# Historical question about MacMahon theorem, Wronski relation etc…

Seemingly the same fact goes under several names: MacMahon master theorem, Wronski relation, unnamed fact about symmetric functions - I wonder what is history and what should be correct name'' ?

Let me remind the facts and formulate questions more precisely:

Denote by $e_k$ elementary and $S_k$ complete symmetric functions, then it is well-known that

Fact (with NO name (e.g. Wikipedia) ): $\sum_i S_{k-i} e_i = 0$ or in terms of generating functions: $(\sum_{i=0...\infty} t^i S_i ) (\sum_{i=0...n} t^i e_i ) = 1$.

Proof: $1/(\sum_{i=0...n} t^i e_i )= 1/\prod_i (1-t\lambda_i)$ and using expansion in the geometric progression we get $(\sum_{i=0...\infty} t^i S_i )$.

Question 1 It seems this relation is too simple to bear someone's name, correct ? Nevertheless is its history known? Who was (were) the first to mention?

Let $A$ be a matrix.

Fact (MacMahon Master theorem or Wronski relation). $det(1-tM)(\sum_i Tr t^i S^i(M)) =1$, where $S^i$ are symmetric tensor power of $M$.

Proof. Assume $A$ is diagonalizible, then it is can be proved in the eigenbasis and is exactly the previous fact. Any complex matrix can be approximated by diagonalizble and since it is algebraic identity this is enough.

It is clear how MacMahon enters this story - he found combinatorial application and describes them in his book, see e.g. Wikipedia

The name of Wronski is used e.g. in papers:

Toru Umeda Application of Koszul complex to Wronski relations for U(gl)

or in many papers by Russian authors on quantum groups e.g. A. Isaev, O. Ogievetsky, P. Pyatov, On quantum matrix algebras satisfying the CayleyHamilton-Newton identities page 9.

Question 2: How Wronski enters the story ? Should the relation for complete and elementary symmetric functions be also called "MacMahon" or "Wronski" relation, since it is equivalent to their theorem ?

PS

The motivation for asking is writting some Wikipedia article (by the way comments are welcome) as well proper refering in my own articles.

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I think the $e_i$ are usually not defined with the signs you've given them. Also, you use both $A$ and $M$ for what I think are the same matrix, and the version of the MacMahon Master Theorem I'm familiar with has $t$ a diagonal matrix of indeterminates rather than a scalar indeterminate. –  Qiaochu Yuan Aug 4 '12 at 14:50
Oops I missed signs, and A =M thanks! Concerning t or diagonal T it is not important. You can absorb them in matrix M. One adds "t" just for convergence. –  Alexander Chervov Aug 4 '12 at 15:20

I will answer only the second question. Full disclosure: I wrote much of the WP MacMahon Master Theorem article.

No, you should not call a trivial symmetric function identity after MacMahon (which was already used by Euler and Newton, see e.g. Newton's identities). His theorem is much deeper than you make it seem, due to its relations to combinatorics. This linear algebra proof which you view as trivial is in fact a relatively recent observation, which (to my knowledge) appeared in print only when the MMT was generalized to various non-commutative rings, etc.

To make a general point, the history of mathematics is complicated and should be treated in context rather than with the present day hindsight. My favorite example is Cayley's formula which was originally proved by Carl Borchardt (i.e. before Cayley) as an elementary consequence of Kirchhoff's determinant formula. This is also how we teach it these days. So, should we rename Cayley's formula after Kirchhoff or Borchardt? The best way is to leave things as is - it was Cayley's paper (which references Borchardt's), which, after all, made a major impact on the developments of combinatorics.

More egregiously so, Cauchy's identities for symmetric functions appear in the same setting as the product formulas you mention. Richard Stanley told me that he checked every page of Cauchy's collected papers and is now convinced that Cauchy never proved these. So, should we rename them? No, I say keep the name. We are used to it by now. Same with "MacMahon" and "Wronski". There is really no need to name and re-name things unless the current terminology leads to some kind of confusion... (see here for a story of one such renaming which possibly saved lives).

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Thank you very much for your kind reply. Macmahon application to Dixon identity is great, but I think the fact itself and proof I mentioned were known to many people without knowledge about Macmahon unfortunately. E.g. the fact is mentioned in "Loop groups" by Pressley Segal as a footnote (may be Russian translation only? ). My master thesis contain this proof and actually more general facts arxiv.org/abs/q-alg/9703017 –  Alexander Chervov Aug 4 '12 at 11:25
I see. Yes, this often happens when same/similar result is being obtained in parallel in different field. MacMahon's contributions were in the 1920s. I stand by my view that no history of MMT and symmetric functions needs revision, but would be interested to learn more about Wronski's work. –  Igor Pak Aug 5 '12 at 5:03
@Igor I certainly agree with you, that revision is useless. I'll try to find out about Wronski and hopefully add a comments later. –  Alexander Chervov Aug 8 '12 at 5:58