Representation theory in physics

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I'm sorry if this is somewhat a dumb question.

First: "Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces"

I know little about particle physics, but to what I know, physicists only deal with the groups of (linear) symmetric operators acting on vector space of states.

So in fact physicists are dealing with the italic part of the representation theory. Why would them bring it in? What is a significance of the action "representing an element of a group as linear transformation" in the work of physicists, who are already dealing with groups of linear transformations?

Answer 1

A physicist may indeed be dealing with a specific group of linear transformations of a vector space, but to understand this you would want to break it down into the simplest pieces, which are irreducible representations. And the representation theory will tell you about what these can be.

Answer 2

The definition itself is of little use to physicists, except that by realizing that this is what we are doing and that it is the same as what mathemeticians mean in a well-studied area of math we can then get the benefit of the body of work by mathematicians, as applied to physics questions.

For example, one important endeavor in Grand Unified Field Theory is (or was) discovering composite particle content of a theory with interactions between two irreducible representations of a Lie group, by finding the decomposition into irreps of the Kronecker product of those two irreps. This problem is tough for physicists (I know, I did a paper on it in 1980 but could not come up with good algorithms for the exceptional groups). There could easily be some mathematician who has solved this problem well.

An important analogous situation is when C. N. Yang realized that physicists' Gauge theory is the same as mathematician's fiber bundles. This opened up a colossal amount of progress for both gauge theory and fiber bundle theory, because smart people looking at the same issues from different viewpoints is often a good thing.

Answer 3

Here is a (mathematical) addendum to Mark Fischer's answer.

In main, there are two general combinatorial models for decomposition of tensor products of irreducible representations both generalizing the classical Littlewood-Richardson Rule from $SL(n, C)$-representations to all other complex (or compact, whatever you prefer) semisimple Lie groups, including the exceptional ones. Both models are based on "counting" some geometric/combinatorial objects different from Young tableaux:

a. Littlemann's LS path model:

P. Littelmann, Paths and root operators in representation theory, Annals of Math. (2) Vol. 142 (1995) no. 3, p. 499-525.

b. Berenstein-Zelevinsky polytopal model:

A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Inventiones Math., Vol. 143 (2001), p. 77-128.

Even though both models have something in common, I do not know how to relate them directly.

My knowledge of the literature is limited, but the LS model was further modified by Stembridge, Leinart and Postnikov to make it more combinatorial (Littlemann's model counts certain piecewise-linear paths). Maybe somebody wrote software using this modification, I do not know. (Stembridge wrote software for other Lie-theoretic computations, see here.)

The BZ-model is also quite algorithmic (one has to count number of integer points in some polytopes), but I do not know if it was implemented either.

However, since 1996 there is a freely available program for computing tensor product decomposition multiplicities (among many other things), called Lie, see here.