APPLICATION OF ALGEBRAIC COMBINATORICS TO FINITE SPIN SYSTEMS |

W. FlorekInstitute of Physics, A. Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland |

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Received: June 11, 2001; in final form July 13, 2001 |

A finite spin system invariant under a symmetry group G is a very
illustrative example of a finite group action on mappings f:X-->Y (X
is a set of spin carriers, Y contains spin projections for a given spin
number s). Orbits and stabilizers are used as additional indices of the
symmetry adapted basis. Their mathematical nature does not decrease a
dimension of a given eigenproblem, but they label states in a
systematic way. It allows construction of general formulas for vectors
of symmetry adapted basis and matrix elements of operators commuting
with the action of G in the space of states. The special role is played
by double cosets, since they label nonequivalent (from the symmetry
point of view) matrix elements < x|H|y> for an operator H
between Ising configurations |x>,|y>. Considerations presented in
this paper should be followed by a detailed discussion of different
symmetry groups (e.g.) cyclic or dihedral ones) and optimal
implementation of algorithms. The paradigmatic example, {i.e.} a finite
spin system, can be useful in investigations of magnetic macromolecules
like Fe_{6} or Mn_{12}acetate. |

DOI: 10.12693/APhysPolA.100.3 PACS numbers: 02.20.Bb, 75.10.Jm, 75.75.+a |