Combinatorics of Interacting Particles and Applications

我的研究项目探讨了组合学、表示理论、统计物理和可积系统交叉的结构。总体目标是通过在它们之间插入的组合对象来探索某些晶格模型和正交多项式之间的联系。的相关性。该建议的核心是多物种不对称简单排斥过程（ASEP），这是一个一维精确可解的状态机制模型，被认为是非平衡系统的典型例子。ASEP在交通流和蛋白质合成中的翻译方面有应用。ASEP还具有丰富的组合结构和与Askey-Wilson、Macdonald、Koornwinder等正交多项式的深层联系。这些非凡的多项式在表示理论、代数组合和代数几何中起着至关重要的作用，并得到了广泛的研究，尽管仍有许多重要的开放性问题存在。背景。麦克唐纳多项式（A型）是一个双参数族，它构成了对称函数环的基。由于其固有的复杂性，大量的工作一直致力于理解它们的组合学。在过去的十年中，统计力学和麦克唐纳多项式之间的联系已经在几种情况下进行了探索。特别地，我们发现Macdonald多项式的专门化与圆上多种ASEP的配分函数成正比。在最近的工作中，我们用Corteel和Williams找到了这些对象在多行队列中的公式。最近研究发现，BC型根系的Koornwinder多项式（Macdonald）与开放边界的多物种ASEP的概率密切相关。在获得多物种情况下的概率公式方面几乎没有取得进展。在过去的工作中，我和我的合作者发现了两种情况下的公式，从而解决了已经开放了十多年的问题。Macdonald多项式和Koornwinder多项式与ASEP和其他晶格模型的联系激发了本研究计划的目标。目标。主要目标包括：目标1。研究了使用多行队列的修正Macdonald多项式[C2]，最近发现的拟对称Macdonald多项式[C1]，以及新发现的与TAZRP的联系。目标2。研究具有开放边界和Koornwinder多项式的多物种ASEP的组合，以期(1)得到该ASEP概率的组合公式，(2)通过ASEP发现Koornwinder多项式的组合公式。这些目标的成功将导致对显著的麦克唐纳多项式的理解取得重大进展，并将进一步对可积系统领域的晶格模型的研究产生重大影响。注意:[C ?指的是我的CCV中的出版物

My research program explores structures that lie at the intersection of combinatorics, representation theory, statistical physics, and integrable systems. The overarching goal is to explore the connections between certain lattice models and orthogonal polynomials through the combinatorial objects that interpolate between them. Relevance. At the center of this proposal is the multi-species asymmetric simple exclusion process (ASEP), which is a 1D exactly solvable stat mech model that is considered a paradigmatic example of non equilibrium systems. The ASEP has applications to traffic flow and translation in protein synthesis. The ASEP also has rich combinatorial structure and deep connections to orthogonal polynomials such as Askey-Wilson, Macdonald, and Koornwinder. These remarkable polynomials play a crucial role in representation theory, algebraic combinatoric, and algebraic geometry, and have been widely studied, though as yet a great many important open questions remain. Background. Macdonald polynomials (of type A) are a two-parameter family that forms a basis for the ring of symmetric functions. A vast body of work has been devoted to understanding their combinatorics due to their inherently complicated nature. The link between statistical mechanics and Macdonald polynomials has been explored in several contexts over the past decade. In particular, it was found that a specialization of the Macdonald polynomial is proportional to the partition function of the multi-species ASEP on a circle. In recent work, with Corteel and Williams formulas were found for these objects in terms of multiline queues. It was found recently that Koornwinder polynomials (Macdonald of type BC root system) are deeply connected to the probabilities of the multi-species ASEP with open boundaries. Extremely little progress has been made on obtaining formulas for probabilities in the multi-species case. In past work, I and my collaborators discovered formulas in the two-species case, thus solving questions that had been open for over ten years. The connections of Macdonald and Koornwinder polynomials to the ASEP and other lattice models has motivated the objectives of this research program. Objectives. The main goals include: Objective 1. Studying the modified Macdonald polynomials using multiline queues [C2], the recently discovered quasisymmetric Macdonald polynomials [C1], and a newly discovered connection to the TAZRP. Objective 2. Studying the combinatorics of the multi-species ASEP with open boundaries and Koornwinder polynomials in order to (1) obtain combinatorial formulas for probabilities of this ASEP, and (2) discover formulas for Koornwinder polynomials via the ASEP.  The success of these objectives will result in major advancement in the understanding of the remarkable Macdonald polynomials and will furthermore have significant impact on the study of lattice models in the field of integrable systems. Note: [C?] refers to publications in my CCV