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My question is in the title: Why do physicists refer to irreducible representations (irreps) as "charges" or "charge sectors"?

For concrete examples, irreps are referred to as "charge sectors" just above Eq. (2.22) on page 7 of this paper about reference frames and "charges" in the 5th line from the end of the abstract (and then throughout) this paper on symmetric quantum circuits.

On one hand, the Hilbert space $H$ describing a physical system can be split up into a direct sum of subspaces that carry possibly different irreps. Physical states that live in these subspaces are confined to this subspace (by definition) when they are acted on by the representation of some relevant symmetry acting on $H$. On the other hand, I understand a charge as some quantity that is conserved in a system. Conservation and invariance are notions of the idea. So is calling an irrep a charge somehow getting at the fact that an observable is block diagonal with respect to the decomposition of $H$ according to the invariant subspaces?

Another point of confusion: Should I think of a charge as an operator (observable), an observable's eigenvalue, or a subspace of a Hilbert space? I think of a charge as a conserved observable quantity so it would make sense for a charge to be represented as an operator.

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If a physical system has a continuous symmetry, a Lie group acts on it, cf. e.g. this Phys.SE post. The symmetry generating elements are often realized as certain quantities/operators in the theory that form a defining representation of the Lie algebra, e.g. via the Noether method.

  1. For a compact Abelian Lie group $U(1)^n$, the irreducible representations (irreps) are classified by integer eigenvalues/charges of properly normalized Lie algebra generators/charge operators.

  2. For a complex semisimple Lie algebra, the charge operators refer to Casimirs, and the charges to their eigenvalues, which classify the irreps, cf. e.g. this and this Phys.SE posts.

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    $\begingroup$ Can you add something that links these math results to what we normally think of as charge in physics, i.e. properties of objects that determine how they interact with a field or with each other? $\endgroup$
    – Marius Ladegård Meyer
    Commented May 13 at 9:02
  • $\begingroup$ I updated the answer. $\endgroup$
    – Qmechanic ♦
    Commented May 13 at 10:43

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