# Young's natural representation of the symmetric group

The literature on the representation theory of the symmetric group contains some terminology that I find puzzling, and I am wondering if someone here knows the full story.

One of the standard ways to construct the irreducible representations of the symmetric group is to define Specht modules. This construction produces an explicit basis for the modules. If one now writes down the representing matrices with respect to this basis, the result is often referred to as Young's natural representation.

From a modern point of view at least, this terminology seems a little strange because it attributes "the same thing" to both Specht and Young. Now one possible explanation is that when Specht and Young were working on this stuff, it didn't seem like the same thing, and it's only today that they look the same. Indeed, in Specht's paper, he writes:

Zwischen den Youngschen Arbeiten und der vorliegenden bestehen daher kaum irgendwelche Zusammenhaenge ausser den rein ausserlichen, die darauf beruhen, dass die hier verwendeten kombinatorischen Hilfsmittel haufig auch von Herrn A. Young, freilich zu ganz anderen Zwecken herangezogen werden.

My German is poor but I think this translates to:

Between Young's work and the present work there exist hardly any connections except the purely superficial one that the combinatorial tools used here are also used by Mr. A. Young, albeit for entirely different purposes.

I tried to look up Young's papers, but found them daunting, and in particular I could not immediately locate anything that looked like "Young's natural representation." Apparently I'm not the only one who is daunted by Young's papers, because here's a quote from some lecture notes of G. D. James:

The representation theory of the symmetric groups was first studied by Frobenius and Schur, and then developed in a long series of papers by Young. Although a detailed study of Young's work would undoubtedly pay dividends, anyone who has attempted this will realize just how difficult it is to read his papers. The author, for one, has never undertaken this task, and so no reference will be found here to any of Young's proofs, although it is probable that some of the techniques presented here are identical to his.

So my question is, can someone point specifically to a place in Young's papers where he discussed what we would nowadays call "Young's natural representation"? And does anyone know the history of how the term "Young's natural representation" came to have its current meaning?

Thanks to Richard Stanley for the pointer to Garsia and McLarnan's paper, Relations between Young's natural and the Kazhdan–Lusztig representations of $$S_n$$, Advances in Math. 69 (1988), 32–92.

Young's fourth paper ("QSA IV") is:

Alfred Young, On Quantitative Substitutional Analysis (Fourth Paper), Proc. London Math. Soc. (2) 31 (1930), no. 4, 253–272.

Note that the year of publication is slightly confusing because the running head of the paper itself says "Nov. 14, 1929" but the volume of the journal was actually published in 1930. Following MathSciNet, I have given the year as 1930, but I have also seen citations of the paper that give 1929 as the year.

I have stared at QSA IV for some time but have failed to fully decipher the notation, so I hesitate to personally vouch for the claim that it describes the same matrix representation that one gets by taking (the usual basis for) Specht modules. However, in addition to Garsia and McLarnan, the book Substitional Analysis by Daniel Rutherford—which by the way is a very useful guide to Young's work—also states that QSA IV describes a recipe for (what we now call) Young's natural representation, so I believe that this claim is true.

It is understandable to me that Specht regarded his work as different from Young's. What I can say from my (limited) understanding of QSA IV is that Young did not construct anything resembling Specht modules, and that Young's recipe for constructing representing matrices came from considering the action of the symmetric group on (what we would now call) primitive idempotents.

There is an interesting remark that Garsia and McLarnan make in their paper (writing in 1988):

Very few authors today have much familiarity with Young's natural representation. The various presentations of Specht modules and the work of Garnir tend to hide the simplicity and beauty of Young's construction. … Young's natural can be constructed at once by a very simple combinatorial procedure which applies to all permutations. Moreover, the proof that the procedure is valid is actually quite short and elementary.

One reason that Young's construction of his natural representation "fell off the radar" for a while may be that the exposition in Rutherford's aforementioned book does not follow Young's construction exactly. Young derives the natural representation first and only later derives the orthogonal representation, whereas Rutherford does it the other way around, making the natural representation seem like an afterthought. In his review of Rutherford's book in the Bulletin of the American Mathematical Society, G. de B. Robinson even goes so far as to say:

[T]he natural representation appears as an anti-climax. Though reference to it had to be included, this reviewer would have preferred that it be in an appendix. The material of §§28–31 has historical and actual value, but it serves to obscure the magnitude of Young's real achievement, the orthogonal representation.

I took a quick look at Robinson's own book on the representation theory of the symmetric group and I think he does not bother at all with Young's natural representation. Anyway, it seems that for the reader who wants to understand Young's natural representation without going through Specht modules, Garsia and McLarnan's account is the most readable one.

• Rutherford’s book contains details on Young’s representation, and is reasonably easy to read. – Dima Pasechnik Nov 17 '18 at 2:59