与磁海森堡环相关的自伴对称算子
Self-adjoint symmetry operators connected with the magnetic Heisenberg ring
摘要：
本文研究了群环C [S_N]中作用在N个格点的一维自旋1/2海森堡磁环的希尔伯特空间H上的对称算子a。我们研究这样的对称算子是自伴的（在本文中定义的意义上），从而产生海森堡模型的观测值。我们证明了如下结果：（i）从S_N的每个子群的每个不可约特征标可以构造一个自伴幂等对称算子。这导致了大量的可观测量。特别是，每个对易对称都会产生这样的幂等元。(ii)C [S_N]的极小右理想R的所有生成幂等元的集合包含一个且仅一个自伴幂等元。(iii)每一个自伴幂等元e都可以分解为本原幂等元e = f_1 +. + f_k，它们也是自伴的和成对正交的。我们给出了一个计算这种分解的计算机算法。此外，我们还提出了三种新的算法，它们有助于利用S_N的离散傅里叶变换计算自伴算子。在我们的调查中，我们使用计算机计算的手段，我们的数学软件包PERMS和HRing。
英文摘要：
We consider symmetry operators a from the group ring C[S_N] which act on the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites. We investigate such symmetry operators a which are self-adjoint (in a sence defined in the paper) and which yield consequently observables of the Heisenberg model. We prove the following results: (i) One can construct a self-adjoint idempotent symmetry operator from every irreducible character of every subgroup of S_N. This leads to a big manifold of observables. In particular every commutation symmetry yields such an idempotent. (ii) The set of all generating idempotents of a minimal right ideal R of C[S_N] contains one and only one idempotent which ist self-adjoint. (iii) Every self-adjoint idempotent e can be decomposed into primitive idempotents e = f_1 + ... + f_k which are also self-adjoint and pairwise orthogonal. We give a computer algorithm for the calculation of such decompositions. Furthermore we present 3 additional algorithms which are helpful for the calculation of self-adjoint operators by means of discrete Fourier transforms of S_N. In our investigations we use computer calculations by means of our Mathematica packages PERMS and HRing.
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