Combinatorial representation theory applied to Schubert calculus and Markov chains

该奖项支持Pi在代数组合数学领域的研究。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。出现的组合问题涉及表象理论(对称性研究)、物理学(粒子的能级及其结构系数)和几何(空间曲线的交点)。在这项研究中使用的主要工具是结晶图，它描述了这些结构在“零温度极限”，但又封装了所有重要的性质。由于组合学是具体的，它很容易进行计算研究。该项目衍生的算法的健壮实现将导致为开源计算机代数系统Sage开发新的代码。通过SAGE传播这一新软件不仅将推进拟议的研究计划，而且已经在数学和计算机科学的不同领域之间产生了交叉影响(例如通过ICERM的一个学期计划)，预计这种情况将继续下去。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类似于半标准Young情景的与弱和强Bruhat序相关的新情景。晶基理论有着用组合术语回答表示理论问题的悠久传统，例如张量积的多重性或正特征展开。将晶体碱应用到弱表理论中，Pi和她的合作者能够回答关于国旗Gromov的结构常数--Witten不变量和量子Schubert结构系数--的某些长期存在的问题。PI试图完善这一理论，并将其应用于强大的场面。这将证明，例如，k-Schur函数是通常Schur函数的仿射版本，并且形成仿射Grassman的上同调的Schubert类的一组代表，关于Schur函数是正扩张的。完整的理论将有助于研究推广的Littlewood-Richardson系数、融合系数和它们的Q-类似物以及一维和的组合表达式，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发展了由R-平凡么半群支配的马尔可夫链的理论。这导致了大量关于特征值、重数和混合时间的新猜想，以及新的马尔可夫链(例如，关于偏序集的线性扩张)，也将被进一步研究。

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.