Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation
Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation
复制标题
DOI：
10.1016/j.aim.2022.108453
复制
发表时间：
2019-05
影响因子：
1.7
通讯作者：
E. Bisi;Nikos Zygouras
中科院分区：
文献类型：
--
作者：
E. Bisi;Nikos Zygouras
研究了经典群GL n（C），SO 2 n+ 1（C），Sp2 n（C），SO 2 n（C）和奇辛群Sp2 n+ 1（C）的不可约特征标的组合结构，发现了与最后一次通过渗流（LPP）概率模型的新联系.将这些特征的表达式作为Gelfand-Tsetlin模式的母函数进行扰动，得到了Sp 2n（C）和SO 2n + 1（C）特征之间以及SO 2n（C）和SO 2n + 1（C）特征之间的两类对称多项式.我们确定的第一个家庭作为一个参数专业化的Koornwinder多项式，因此，我们提供了一个新的组合结构，另一方面，第二个家庭似乎是新的。接下来，我们开发了一种方法的Gelfand-Tsetlin模式分解，以建立所有这些多项式之间的身份，在不可约字符的情况下，可以被视为分支规则。通过这些公式，我们将正交和辛特征，以及更一般地将插值多项式连接到具有各种对称性的LPP模型，从而超越了与Baik和Rains最初发现的经典Schur多项式的联系（杜克数学杂志，2001年）。考虑LPP模型的标度极限，我们最终解释了为什么随机矩阵理论的Tracy-Widom GOE和GSE分布允许Fredholm行列式和Fredholm Pfrons公式。
We study the combinatorial structure of the irreducible characters of the classical groups GL n (C), SO 2 n+ 1 (C), Sp 2 n (C), SO 2 n (C) and the “non-classical” odd symplectic group Sp 2 n+ 1 (C), finding new connections to the probabilistic model of Last Passage Percolation (LPP). Perturbing the expressions of these characters as generating functions of Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that interpolate between characters of Sp 2 n (C) and SO 2 n+ 1 (C) and between characters of SO 2 n (C) and SO 2 n+ 1 (C). We identify the first family as a one-parameter specialization of Koornwinder polynomials, for which we thus provide a novel combinatorial structure; on the other hand, the second family appears to be new. We next develop a method of Gelfand-Tsetlin pattern decomposition to establish identities between all these polynomials that, in the case of irreducible characters, can be viewed as branching rules. Through these formulas we connect orthogonal and symplectic characters, and more generally the interpolating polynomials, to LPP models with various symmetries, thus going beyond the link with classical Schur polynomials originally found by Baik and Rains (Duke Math. J., 2001). Taking the scaling limit of the LPP models, we finally provide an explanation of why the Tracy-Widom GOE and GSE distributions from random matrix theory admit formulations in terms of both Fredholm determinants and Fredholm Pfaffians.