arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 0901.0323math-phhep-thmath.MPmath.PRnlin.SI
收藏

可积族、矩阵模型和τ-函数的卷积对称性

Convolution symmetries of integrable hierarchies, matrix models and τ-functions

J. Harnad, A. Yu. Orlov

备注：
29 pages. The paper has been revised to correct a number of typos, add details in the proofs of Propositions 4.1, 4.2 and 4.4, and clarify the introduction of convolution symmetries. This version will appear in a special MSRI volume based on the Fall 2011 semester devoted to Random Matrix Theory (eds. P. Forrester and P. Deift)
发表信息：
Random Matrix Theory, Interacting Particle Systems and Integrable Systems (ed. Percy Deift and Peter Forrester), MSRI Publications {\bf 65} 247-275 (2014)
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

KP型和2KP-Toda型可积族的广义卷积对称将希尔伯特空间$\HH= L^2(S^1)$元素的傅立叶系数乘以一个指定的常数序列。这导致了希尔伯特空间格拉斯曼$\Gr_{\HH_+}(\HH)$上的相应变换，从而导致了决定KP和2-Toda族解的Sato-Segal-Wilson τ-函数上的相应变换。相应的作用在相关的费米子Fock空间也是对角的标准正交基确定占领网站和分区标记。定义这些交换流的初始点的元素元素$W \in \Gr_{\HH_+}(\HH)$的普吕克坐标是相关τ函数的单Schur函数和双Schur函数中的系数，因此在此作用下乘以相应的对角因子。将这种变换应用于矩阵积分，我们得到了新的外耦合型矩阵模型，它们也是KP或2KP-Toda τ-函数。类似地得到了τ函数的更一般的多重积分表示及其有限行列式表达式。

英文摘要：

Generalized convolution symmetries of integrable hierarchies of KP and 2KP-Toda type multiply the Fourier coefficients of the elements of the Hilbert space $\HH= L^2(S^1)$ by a specified sequence of constants. This induces a corresponding transformation on the Hilbert space Grassmannian $\Gr_{\HH_+}(\HH)$ and hence on the Sato-Segal-Wilson τ-functions determining solutions to the KP and 2-Toda hierarchies. The corresponding action on the associated fermionic Fock space is also diagonal in the standard orthonormal base determined by occupation sites and labeled by partitions. The Plücker coordinates of the element element $W \in \Gr_{\HH_+}(\HH)$ defining the initial point of these commuting flows are the coefficients in the single and double Schur function of the associated τfunction, and are therefore multiplied by the corresponding diagonal factors under this action. Applying such transformations to matrix integrals, we obtain new matrix models of externally coupled type that are hence also KP or 2KP-Toda τ-functions. More general multiple integral representations of τfunctions are similarly obtained, as well as finite determinantal expressions for them.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑