August 11, 2025Open Access

Scaling of symmetry-restricted Lie groups

MMMaximilian Balthasar ManskyLMU Klinikum MMMiguel Armayor MartinezLMU Klinikum ASAlejandro Bravo de la SernaTechnical University of Munich

Key Points

The study reveals how subspaces inherit topological properties from the group SU(n), enhancing our understanding of quantum systems.
Through a combinatorial approach, the dimensions of these subspaces were computed when represented by permutation groups and Hamiltonians.
Lie algebras for permutation-invariant groups were constructed using the Kronecker product of su(2) matrices, showing a novel method for analysis.
Numerical results confirmed the theoretical predictions about the dimensions of symmetry-restricted subspaces, supporting their practical relevance.

Abstract

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.

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