def to_character(self): r""" Return the character of the representation. EXAMPLES: The trivial character:: sage: rho = SymmetricGroupRepresentation([3]) sage: chi = rho.to_character(); chi Character of Symmetric group of order 3! as a permutation group sage: chi.values() [1, 1, 1] sage: all(chi(g) == 1 for g in SymmetricGroup(3)) True The sign character:: sage: rho = SymmetricGroupRepresentation([1,1,1]) sage: chi = rho.to_character(); chi Character of Symmetric group of order 3! as a permutation group sage: chi.values() [1, -1, 1] sage: all(chi(g) == g.sign() for g in SymmetricGroup(3)) True The defining representation:: sage: triv = SymmetricGroupRepresentation([4]) sage: hook = SymmetricGroupRepresentation([3,1]) sage: def_rep = lambda p : triv(p).block_sum(hook(p)).trace() sage: map(def_rep, Permutations(4)) [4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0] sage: [p.to_matrix().trace() for p in Permutations(4)] [4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0] """ from sage.groups.perm_gps.permgroup_named import SymmetricGroup Sym = SymmetricGroup(sum(self._partition)) values = [ self(g).trace() for g in Sym.conjugacy_classes_representatives() ] return Sym.character(values)
def to_character(self): r""" Return the character of the representation. EXAMPLES: The trivial character:: sage: rho = SymmetricGroupRepresentation([3]) sage: chi = rho.to_character(); chi Character of Symmetric group of order 3! as a permutation group sage: chi.values() [1, 1, 1] sage: all(chi(g) == 1 for g in SymmetricGroup(3)) True The sign character:: sage: rho = SymmetricGroupRepresentation([1,1,1]) sage: chi = rho.to_character(); chi Character of Symmetric group of order 3! as a permutation group sage: chi.values() [1, -1, 1] sage: all(chi(g) == g.sign() for g in SymmetricGroup(3)) True The defining representation:: sage: triv = SymmetricGroupRepresentation([4]) sage: hook = SymmetricGroupRepresentation([3,1]) sage: def_rep = lambda p : triv(p).block_sum(hook(p)).trace() sage: map(def_rep, Permutations(4)) [4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0] sage: [p.to_matrix().trace() for p in Permutations(4)] [4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0] """ from sage.groups.perm_gps.permgroup_named import SymmetricGroup Sym = SymmetricGroup(sum(self._partition)) values = [self(g).trace() for g in Sym.conjugacy_classes_representatives()] return Sym.character(values)