标题： 排列不变性，分划代数和大$N$矩阵模型 显示英文标题

标题： Permutation invariance, partition algebras and large $N$ matrix models

摘要： 在本论文中，我们将研究具有离散规范群 $S_N$的矩阵模型。 我们将这些矩阵模型置于广义的Schur-Weyl对偶框架中，在此框架中出现称为分拆代数的对偶代数。 这些代数是对称群代数的推广——它们是半单的有限维结合代数，其基由图示标记。 我们回顾分拆代数的结构和表示理论。 然后利用这些代数计算 $S_N$不变观测量的期望值。 这是研究其费曼图展开中新的几何结构出现的一个步骤。 矩阵模型也以矩阵振子的量子力学模型形式出现。 我们探讨Schur-Weyl对偶框架对具有排列对称性的矩阵量子力学的影响。 显示英文摘要

摘要： In this thesis we will study matrix models with discrete gauge group $S_N$. We will put these matrix models into a generalized Schur-Weyl duality framework where dual algebras, known as partition algebras, emerge. These form generalizations of the symmetric group algebras -- they are semi-simple finite-dimensional associative algebras with a basis labelled by diagrams. We review the structure and representation theory of partition algebras. These algebras are then used to compute expectation values of $S_N$ invariant observables. This is a step towards studying the emergence of new geometric structures in their Feynman diagram expansion. Matrix models also appear in the form of quantum mechanical models of matrix oscillators. We explore the implications of the Schur-Weyl duality framework to matrix quantum mechanics with permutation symmetry.