It’s often said that part of the unique character of mathematics is that it builds on itself. Old results may be forgotten, but are only very rarely found to be incorrect. But changes in the ‘expected general background’ make many old papers impenetrable. For example, a long standing conjecture of Foulkes was introduced in a paper with the title ‘On the concomitants of the quintic and sextic up to degree four in the coefficients of the ground form‘. How many algebraists working today would immediately know even roughly what it is about? Of them, I’m sure still fewer would expect to read his paper with any pleasure. Certainly I cannot.
The purpose of this post is to do a small, but detailed, translation exercise on a paper of Newell from 1951, with the title ‘A theorem on the plethysm of -functions‘. This is a relatively late publication from the stable of British algebraists working on invariant theory, so one might hope it would be fairly accessible.
Fix and let be a partition of . Newell’s paper is cited in an important paper of Weintraub for the pair of results stated verbatim below:
Weintraub defines all his notation (which was standard for the time) clearly. For example, denotes the irreducible representation of the symmetric group labelled by the partition , and is the analogue of the plethysm product for the symmetric group, with the variables swapped. Writing for the Schur function labelled by the partition , an equivalent statement in the language of symmetric functions is:
Even a careful visual inspection of Newell’s paper reveals nothing that looks remotely like Weintraub’s statement (or my restatement). But in fact, the upper displayed equations are a special case of Newell’s Theorem 1, stated verbatim below:
If where and are integers then for any integer
To be fair to Newell, is defined earlier: one reads ‘… where is defined from the multiplication of -functions by means of ‘. So alles klar?
Translation up to Theorem 1
Much of Newell’s notation was standard at its time: is the Schur function and is the plethystic product corresonding to Weintraub’s . (So again, the order is reversed compared to .) The hypothesis may still seem a little mysterious to modern eyes, since a general plethysm is certainly not multiplicity-free: nowadays we might write ‘ where for each partition of ‘. Only the remains to be understood: Newell defines for natural numbers , but then uses general partitions as the coefficients. Even in the special case, and with the benefit of knowing what is meant, his definition seems highly unclear to me.
A clue to the correct interpretation is given by the start of the sentence defining quoted above: ‘It is known [(2) 349] that if then …’. This must be parsed bearing in mind the summation convention that the repeated letters and are summed over. So, in modern language, it says: ‘if then
(I have included the parentheses around as a small editorial mercy.) After staring at this for a while, I decided it must express the following symmetric function identity:
where is the sum of all Schur functions labelled by partitions obtained from by removing a box. (This identity is proved below, using the symmetric group.) Reciprocally, we have
Thus is the adjoint to multiplication by , and Newell’s are the corresponding coefficients. Generalizing freely, we can guess the correct definition of .
Definition. Given partitions we define
If we believe this is correct, then after one more piece of guesswork where we amend in the statement of Theorem 1 to (making it consistent with the above, and also with the analogous Theorem 1A), the conclusion of Theorem 1 becomes
The left-hand side is
Therefore Newell’s Theorem 1 can be stated as follows:
If appears in then has at most parts. On the other hand, by Pieri’s rule, is the sum of all partitions obtained from by adding boxes, no two in the same column. Therefore, in the special case when , we have
Since , the unit symmetric function, we obtain
This is equivalent to the first displayed equation in my restatement of Weintraub’s version of Newell’s result. The second displayed equation can be translated similarly.
Proof of Theorem 1
We prove Newell’s theorem in the symmetric group, replacing with for consistency with Weintraub’s statement. Let be the collection of all set partitions of into sets each of size , and let denote the corresponding permutation module for with basis . In Weintraub’s notation, the permutation character of is .
Consider the restricted module , where permutes and permutes . The maximal summand of on which acts as the sign representation is spanned by the symmetrized set partitions for , where
Observe that unless has at most one entry from in each of its constituent -sets. If and are two such subsets, with entries and in respectively, then
It follows that
where is a sum of irreducible characters of not of the form . By Frobenius reciprocity,
for each partition of . This is equivalent to
which is obtained from my restatement of Newell’s theorem by taking the inner product with . The companion result follows by a sign twist.
No wonder old results are frequently reproved: its invariably less work than reading the original.