Scaling of symmetry-restricted Lie groups

Abstract Symmetries are omnipresent in physics and have been used to reduce the number of degrees of freedom of systems. In this work, we investigate the properties of M SU ( n ) , M -invariant subspaces of the special unitary Lie group SU( n ). This group is relevant to quantum computing and quantum systems in general. We demonstrate that for certain choices of M , the subset M SU ( n ) inherits many topological and group properties from SU( n ). We then present a combinatorial method for computing the dimension of such subspaces when M is a representation of a permutation group acting on a tensor product of spaces G SU ( n ) , or a Hamiltonian H ( n ) SU ( n ) . The Kronecker product of su ( 2 ) matrices is employed to construct the Lie algebras associated with different permutation-invariant groups G SU( n ). Numerical results on the number of dimensions support the developed theory.