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Topological groups, Lie groups and representation theory

From 1923 to 1938, Weyl developed the theory of compact groups, in terms of matrix representations. In the compact Lie group case he proved a fundamental character formula.

These results are foundational in understanding the symmetry structure of quantum mechanics, which he put on a group-theoretic basis. This included spinors. Together with the mathematical formulation of quantum mechanics, in large measure due to John von Neumann, this gave the treatment familiar since about 1930. Non-compact groups and their representations, particularly the Heisenberg group, were also streamlined in that specific context, in his 1927 Weyl quantization, the best extant bridge betweenclassical and quantum physics to date. From this time, and certainly much helped by Weyl's expositions, Lie groups and Lie algebras became a mainstream part both of pure mathematics and theoretical physics.

His book The Classical Groups reconsidered invariant theory. It covered symmetric groups, general linear groups, orthogonal groups, and symplectic groups and results on their invariants and representations.