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Theory of symmetry classes
N Metropolis1, G C Rota, J A Stein
1Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
Summary
This study introduces intuitive characterizations for symmetry classes in group theory. It provides two novel methods for describing elements within these classes, simplifying complex mathematical structures.
Area of Science:
- Mathematics
- Group Theory
- Representation Theory
Background:
- Existing work on irreducible representations of the symmetric group (Sn) and general linear group (GLn) lacks intuitive characterization.
- A need exists for simpler ways to understand and define symmetry classes within these groups.
Purpose of the Study:
- To provide simple, intuitive characterizations for symmetry classes of the symmetric group and general linear group.
- To offer two distinct methods for describing elements belonging to any given symmetry class.
Main Methods:
- Systematically distinguishing between permutations of variables and permutations of places.
- Developing characterizations based on solutions to linear equations and linear combinations of simple elements.
Main Results:
- Two intuitive characterizations for symmetry classes are presented.
- Elements are described either as solutions to explicit linear equations or as linear combinations of generalized decomposable skew-symmetric tensors.
Conclusions:
- The provided characterizations offer a more accessible understanding of symmetry classes.
- These methods simplify the description of elements within complex group structures, advancing representation theory.
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