# 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.

-
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