# Fundamental representation

In mathematics, a fundamental representation is a representation of a mathematical structure, such as a group, that satisfies the following condition:

All other irreducible representations of the group can be found in the tensor products of the fundamental representation with many copies of itself.

In general, the fundamental representation is NOT unique, even up to representation equivalence. In general, any faithful irreducible representation is a fundamental representation.

For example, the fundamental representation of SO(n) or SU(n) are n-dimensional vector spaces, and the fundamental representation of E8 is 248-dimensional.

This notion can easily be illustrated in the case of finite groups and representations of those over the complex numbers. It is also interesting in the case of compact groups; and more generally for semisimple groups; these being cases in which representations are completely reducible. In greater generality it is not expected to happen in the way that all irreducible representations occur as direct summands; rather (at best) as subquotients.

While certain cases are therefore clear, the general case of the definition is less so; it may require formulation in terms of category theory. Another approach would be to specify that the matrix coefficients of the representation are a sufficiently rich set of functions on the group. This is the older way of looking at it.

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