(a) What is a group homomorphism? (b) What is a group representation? What is the dimension of a group representation?

(a) What is a group homomorphism?

(b) What is a group representation? What is the dimension of a group representation?

(c) What is an irreducible group representation?

(d) What is a unitary group representation? Give an example. Why are such representations important in quantum mechanics?

Expert Solution

(a) A group homomorphism is a map between two groups such that the group operation is preserved: for all , where the product on the left-hand side is in and on the right-hand side in .

(b) A representation of a group is a group action of on a vector space by invertible linear maps.

(c) An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces.

(d) In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.

genius_generous answered 2 years ago

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(a) What is a group homomorphism? (b) What is a group representation? What is the dimension of a group representation?

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