Authors: Maximilian Balthasar Mansky | Miguel Armayor Martinez | Alejandro Bravo de la Serna | Santiago Londoño Castillo | Dimitra Nikolaidou | Gautham Sathish | Zhihao Wang | Sebastian Wölckert | Claudia Linnhoff‐Popien
Source: Physica Scripta (article, 2025-07-17)
Field: Chemistry — Spectroscopy

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> <mml:mi mathvariant="normal">SU</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> </mml:math> , the subset <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> <mml:mi mathvariant="normal">SU</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> inherits many topological and group properties from SU( n ). We then present a combinatorial method for computing the dimension of such subspaces when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> </mml:math> is a representation of a permutation group acting on a tensor product of spaces <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi mathvariant="normal">SU</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfenced> </mml:math> , or a Hamiltonian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mfenced close=")" open="("> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mi mathvariant="normal">SU</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfenced> </mml:math> . The Kronecker product of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="fraktur">su</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> 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. Algorithm ; Artificial intelligence ; Linear subspace ; Computer science ; Geometry ; Mathematics Molecular spectroscopy and chirality ; Algebraic structures and combinatorial models ; Advanced NMR Techniques and Applications