Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
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DOI：
10.1007/s00222-013-0485-9
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发表时间：
2013-10
影响因子：
3.1
通讯作者：
N. O’Connell;T. Seppäläinen;Nikos Zygouras
中科院分区：
文献类型：
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作者：
N. O’Connell;T. Seppäläinen;Nikos Zygouras
我们证明了 A.N. 引入的 RSK 对应关系的几何提升。 Kirillov (Physics and Combinatorics. Proc. Nagoya 2000 2nd Internat Workshop, pp. 82–150, 2001) 是关于其域上的自然乘积测度的体积守恒，并且在这一背景下，Givental 的 Whittaker 函数积分公式中的被积函数自然出现。除了进一步证明 Whittaker 函数是这种情况下 Schur 多项式的自然类似物之外，我们的结果还为无规聚合物的研究提供了一个新的“组合”框架。当输入矩阵由随机逆伽玛分布权重组成时，由这些权重构造的聚合物配分函数的概率分布可以用 Whittaker 函数明确地写出。接下来，我们将几何 RSK 映射限制为对称矩阵，并证明体积保持属性继续成立。我们确定了具有逆伽玛权重的聚合物配分函数的概率定律，该权重被限制为关于主对角线对称，并且在主对角线上有一个附加因子。研究的第三个组合映射是三角形阵列的几何 RSK 映射的变体，它再次被证明是体积保持的。这就得出了聚合物模型概率分布的公式，其路径被限制在对角线以下。我们还表明，本文中 Cauchy-Littlewood 恒等式的类似物相当于 Bump（数论、迹公式和离散群，第 49-109 页，1989 年）以及 Bump 和 Friedberg（纪念 Piatetski-Shapiro 的 Festschrift，第二部分，第 47-65 页）猜想的 Whittaker 积分恒等式的集合， 1990）并由 Stade 证明（Am. J. Math. 123:121–161, 2001；Israel J. Math. 127:201–219, 2002）。我们的方法导致了这些恒等式的新“组合”证明和概括，并对参数进行了一些限制。
We show that the geometric lifting of the RSK correspondence introduced by A.N. Kirillov (Physics and Combinatorics. Proc. Nagoya 2000 2nd Internat Workshop, pp. 82–150, 2001) is volume preserving with respect to a natural product measure on its domain, and that the integrand in Givental’s integral formula for-Whittaker functions arises naturally in this context. Apart from providing further evidence that Whittaker functions are the natural analogue of Schur polynomials in this setting, our results also provide a new ‘combinatorial’ framework for the study of random polymers. When the input matrix consists of random inverse gamma distributed weights, the probability distribution of a polymer partition function constructed from these weights can be written down explicitly in terms of Whittaker functions. Next we restrict the geometric RSK mapping to symmetric matrices and show that the volume preserving property continues to hold. We determine the probability law of the polymer partition function with inverse gamma weights that are constrained to be symmetric about the main diagonal, with an additional factor on the main diagonal. The third combinatorial mapping studied is a variant of the geometric RSK mapping for triangular arrays, which is again showed to be volume preserving. This leads to a formula for the probability distribution of a polymer model whose paths are constrained to stay below the diagonal. We also show that the analogues of the Cauchy-Littlewood identity in the setting of this paper are equivalent to a collection of Whittaker integral identities conjectured by Bump (Number Theory, Trace Formulas, and Discrete Groups, pp. 49–109, 1989) and Bump and Friedberg (Festschrift in Honor of Piatetski-Shapiro, Part II, pp. 47–65, 1990) and proved by Stade (Am. J. Math. 123:121–161, 2001; Israel J. Math. 127:201–219, 2002). Our approach leads to new ‘combinatorial’ proofs and generalizations of these identities, with some restrictions on the parameters.