A unitary representation is a homomorphism from a group to the group of unitary operators on a Hilbert space, preserving the inner product structure. This means that the action of the group on the space is not only linear but also maintains the lengths and angles, making it a very structured way to understand symmetries in quantum mechanics and other areas of mathematics. Unitary representations play a key role in understanding linear representations and their properties, as well as in the application of Schur's Lemma and orthogonality relations, which provide essential tools for analyzing these representations.
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