Finite-dimensional linear transformations are used a lot in theoretical physics. They form a group with respect to multiplication, and some of these groups are extremely useful. So we are dealing with matrix products. In theoretical physics, you'll sooner or later encounter some of the following notation for multiplication groups of matrices. n always stands for dimension:

  • GL(n,C) complex matrices
  • GL(n,R) real matrices
  • SL(n,C) complex matrices with unit determinant
  • U(n) unitary matrices
  • SL(n,R) real matrices with unit determinant
  • SU(n,R) unitary matrices with unit determinant
  • O(n) orthogonal (unitary and real) matrices
  • SO(n) orthogonal with unit determinant
Examples:
  • The above groups are all subgroups of GL(n,C).
  • SL(n,R), O(n) and SO(n) are subgroups of GL(n,R).
  • SO(n) is a subgroup of O(n), same with SU(n) and U(n)
  • SO(3) is the group of rotations in 3-dimensional space.
  • U(1) is the gauge group of quantum electrodynamics
  • SU(3) is the gauge group of quantum chromodynamics
  • SU(2) x U(1) is the gauge group of electroweak theory

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