Symmetry classes connected with the magnetic Heisenberg ring

Abstract

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.