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Representation theory
Representation theory, in its broadest sense, is the art of relating the symmetries of different objects.
To study symmetry in the first place, mathematicians introduced the notion of a group. For instance a square has four bilateral symmetries (reflections across diagonals or across lines connecting opposite borders) and four rotational symmetries (by 0, 90, 180 or 270 degrees). Together these eight symmetries form a group.
Incorrigibly Plural, by Tessa Coe. Image courtesy of the artist.
How many symmetries?
Symmetry appears in many different guises. Galois discovered the right way to understand symmetries of a polynomial equation. For example, the equation X^4=2 has four solutions (in the complex numbers), and it turns out that there are eight symmetries of these solutions. These symmetries form a group which has exactly the same structure as the group of symmetries of the square.
Representation theory is a vast subject enjoying a close relationship with topology, geometry, number theory, combinatorics and mathematical physics.