8 research outputs found
Self-adjoint symmetry operators connected with the magnetic Heisenberg ring
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.Comment: 13 page
Hiding canonicalisation in tensor computer algebra
Simplification of expressions in computer algebra systems often involves a step known as "canonicalisation", which reduces equivalent expressions to the same form. However, such forms may not be natural from the perspective of a pen-and-paper computation, or may be unwieldy, or both. This is, for example, the case for expressions involving tensor multi-term symmetries. We propose an alternative strategy to handle such tensor expressions, which hides canonical forms from the user entirely, and present an implementation of this idea in the Cadabra computer algebra system