Combinatorial representation theory applied to Schubert calculus and Markov chains

该奖项支持PI在代数组合学领域的研究。这是对离散对象和它们之间的映射以及控制这些对象结构的代数性质的研究。出现的组合问题与表示理论（对称的研究），物理学（粒子的能级及其结构系数）和几何（空间曲线的相交）有关。本研究中使用的主要工具之一是晶体图，它在“零温度极限”下描述这些结构，但却捕获了所有重要的性质。由于组合学是具体的，它非常适合于计算研究。该项目衍生的算法的健壮实现将导致开源计算机代数系统Sage的新代码的开发。通过Sage传播这款新软件不仅将推进拟议的研究计划，而且已经导致了数学和计算机科学各个领域之间的交叉受精（例如通过ICERM的一个学期计划），预计这将继续下去。这个项目涉及研究生的培养。仿射舒伯特演算的组合研究导致了与弱和强布鲁哈特顺序相关的新表，类似于半标准杨表。晶体基理论有悠久的传统，用组合术语来回答表示理论问题，如张量积的多重性或正特征展开。将晶体基应用于弱表形理论，PI和她的合作者能够回答关于旗Gromov- Witten不变量和量子舒伯特结构系数的结构常数的某些长期存在的问题。PI试图完成这一理论，并将其应用于强大的场景。这将证明，例如，k-舒尔函数，它是通常舒尔函数的仿射版本，并形成了仿射格拉斯曼上同调的舒伯特类的一组代表，在舒尔函数中正展开。完整的理论将有助于研究Littlewood- Richardson系数、聚变系数及其q-类似物的组合表达式，以及一维和，使用几何和物理驱动的代数，以及与Macdonald理论、Demazure晶体和Kirillov- Reshetikhin晶体的关系。此外，PI和她的合作者最近成功地开发了一个由r平凡单群控制的马尔可夫链理论。这导致了大量关于特征值、多重性和混合时间的新猜想，以及新的马尔可夫链（例如，在偏置集的线性扩展上），这些也将进一步研究。

The award supports PI's research in the area of algebraic combinatorics. This is the study of discrete objects and maps between them together with algebraic properties that govern the structures of these objects. The combinatorial problems which arise are related to representation theory (study of symmetries), physics (energy levels of particles and their structure coefficients), and geometry (intersections of curves in space). Among the main tools used in this research are crystal graphs, which describe these structures in the "zero temperature limit" and yet encapture all of the important properties. Since combinatorics is concrete, it is very amenable to computational investigations. The robust implementation of algorithms derived from the project will lead to the development of new code for the open-source computer algebra system Sage. The dissemination of this new software through Sage will not only advance the proposed research program, but has already led to cross-fertilization between various areas in mathematics and computer science (for example through a semester program at ICERM), and this is expected to continue. This project involves the training of graduate students.The combinatorial study of affine Schubert calculus has led to new tableaux related to weak and strong Bruhat order in analogy to semistandard Young tableaux. Crystal base theory has a long tradition of answering representation theoretic questions in combinatorial terms, such as multiplicities of tensor products or positive character expansions. Applying crystal bases to the theory of weak tableaux, the PI and her collaborators were able to answer certain longstanding questions about structure constants of flag Gromov--Witten invariants and quantum Schubert structure coefficients. The PI seeks to complete this theory and apply it to strong tableaux. This would prove, for example, that k-Schur functions, which are affine versions of the usual Schur functions and form a set of representatives for the Schubert classes of the cohomology of the affine Grassmannian, expand positively in terms of Schur functions. The complete theory will facilitate the study of combinatorial expressions for generalizations of Littlewood--Richardson coefficients, fusion coefficients, and their q-analogues as well as one-dimensional sums, using algebras motivated by geometry and physics and relations to Macdonald theory, Demazure crystals, and Kirillov--Reshetikhin crystals. In addition, the PI and her collaborators have recently successfully developed a theory of Markov chains governed by R-trivial monoids. This has led to an abundance of new conjectures about eigenvalues, multiplicities, and mixing times as well as new Markov chains (for example, on linear extensions of posets) that will also be studied further.