Averaging method in combinatorics of symmetric polynomials

We elaborate on the recent suggestion to consider averaging of Cauchy identities for the Schur functions over power sum variables. This procedure has apparent parallels with the Borel transform, only it changes the number of combinatorial factors like $d_R$ in the sums over Young diagrams instead of just factorials in ordinary sums over numbers. It provides a universal view on a number of previously known, but seemingly random identities.


Introduction
Borel transform [1] is an important tool in the study of non-perturbative phenomena. It is used to extract information from divergent series which typically arise in perturbative expansions. Non-perturbative partition functions depend on extra parameters, like the choice of integration contour, this dependence seems to disappear in perturbative expansion, but is in fact traded for divergence of the series. New parameters appear as an ambiguity in the Borel transform. All this is well seen already in the simplest example of the exponential Borel transform Now, it is clear that the r.h.s. depends on the integration contour, and there is an ambiguity so that the sum of divergent series is defined modulo this "instantonic" term. This standard argument, lying in the base of entire "resurgence theory" [2] is, however, technically based on exponential functions, and their Gamma and, more general, hypergeometric generalizations. They have, of course, a straightforward q ("quantum") and t deformations [3][4][5], but this is not the only important direction to look at. In modern integrability theory [6], which is a crucially important part of non-perturbative physics, an important role is played by combinatorial generalizations of the factorial, like the quantity d R , which extends the factorial from integers to partitions (Young diagrams). It seems important to look at the versions of Borel transform that deal with such additional combinatorial structure, hence, the name combinatorial Borel transform. Surprisingly or not, exponentials still show up in the combinatorial Borel transforms, through Cauchy identities [7], thus most probably this is not the last step. However, it can prove sufficient for the needs of integrability theory, at least. CBT Figure 1: A picture, illustrating the concept of the combinatorial Borel transform (CBT) of the series. First, we introduce into the original series an auxiliary function Sr{p} such that its average overp with an appropriate measure can change the coefficients in a desired way: say, ar = r! −→ br = 1 in (1) or, more general, ar = d m r = (r!) −m −→ br = ardr = d m+1 r = (r!) −m−1 . For traditional applications, the main point is that the underlined sum can converge better than the original one in the left upper corner due to Sr{0} = br cr ∼ 1 r! , but for us this is just one application of many. Then, we extend the summation domain {r} −→ {R} and introduce an additional auxiliary function S R {p} so that the sum in the box is easily calculated, for instance, with the help of the Cauchy identity. As an additional bonus, this converts the Cauchy relations into ineteresting formulas for the series F {p} at the right upper corner, which can a priori look non-trivial, but are actually thep averages of simple identities. Technically, what we do is a substitution of the Gamma-function average q r −→ q r Γ(r + 1) = r! · q r which inverts the left vertical arrow in the usual Borel transform, by a smoother and richer Gaussian average, at the price of extending the sum over integers r to that over all integer partitions (Young diagrams) R. Such generalization is not unique, which opens a way to obtain a whole class of formulas by slight changes of the Gaussian measure.
As to possible techniques which could provide a route to the combinatorial Borel transforms, in [8] it was suggested to consider an interpolation between two different kinds of sums over all Young diagrams by taking a specially adjusted average of the Schur functions over the power sums p k . The emphasis, however, can be put not so much on averaging, but on coexistence of generating functions differing by the non-trivial combinatorial weights d R . It is a common place to consider such families differing by the number of factorials in the weights: these are just related by the standard exponential Borel transform. It is an interesting direction to study a more tricky combinatorial difference. The goal of the present paper is to exploit this idea and extend it further in various directions. The paper is organized as follows. In section 2, we explain the basic of our approach, and demonstrate how the averaging method works for evaluating sums. In section 3, we extend the method to reproduce multiple combinatorial sums. In section 4, we discuss evaluating bilinear averages in parallel with the property of strong superintegrability in matrix models. Section 5 contains short notes on relations of the sums considered in the paper with the Ramanujan sums. At last, section 6 discusses the cases when the sums are divergent, while section 7 contains some concluding remarks.
Notation. We use the notation S R {p k } for the Schur functions, which are symmetric functions of x i , and are graded polynomials of power sums p k := i x k i . They are labeled by partitions, or, equivalently, by Young diagrams R with l R parts: The corresponding skew Schur functions labeled by two partitions R and Q ∈ R are denoted as S R/Q {p k }, while the Macdonald polynomials are denoted through M R {p k }. We also denote R ∨ the conjugate partition (Young diagram).
Our main quantity in this paper is d R , which is equal to S R {δ k,1 } in terms of Schur functions, and is equal to dim R /|R|!, where dim R is the dimension of representation R of the symmetric group S |R| . |R| = l R i R i is here the size of partition R with l R parts. The theory of symmetric functions can be found in [7], and the theory of symmetric groups, in [10]. We also use notation R for the sum over all partitions R including the empty set, while R m means summing over all partitions of integer m.
2 Basic example of averaging method Eq.(3) counts Young diagrams, or the ordered integer partitions, while (4) is the Cauchy identity for the Schur functions, i.e. for the characters of linear groups GL N . Technically, upon choosing p k =p k = δ k,1 in (4), the difference between the two formulas is in the power of d R := S R {δ k,1 }: they are the sums of d 0 R and d 2 R , accordingly. Generally, |R|!·d R = dim R are integer-valued dimensions of representations of the symmetric group S |R| [10], and the sum looks somewhat terrible. The first terms in (6) are (4!) m q 4 + 6 m + 2 · 5 m + 2 · 4 m + 2 (5!) m q 5 + + 16 m + 2 · 10 m + 2 · 9 m + 4 · 5 m + 2 (6!) m q 6 + 2 · 35 m + 2 · 21 m + 20 m + 2 · 15 m + 4 · 14 m + 2 · 6 m + 2 (7!) m q 7 + + 90 m + 2 · 70 m + 2 · 64 m + 2 · 56 m + 42 m + 2 · 35 m + 2 · 28 m + 2 · 21 m + 2 · 20 m + 2 · 14 m + 2 · 7 m + 2 The more surprising is that, at some particular values of m, one gets sensible formulas. The best known is the quadratic case, m = 2, where the generic theory of finite groups implies that the sum of the squared dimensions is always a dimension of the group. In the case of S r , this gives and leads to the particular case of (4) For m = 0, one gets a far more transcendental, still comprehensible formula (3), which just counts the numbers of representations of S |R| .
Amusingly, the case of m = 1 is also simple: gives rise to sequence A000085 from [9] which counts the numbers of Young tableaux of a given size r, and this is in perfect accordance with the fact that the dimension dim R is equal to the number of Young tableaux associated with the diagram R. For the exponential generating function of this sequence, see [9, sequence A000085 and references therein], Moreover, this sum can be lifted to a sum of single(!) Schur functions In the language of [8], this follows from taking average of the Cauchy identity (4) over the variablesp k with the measure Hereafter, the average with a given measure dµ over variables p k , k = 1, 2, . . . is defined as and normalized in such a way that 1 = 1.
Average (13) converts the Schur polynomials into 1, S R {p} A = 1, and is Gaussian, so it is trivial to take averages of the r.h.s. of (4), is just the r.h.s. of (12). The sum (11) can be also represented as i.e.
Note that while (3) is a generating function for numbers of Young diagrams, (11) is an exponential (i.e. weighted with additional factors 1 r! ) generating function for numbers of Young tableaux. Another important difference between counting diagrams and tables is that the latter, being related to dimensions, admit a quantum deformation, see sec.3.3 below.
According to [9], no interpretations for ξ m are yet known for m ≥ 3.

Simple generalizations 3.1 Derivation of various sum formulas by the averaging method
The averaging method can be easily modified to handle more examples. Say, we can ask the average S R = 1 only for even diagrams R, i.e. with all rows of even lengths, and vanishes otherwise. The corresponding measure is still Gaussian, and the counterpart of (12) is R even Let us demonstrate how to prove that S R B is non-zero and equal to 1 only for even partitions. To this end, we note that the measure B is obtained from A by the shift of times p k → p k − (−1) k . In other words, we have The choice of p k = 1 is equivalent to choosing only one non-zero symmetric variable x 1 = 1, and the sum P (−1) |P | S R/P {1} is not zero only when R is an even partition, which follows from the representation of the skew Schur function S R/P = T x T with the sum running over all tableaux T of shape R − P , see details in [7,Eq.(5.12)].
All further examples in this section are proved analogously. Note that checking this kind of claims is much easier than proving them: one can just calculate simple Gaussian averages of the Schur functions at any concrete example. Now, in order to pick up only even columns, one need to make transposition, which is equivalent to changing signs of all the time variables: the relevant measure, which gives S R = 1 for all even-column diagrams and zero otherwise is and R ∨ even If c R counts the number of rows of odd length in R, then, introducing a formal parameter t, Counting the number of columns of odd length is provided by the same formula with all the signs of p k inverted, see (34) below.

Extension to skew functions
Now, one can easily obtain similar formulas for sums of the skew Schur functions S R/P . To this end, one needs a slight generalization of the Cauchy identity (4): Taking the average of this identity overp k with measure (13), one obtains Let us prove that the r.h.s. of this formula is given by with literally the same p-dependent factor as in (12). To this end, we convert this formula with the Schur functions S P {p }. This gives where we once again used the Cauchy identities (4) and (27) at the l.h.s. and at the r.h.s. Now it remains to calculate the Gaussian integral at the l.h.s. as it was done in (12), and also to use (12) to evaluate the sum at the r.h.s. Hence, formula (29) is correct, and we finally obtain from (28) We can now make one more check that (31) is correct. Convert this formula with S P {p } and use the Cauchy identity (27) and the definition of the skew Schur polynomial Both sums can be calculated using (12) in order to validate this formula. We emphasize once again that, according to this argument, the p-dependent factor at the r.h.s. is literally the same as in (12), and the same will be true for the skew Schur counterparts of all other identities in this section.

A collection of sum formulas
To summarize, we can reproduce the whole collection of the single-Schur sums from [7, secs.1.5,3.4,3.5].
They are naturally divided in three blocks, and the formulas are given also in terms of symmetric variables x i such that p k = i x k i .
1. The first one is already familiar where c R (r R ) is the number of rows (lines) of odd length, and ν R = i (i − 1)R i . The first three lines follow from the last two lines upon putting t = 0, 1.

2.
The second block involves the skew Schur functions: for arbitrary Q.
3. The third direction is generalizations of these formulas to the Macdonald polynomials. We do not provide relevant averages for this case in order to avoid overloading the text with unnecessary complicated formulas. Still, we list the results. Denote The full product H R is just the coefficient that stands in the sum of the Cauchy identity:

The Gaussian measure
The main Gaussian measure (13) itself enjoys a curious property: it can be expanded into a simple sum over the Schur functions containing only self-conjugated Young diagrams: where h R is the number of hooks that form the Young diagram R, and ( α| β) is the Fröbenius notation of the Young diagram [7].

Calculating various sums
Note that the averaging procedure allows one to evaluate various sums of combinatorics quantities by evaluating the Gaussian moment. For instance (using the Fröbenius formula [10]), i.e. (see also [7, Example 11, sec.I.7]) are products of the Gaussian moments. Here ∆ is a partition with parts δ i , p ∆ := l∆ i=1 p δi , and ψ R (∆) is the character of the symmetric group S |R| in the representation R. Also etc.
where N R P Q are the Littlewood-Richardson coefficients, and we used that [7] S R/Q = P N R P Q S P (56)

Strong superintegrability and W -operators
As we saw in the previous sections, the Schur functions form a full set of polynomials that have simple averages, this phenomenon is called superintegrability [11]. Similarly, one can construct a full set of bilinear combinations with simple averages too. This phenomenon is called strong superintegrability, and first we remind how this phenomenon looks like in the case of the Gaussian Hermitian matrix model.

Strong integrability in Gaussian Hermitian matrix model
The bilinear correlators in the Gaussian Hermitian matrix model are generated by the action of the W -operatorŝ W (−) on the Schur function S R as functions of P k := Tr H k [12], where H is the matrix that is integrated over in the matrix model, and, by the matrix derivative, we imply the derivative w.r.t. matrix elements of the transposed matrix: ∂ ∂H ij = ∂ ∂Hji . As we demonstrated in [13], the averages (57) can be reduced to a correlator of the form where the polynomials K R form a complete basis, and celebrate the property Examples of these polynomials can be found in [13,Appendix]. They are obtained by using integration by parts and the action of the S Q {Ŵ (−) k }-operators on the Gaussian measure. Note that, throughout the paper [13], we discussed another basis of polynomials, K ∆ , the two being related by the Fröbenius formula where z ∆ is the standard symmetric factor of the Young diagram (order of the automorphism) [10].

Strong superintegrability in the p-Gaussian model
In the p-Gaussian model with the measure (13), which is under our consideration in this paper, there is a formula for the bilinear correlator similar to (57), however, in this case, one has to deal with the operators 3 W (0) ∆ . They are defined asŴ where Λ is a matrix such that the p k -variables are parameterized as p k = Tr Λ k , and the invariant operatorsŴ ∆ require the normal ordering, otherwise they would depend on the size of the Λ matrix [14]. In the case of operatorsŴ where the eigenvalue φ R (∆) is manifestly expressed through the symmetric group characters ψ µ (∆) by [15][16][17] Now using the identity which follows from the orthogonality relation one finally comes to the formula Now, using integration by parts, one could again recast the action of the S P (Ŵ (0) k )-operators into the action to the measure (13) in order to produce a complete set of polynomials K P {p k } of p k : where we use the left arrow to denote the operator acting to the left. Thus, one could construct this way the However, there is a problem in doing this. To see the problem, let us consider the simplest case of the operatorŴ (0) [1] , which is manifestly given bŷ When integrating by parts, this operator acts to the measure (13) as The last sum requires some regularization ∞ a=1 a −→ σ reg , and, when acting to the measure, this operator produces a function, not a polynomial. Still, the Schur functions are eigenfunctions of this operator. If one acts to the Gaussian measure (13) with this operator, one obtains i.e.
At the second level, the formulas are even more involved. Indeed, consider the operator [2] , which is given by In this case, there are no infinite sums emerging from differentiating within the operator, however, in contrast with the standardŴ (0) [2] -operator [14,18], whose eigenfunctions are the Schur functions, the eigenfunctions of [2] do not form a complete basis in the space of graded polynomials of p k . For instance, at grading 3, where the matrix is degenerate: it has rank 2. Here we labelled the indices: The action of [2] to the Gaussian measure (13) is now cubic in p k : i.e.
This is a general rule: any K R is a polynomial of degree |R|+1 in p k , however, it involves p k with arbitrary large index k, and also involves infinite sums, which require a regularization. All this requires a careful treatment, which we postpone for a separate publication.

Relation of sum formulas to Ramanujan sums
Another amusing fact about the two nice cases (9) and (11) where the sum goes over a, which are coprime with m, and it is always a positive integer. Counterparts of (9) and (11), not literal, still surprisingly similar appear in two ways: with plethystic and ordinary exponentials. Defining the plethystic exponential of a function f (x) as is the quantity (6). In terms of ordinary exponentials, one gets In fact, and are always expressed through the exponentiated finite sum over divisors of m. Thus, these are simple formulas, which, unfortunately, have nothing to do with our ξ m at m > 2.

Hurwitz tau-functions
The combinatorial Borel transform allows one to enrich the set of the Hurwitz τ -functions [14,20] to make the power of d R and the number of Schur functions independent: suppose m > n, then Averaging here is performed over the m − n sets of time variables. For example, a direct counterpart of (1) would be At the same time, from what we know about the spectrum of dimensions, It is important that this is a discrete spectrum and multiplicities stabilize at large r. The smallest dimensions, like listed in (89) are made from polynomials like r m m! 1 + O(r −1 ) , but at another end of the spectrum the growth is already factorial. In particular, the biggest dimension D r of all dim R at the given level |R| = r is restricted at large r by [21]: where α 1,2 are some constants. Still this growth is much weaker than r! in the numerator of (88). Thus contributions with all m are divergent at large r = |R|, and one can expect the whole series to be ambiguous. The same is true for any particular term in the expansion (88). We will deal with the divergent sums ξ −ν = R q |R| d ν R with negative powers of d R as with formal power series. Note that they have integer-valued coefficients, provided by inverse powers of representation dimensions d R = |R|! dim R with multiplicities read from (7). Their plethystic logarithms Ψ ν (k) also have an interesting interpretation. Indeed, consider Remarkably, all the coefficients Ψ ν (k) are positive integers, and they can be interpreted as the numbers of subgroups of index k in the fundamental group of a certain fiber space [22]. They are a little more complicated (multi-linear) combinations of powers of above-mentioned integers d R = |R|! dim R so that in the decomposition the spectrum η n of Ψ ν (k) is more complicated than the spectrum d −1 R of ξ −ν , with some of elements η n coming from products of d −1 R , and the coefficients c n (k) are not obligatory positive (despite Ψ ν (k) is positive), and, hence, they can not play a role of multiplicities.
One can compare the spectrum of ξ −ν (the values of d −1 R ), with the spectrum of Ψ ν (k) appearing in (92) (see Table 3 of [22], note that Ψ ν (k) are denoted as R ν (k) in [22]): where the quantities in the boxes are the products of the powers of the primary elements of the spectrum from (93), and for each Ψ ν (k) only η n corresponding to non-zero c n in (92) are specified.

Conclusion
In this paper we raised the trick of [8] to the level of a method, applicable to a large variety of functions and providing a unified treatment of the so far sporadic set of amusing identities. We emphasize its conceptual relation to the Borel transform, which is used in physics to explain and put under control the ambiguity of divergent series. As we demonstrate, the abilities of our suggested combinatorial Borel transform are much wider and applications are not restricted to divergent series. There are plenty of different directions to develop these ideas, and also a number of conceptual questions to resolve, of which we mention just two.

Basic open questions:
1) When (for what conditions S R = α R ) does a simple averaging procedure exist? When is the measure Gaussian?
2) How does a trivial Gaussian measure like (13) converts non-trivial series like F {p} = R q |R| d 2 R S R {p} into a triviality: F {p} = e q ? What is the role of triangularity (only p k with k ≤ |R| affect the coefficient of q |R| )? To what extent and in what sense can the averaging procedure (the combinatorial Borel transform) be invertible?
Clearly this integration is already radical enough to allow for any simple inversion.
3) In the paper, we only started exploration of the combinatorial Borel transform. In particular, we did not discuss the role of analytic continuation within this context. Moreover, we did not discuss in detail when the sums under consideration have different possible types of growth, in particular, a factorial divergence and other types of divergencies (for exception of comments in sec.6). This is especially important in the case of the Hurwitz τ -functions in order to reveal the non-perturbative nature of the large order behaviour in these.
We hope that this new, more general look on a combinatorial generalization of the Borel transform would lead to new insights or, at least, to new amusing relations and formulas, which look artificial from other points of view. There are many of this kind, besides the ones that we mentioned, and now we have a new option to interpret and understand at least some of them.