Equating Schur functions

We wonder if there is a way to make all Schur functions in all representations equal. This is impossible for fixed value of time variables, but can be achieved for averages. It appears that the corresponding measure is just Gaussian in times, which are all independent. The generating function for the number of Young diagrams does not straightforwardly appear as a product, but is reproduced in a non-trivial way.

1. Schur functions and their close relatives are acquiring a sort of a central position in advanced theoretical physics.This is related to their role in description of interplay between representations of linear and symmetric groups on one side and (super)integrability properties of the non-perturbative partition functions on the other.
Schur functions S R {p} depend on infinite set of time-variables p k , k = 1, . . ., ∞ and are labeled by Young diagrams R -the ordered integer partitions of their size 2. In this paper we pose a very simple question.The number of Young diagrams, i.e. of partitions or of irreducible representations of linear sl(∞) and symmetric S(∞) groups, have a very simple generating function It is natural to ask, if this sum can be somehow related to Cauchy identity [1] R which has the same summation domain, but the weight includes a product of two Schur functions.The questions is, can we make this weight unity?At the level |R| = 1 this looks simple: S [1] = p 1 and it is enough to put p 1 = 1.The difficulty becomes obvious already at the level |R| = 2, where there are just two diagrams and two Schur functions Then Thus there is no way to choose times so that all S R {p} = 1.In general the coefficient of r! , and all p 1 = √ r! with different r are different.
3. However, in quantum field theory there is a simple way out: instead of evaluating S R at given values of arguments p k one can consider averages.The question then is if we can find a measure dµ{p}, such that In our above example we get We note in passing that the boldfaced sub-sequence is dear to the heart of every string physicist.Also amusing is that the entire first line p n 1 appears consisting of the numbers of Young tableaux -this is the sequence A000085 from ref. [2], where a number of alternative interpretations is also listed.There is no obvious reason for these remarkable properties in the given context.
• Looking at this table, one can observe the simple recursions for odd times and for even times.The one for m = 1 is a well known identity for the numbers of the Young tableuax.
Note that the first lines should be multiplied, not added, in order to coincide with the last line.
6.This note reports a funny observation about eliminating the factors d R = S R {δ k,1 } from Cauchy identity and efficiently averaging Schur functions to unities -all at once.Somewhat surprisingly, Gaussian measures appear sufficient to solve this problem.It can possess interesting generalizations to Jack and Macdonald polynomials [5] and, most interesting to the still hypothetical 3-Schur functions [6], associated to plain partitions, when Cauchy identity would get related to the MacMahon generating function n (1 − q k ) −k .As a bonus we get an integral formula for the number of Young tableaux -they appear to be simple Gaussian averages.Intriguing are the possible relations to matrix models [4] and to their superintegrability properties [7] -but this requires restriction to finite matrix size N (what nullifies some of the Schur functions) and a matrix-model interpretation of our new measures, which are Gaussian in times rather than in matrix and eigenvalue variables X = diag(x i ), usually used in matrix models a la [7] to parameterize the Miwa locus p k = tr X k .

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It is straightforward to consider further examples at other levels |R|, what leads to the following conclusions about the relevant measure:• The variables p k are all independent: .the measure is a product dµ{p} = k dµ k (p k )• The first averages, which are implied by(4