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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 :).

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    $\begingroup$ H. Weyl, Group theory and quantum mechanics, Chapter V. $\endgroup$ Nov 15, 2022 at 13:47
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    $\begingroup$ An interesting book is A Course on the Applications of Group Theory to Quantum Mechanics by Irene Verona Schensted. She was married to Craige Schensted, famous for his contribution to the RSK algorithm in algebraic combinatorics. $\endgroup$ Nov 15, 2022 at 16:19
  • $\begingroup$ @RichardStanley, this book seems out of print, alas. $\endgroup$
    – Malkoun
    Nov 17, 2022 at 17:40

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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.

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  • $\begingroup$ Thank you! By the way, the set of rules for decomposing the tensor product of 2 irreps look a bit different in the 2 sets of notes you have provided. Are they equivalent? In one set of notes, there seem to be a column counting rule and a row counting rule (which I didn't understand very well, in terms of wording). In the other, one starts from the top right square and goes to the left traversing the 1st row, then goes to the 2nd row on its right-most square and goes to the left traversing the 2nd row and so on (all the while counting the total number of a's and b's). Are they equivalent? $\endgroup$
    – Malkoun
    Nov 16, 2022 at 2:15

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