I am interested in knowing more about applications of Young diagrams and Young tableaux to Quantum Mechanics. A friend of mine suggested as a reference the following book:
Wybourne, B.G.; "Symmetry Principles and Atomic Spectroscopy"; Wiley–Interscience, New York, 1970.
I ordered the book, but in the mean time, could someone perhaps suggest some other reference(s) possibly please?
Motivation: roughly speaking, associated to a Young diagram, I have a constructed a smooth Weyl and SU(2) equivariant with domain the configuration space of $n$ distinct points in $\mathbb{R}^3$ and with target space the space of all functions from the finite set of all semistandard Young tableaux (corresponding to the given Young diagram) to some complex projective space (whose dimension can be extracted from the data).
I would like to know if such maps could have applications to Quantum Mechanics, possibly along lines similar to the Berry-Robbins "moving spin basis" in the article
Berry, M. V. & Robbins, J. M.,1997, Indistinguishability for quantum
particles: spin, statistics and the geometric phase Proc. Roy. Soc.
Lond. A453, 1771-1790.
However, I am definitely open to learning about other possible applications to Physics too.
Edit: Wybourne's book mentioned above is a really great reference. I guess the author does skip many proofs but he does explain in later parts how rep theory can describe Hilbert spaces which describe states of n identical electrons, say, within an atom. It was good to understand that! I mean, I know it is basically what @Carlo Beenakker wrote in his nice but short answer below, but from my own perspective, it was only after I was able to get a hold of Wybourn's book and read chunks of it that things finally began to make sense to me :).
Best Answer
Young diagrams or Young tableaux (the latter being diagrams with integers in each box) are used in particle physics to describe the states of indistinguishable fermions or bosons: $n$ indistinguishable particles, each of which can be in one of $m$ states form an irreducible representation of $U(m)$. A Young tableaux in which each box has integer $\leq m$ encodes one representation.
See page 12 and following of these lecture notes, or see Introductory Algebra for Physicists.