与磁性海森堡环有关的对称类
Symmetry classes connected with the magnetic Heisenberg ring
摘要：
本文定义了一维自旋为1/2的海森堡磁环的希尔伯特空间H中的对称类和对易对称，并利用对称群S_N的表示论中的工具，如群环C[S_N]的理想分解、C[S_N]的幂等元、S_N的离散傅里叶变换、Littlewood-Richardson积等，研究了它们.特别是，我们确定最小的对称类和稳定子群的单个本征向量v和子空间U的本征向量的哈密顿的磁铁。确定U的最小对称类是基于一个算法，该算法显式地计算C[S_N]的右理想的非直和的生成幂等元。设U是权为（r_1，r_2）的Hamilton算子的固定特征值μ的特征向量的子空间.如果确定U中每个v的最小对称类，则可以观察到对称行为的跳跃。对于“generic "v，所有最小的对称类都有相同的最大维数d和结构。但是U可以包含线性子空间，在该线性子空间上，v的最小对称类的维数跳到小于d的值。则v的稳定子群可以增加。我们可以精确地计算这种跳跃。在我们的调查中，我们使用计算机计算的数学软件包PERMS和HRing。
英文摘要：
We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups S_N such as decompositions of ideals of the group ring C[S_N], idempotents of C[S_N], discrete Fourier transforms of S_N, Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian of the magnet. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[S_N]. Let U be a subspace of eigenvectors of a a fixed eigenvalue μof the Hamiltonian with weight (r_1,r_2). If one determines the smallest symmetry class for every v in U then one can observe jumps of the symmetry behaviour. For ''generic'' v all smallest symmetry classes have the same maximal dimension d and structure. But U can contain linear subspaces on which the dimension of the smallest symmetry class of v jumps to a value smaller than d. Then the stability subgroup of v can increase. We can calculate such jumps explicitely. In our investigations we use computer calculations by means of the Mathematica packages PERMS and HRing.
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