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Representation Theory

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More about this group

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.

Representation theory is a vast subject enjoying a close relationship with topology, geometry, number theory, combinatorics and mathematical physics.

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.