bijection, generalizing the well-known RSK correspondence, between the tableaux in question and the (r + 1)-ary sequences of length g, and we explore our bijection's combinatorial properties. We also apply similar methods to give a combinatorial interpretation and proof of the fact that, in the modified setting in which r = 1 and several marked points map to the same point in P1, the number of morphisms is still 2g for sufficiently large d.
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    Symmetric functions, shifted tableaux, and a class of distinct Schur Q-functions
    (Colorado State University. Libraries, 2022) Salois, Kyle, author; Gillespie, Maria, advisor; Cavalieri, Renzo, committee member; Hulpke, Alexander, committee member; Cooley, Daniel, committee member
    The Schur Q-functions form a basis of the algebra Ω of symmetric functions generated by the odd-degree power sum basis pd, and have ramifications in the projective representations of the symmetric group. So, as with ordinary Schur functions, it is relevant to consider the equality of skew Schur Q-functions Qλ/μ. This has been studied in 2008 by Barekat and van Willigenburg in the case when the shifted skew shape λ/μ is a ribbon. Building on this premise, we examine the case of near-ribbon shapes, formed by adding one box to a ribbon skew shape. We particularly consider frayed ribbons, that is, the near-ribbons whose shifted skew shape is not an ordinary skew shape. We conjecture with evidence that all Schur Q-functions for frayed ribbon shapes are distinct up to antipodal reflection. We prove this conjecture for several infinite families of frayed ribbons, using a new approach via the "lattice walks'' version of the shifted Littlewood-Richardson rule, discovered in 2018 by Gillespie, Levinson, and Purbhoo.