Sum rules for characters from character-preservation property of matrix models

One of the main features of eigenvalue matrix models is that the averages of characters are again characters, what can be considered as a far-going generalization of the Fourier transform property of Gaussian exponential. This is true for the standard Hermitian and unitary (trigonometric) matrix models and for their various deformations, classical and quantum ones. Arising explicit formulas for the partition functions are very efficient for practical computer calculations. However, to handle them theoretically, one needs to tame remaining finite sums over representations of a given size, which turns into an interesting conceptual problem. Already the semicircle distribution in the large-N limit implies interesting combinatorial sum rules for characters. We describe also implications to W-representations, including a character decomposition of cut-and-join operators, which unexpectedly involves only single-hook diagrams and also requires non-trivial summation identities.


Introduction
As emphasized quite recently in [1] following the consideration in [2], a key feature of the Gaussian measures is that really nice are the averages of characters. In particular, in the Hermitian matrix model [3][4][5][6][7][8][9][10][11][12][13][14][15][16], the average of a character (Schur function) χ R [M ], which is a function of eigenvalues of the matrix variable M , is again a character, actually, a dimensions D R (N ) = χ R [I N ] = χ R {N } of the representation R of gl N . For monomial non-Gaussian measures like e tr M s dM and appropriate choice of integration contour, the coefficient contains χ R {δ k,s } [17]. In the present paper, we concentrate on the case of Gaussian measures with s = 2. We use the square and curled brackets in order to denote the character as a (symmetric) function of matrix eigenvalues (the first Weyl formula) and as a function of time variables tr M k (the second Weyl formula) accordingly. The matrix eigenvalues are often called Miwa variables within this context.

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As an example of (1.1), Tr M 2 = N 2 and (Tr M ) 2 = N imply that which are dimensions of the symmetric and antisymmetric representations [2] and [1,1]. The only non-trivial ingredient of the theory is the coefficient, which is actually a ratio of characters at two peculiar points in the space of time-variables, p k = δ k,2 and p k = δ k,1 . It is this R-dependent coefficient, which makes the matrix model somewhat non-trivial. At the same time, the character-preserving property can be considered as a defining feature of the Gaussian measures, and can serve as a key for the definition of various deformations of the Hermitian model defined by change of the Schur functions to other systems of orthogonal symmetric functions [18,19]. In more detail, the partition function of the Hermitian matrix model [3][4][5][6][7][8][9][10][11][12][13][14][15][16], i.e. the Gaussian average over N × N Hermitian matrices M , can be decomposed into a sum over all Young diagrams R: From now on, for the sake of simplicity, we put µ = 1, it can be easily restored by dimensional analysis. In (1.2), the character χ R {p} is a polynomial of the time variables p k , labeled by the Young diagram R: where, for the Young diagram ∆ parameterized in one of the two ways ∆ = [δ 1 , δ 2 , . . .] = [1 m 1 , 2 m 2 , . . .], and ψ R,∆ is the symmetric group character, and ϕ R,∆ , its sometimes more convenient rescaled. The symbol is used in Hurwitz theory to denote restricting to diagrams of a given size, and |R| is the number of boxes in the Young diagram R. The N -dependence at the r.h.s. of (1.2) comes entirely from the dimensions χ R {N } = D R (N ), which are the values of characters at the locus where all p k = N . Standing in the denominator of (1.2) is an important representation theory quantity often denoted by d R :

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where l R is the number of lines in the Young diagram R. The "classical" tool to deal with the characters is the Cauchy formula and it turns sufficient to solve surprisingly many problems, involving characters. But not all, as we immediately see in what follows, and as is already clear from (1.2), where the summand is still of the second-order in characters, but not just bilinear: it contains one extra character in the numerator and another one, d R , in the denominator.
In the particular case of N = 2, the only relevant are characters of the symmetric representations R = [r], which, when expressed through the two eigenvalues of M , are just Then the l.h.s. of (1.1) is For symmetric representations, the special characters can be easily obtained from the Cauchy formula atp k = x k : which coincides with (1.7). The need for a character in the denominator of (1.2) becomes nearly obvious, if we extend the Hermitian model to the rectangular complex one [20][21][22], where the variable M is N 1 × N 2 rectangular matrix not obligatory square. Then the average should depend on two parameters N 1 and N 2 in symmetric way. Thus one expects, and gets two characters and therefore there should be one in the denominator as well in order to balance the number of characters at the l.h.s. and the r.h.s. In this paper, we review character sum rules arising from the large-N expansion, and describe a more general approach based on use of cut-and-join operators introduced in [23,24] and playing a big role in the theory of Hurwitz tau-functions. We also comment on extension to various models from the unitary (trigonometric, Chern-Simons, . . . family, including torus knot and MacMahon models.

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2 Sum rules from comparison to Harer-Zagier formula Besides (1.2), there are other explicit generating functions like the Harer-Zagier formula [25][26][27] (for increasingly sophisticated multi-trace generalizations see [28,29]). Note that the same coefficient Z [2m] in front of p [2m] = p 2m can be read off from (1.2), and is provided by the sum Now, substituting (2.2) into (2.1), we obtain a non-trivial sum rule for characters: This formula can be definitely also derived from combinatorics, using that i.e. only the hook Young diagrams R contribute, moreover, since χ R {δ k,2 } is non-zero only at even |R|, we can parameterize the hook diagrams as R = [r − d, 1 d ] at even r. For these Young diagrams, where C m n are the binomial coefficients, and i d denotes the integer part of d/2. However, calculating the l.h.s. of (2.3) does require additional summations over r and d, and the derivation of (2.3) becomes not that immediate.
Actually, it is easy to check that

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where we explicitly write down a few typical terms in the expansion at the r.h.s. Expansion coefficients here are: The simplest here is the p 2m 1 term: it comes from the contribution χ R {δ k,1 }p |R| 1 to χ R {p}, and the coefficient cancels the denominator (1.2) so that the remaining sum is calculated with the help of the Cauchy formula, One can say that the p m 2 term is also easy, since it can be obtained by differentiating the Gaussian integral: However, from the point of view of the sum (1.2), this is already a non-trivial sum rule for the dimensions D N = χ R (N ): For m = 1, it is still trivial: but already for m = 2 it is not: The natural question is how one can handle all the variety of the sum rules for characters, which arise in this way. The question becomes even more interesting, because the result (1.2) of [1] possesses wide generalizations: to various deformations (q−, t− of [30][31][32] and many other) of matrix models [19] and, in another direction, to Aristotelian and other tensor models [33][34][35][36][37][38][39]. At the same time, already at the Hermitian matrix model level, it has important applications, the currently fashionable ones being related to localization formulas  in conformally invariant supersymmetric field theories, which reduce perturbative contributions to certain correlators to those in the Gaussian matrix model averages [62][63][64].

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3 Harer-Zagier formula and planar limit Let us restore the µ-dependence in (1.2), and consider the planar large-N ('t Hooft) limit N → ∞, ν := N/µ =fixed. Then, the model is described by the semicircle distribution of eigenvalues This means, in particular, that, in the large-N approximation (for the sake of simplicity, we put ν = 1/4), Consistency with (2.1) is straightforward: its r.h.s. in 't Hooft limit is equal to and this is exactly what one gets by substituting (3.2) into the l.h.s. of (2.1). More interesting is consistency with (1.2). Of course, it is straightforward to get the leading contribution to (1.2), because the leading large-N asymptotics of χ R {N } = d R N |R| + . . . comes from p |R| 1 and thus is proportional to d R = χ{δ k,1 }, which stands in the denominator. Thus the main large-N asymptotics of (1.2) is controlled by the Cauchy formula: However, this is not what we need for comparison with (2.1). Indeed the relevant coefficient

Gaussian averages of exponentials (Wilson loops)
Formulas like (4.3) can become a little less mysterious, if the r.h.s. is written in a somewhat different way. In fact, such a possibility is provided by the theory of exponential correlators.

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According to [80][81][82], the generating function of exponentials in the Hermitian matrix model is given by the simple integral and it provides a short way [28,29] to generalization of the Harer-Zagier formulas like (2.1).
It is related to resolvents by the Laplace transform, For n = 1, this gives (see also [65, eq Note that expanded is the quadratic exponential, not the linear one, because E(s) is treated as a series in s.
Using this formula, we can rewrite the sum rule (4.3) without y(z) as Contributing at the both sides are only odd negative powers of z beginning from z −3 .

Cut-and-join operator
Using the second relation in (1.3), we can trade the denominator in (1.2) for the character ϕ:

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and then use the fact that ϕ is the eigenvalue of the generalized cut-and-join operator [23,24] acts on the time-variablesp k = TrM k . The normal ordering in (6.4) implies that all the derivatives ∂ M stand to the right of all M . Since W ∆ are "gauge"-invariant matrix operators, and we apply them only to gauge invariants, they can be realized as differential operators inp k [23,24]. Then the coefficient in front of p ∆ in Z N {p} can be represented as where e x n denotes projection to grading n, which is needed in (6.7) because the sum over Young diagrams is restricted to a given size (note that p 2 has the grading degree 2). For example,Ŵ [1 2m ] = 1 (2m)! :Ŵ 2m [1] : whereŴ [1] =D 1 = k kp k ∂p k , i.e. multiplies p m 2 by 2m. The normal ordering means that :Ŵ 2m [1] : multiplies it by (2m)!, and therefore the coefficient in front of p [1 2m Alternatively one can rewrite (6.6) as where again the index |∆| means that one should pick up a contribution of particular grading degree.

W -representations in terms of characters
Eq. (6.7) is a kind of a dual to the W -representation [83][84][85][86] of Hermitian partition function which involves a close relativê of the simplest cut-and-join operatorŴ [2] , [87], but in (7.1) this operator is exponentiated in contrast with (6.7). The average of character in the W -representation is equivalent to action of the differential operator χ R {∂/∂t k }. Moreover, since W 2 in (7.1) has non-trivial grading (+2), only the single term of the exponential expansion contributes to the average: Contributions at odd levels |R| are vanishing. At the level |R| = 2 this gives for R = [2] and [1, 1]: For arbitrary even |R| the highest power of N comes from the term N |R| p |R|/2 2 inŴ |R|/2 2 , and is equal to which differs by a factor d R = χ R {δ k,1 } from the item d R N |R| in D R (N ). In general, (1.1) implies that Similarly, for the rectangular complex matrix model These formulas for partition functions were discovered and discussed in [1,2], but, in this section, we want to derive them from the W -representations.

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The key is the generalization of (6.3), which is equivalent tô with the differential operatorχ R = χ R k ∂ ∂p k , which is a dual character, [18] χ R χ R {p}| p=0 = δ R,R (7.11) In the particular case of (7.3) this means that [3] − χ [2,1] ) ·χ [3] + (−χ [3]  As follows from (6.3), the operator W [2] has an eigenvalue ϕ R, [2] , which is equal to Note that the operator at the r.h.s. of (7.12) is not equal just to a diagonal sum R κ R χ R {p}χ R with the eigenvalue κ R . This is because, before putting p = 0, the orthogonality condition (7.11) is not true, e.g.χ [2] χ [3] {p} = p 1 , and [3] − χ [2,1] = 3χ [3] = κ [3] χ [3] (7.14) In general, lifting of (7.11) to the operator level involves decomposition of the commutator into a sum of skew characters, in particular, picking up just the c-number piece at the r.h.s., we get the contribution from Q = R only, i.e.χ R · χ R {p} = χ R /R {p}, what is non-vanishing only for |R | ≥ |R|. We remind that the skew characters are defined by the property and can be expressed via the usual Schur functions, through the Littlewood-Richardson coefficients C R QR , the structure constants of the character multiplication

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Coming back to (7.12), the coefficient in front ofχ Y is given by a recursion formula One can calculate the coefficients α R,R from this formula. Remarkably, these coefficients are non-vanishing only for the single-hook diagrams R = [r, 1 s−1 ] and R = [r , 1 s −1 ], and the final answer is Note that despite only the single hookχ contribute, all χ R , not only single hook are eigenfunctions of this 1 2Ŵ [2] with non-vanishing eigenvalues κ R . Note also that the operator at the l.h.s. of (7.20) contains at most second derivatives, while particular items at the r.h.s. contain derivatives up to order |R|, though all these higher derivatives cancel in the sum. Now we want to do the same for W 2 and W 1 , which have a non-vanishing grading, thus the sums will not be diagonal even in the size of the Young diagrams. Instead, where the sum goes over the Young diagrams R and R which differ by 2 in size. Note that the operator at the l.h.s. contains at most second derivatives, while particular items at the r.h.s. contain derivatives up to order |R |. Still all the higher derivatives cancel in the sum. The coefficients A R,R are linear functions of N , and contributing are only the single-hook diagrams R = [r, 1 s−1 ] and R = [r , 1 s −1 ] so that the answer is   Exponentiation of (7.21) and (7.22) provides an unrestricted sum over R and R : with the coefficients A R,R non-zero only for |R| − |R | being even and non-negative. Eq. (7.7) is the piece of this sum with R = ∅. Note that items with R and R of odd sizes in the expansion of W 2 contribute to the exponential. Likewise, (7.8) is the piece of the sum Exponentiation can be performed with the help of (7.15). From (7.15) and (7.22), we obtain Y Z is the contribution from W 2 1 to the full matrix B Y Z for the exponential e W 1 . Diagrams of the type R are all single-hook, thus the same is true for diagrams Q appearing in the skew characters. However, at the last stage the single-hook characters are multiplied, and Y and Z are already 2-hook diagrams. Likewise, the higher powers W m 1 involve transition matrices B and leads to substitution of the Schur functions by the Macdonald functions and their various limits (like the Jack and Hall-Littlewood polynomials) [18,19]. There is, however, another important direction to generalize: to a unitary 1 or trigonometric model and, further, to the MacMahon model: 3) The former model was a testing area for initial studies of character expansions in matrix models [79,[94][95][96]. It also emerged in the studies of localization of the ABJM theory [92]. The MacMahon model arises in the studies of localization of superconformal gauge theories [60] (there are also instanton correction, which vanish in the large N [93] and other interesting limits). They can be considered as "perturbations" of Hermitian model by bi-trace addition to the action with ν = 0, 1 for trigonometric and MacMahon models respectively, and ζ(s) is the Riemann zeta-function. In the former case, values of the ζ-functions are elementary numbers (modulo powers of π), while, in the latter case, they are transcendental, but in existing applications this difference does not manifest itself. This is because these theories are often studied perturbatively, by expanding the exponentiated bi-trace into a series and taking averages within the Hermitian matrix model. 1 The trigonometric Vandermonde determinant has first emerged within the context of the unitary matrix models [88,89]. However, the "fair" unitary model requires the choice of integration contour along the imaginary axis (in fact, on an imaginary segment) and averaged are bilinear combinations of characters. Hence, we refer to this type of models just as to trigonometric (though maybe more exact is the name "hyperbolic"). Note that these models emerge most naturally as Chern-Simons type matrix models [90,91].

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However, it is much more interesting to look for exact formulas like (1.2) in the trigonometric model itself, without a reference to the Hermitian one. For the trigonometric model, the statement is where the average is taken with the weight When averaging, the argument of character in the integrand is the diagonal matrix with the entries e m i , and, at the r.h.s., the parameters are q = e g 2 /2 , A = q N = e N g 2 /2 , the exponent κ R = (x,y)∈R (y − x) is the same as in (7.13). The time variables in the argument of the character at the r.h.s. lie in the "topological locus" obtained by the q-deformation of p k = N . Thus, the difference from (1.2) are a quantum deformation, and the drastic change of the combinatorial factor from the ratio of characters to the exponential of the eigenvalue of the second Casimir operator in the representation R, κ R . The limit q −→ 1 is trivial: it corresponds to g 2 −→ 0, when the Gaussian exponential e Tr M 2 /2g 2 turns into the δ-function so that the integral in (8.6) gets localised at M = 0, i.e. U = I N , and this relations just gives rise to the identity χ R [I N ] = χ R {N }. In the opposite limit of g 2 −→ ∞, the Gaussian exponential disappears, so the measure reduces to the Haar measure, but the integral diverges, and so does the r.h.s. of (8.5), where q −→ ∞. One could instead consider true unitary integrals with pure imaginary m i and unimodular q, but, in this case, non-vanishing are only balanced averages, with the same number of U and U † , which now differs from U and rather equal to U −1 . This is a more interesting and complicated case, with the 't Hooft-de Wit anomalies and other peculiarities, see [79,94] and references therein.
We can now compare the implications of (8.4) at ν = 0 with those of (8.5). As the simplest example consider On the other hand, the same quantity is just the ratio of the Hermitian model averages

Knot matrix models
There is another interesting way to deform the trigonometric model [97,98]: Like the trigonometric model (8.6), it also preserves characters, moreover, (8.5) remains true, what changes is only the value of q = exp g 2 2ab . This formula "spontaneously breaks" the a ↔ b symmetry, and has the corresponding counterpart with the same q. This measure appears in description of the HOMFLY polynomials for torus knots with a and b coprime: The character at the l.h.s. depends on e M and, before (9.2) can be applied, one needs to express it through characters of e M/a . This decomposition involves a combination of characters for Young diagrams of the size a|R| with peculiar Adams coefficients c. After this, substitution (9.2) converts (52) into the Rosso-Jones formula [99,100]: is a Laurent polynomial of q and A (it remains such for arbitrary knots, not only torus). Within this framework, the equivalence of (9.2) and (9.3) reflects the Reidemeister equivalence of a-strand and b-strand realizations of the same torus knot Torus [a,b] . For attempts to extend matrix model description beyond the torus knots see [101].
When a and b have a non-trivial common divisor, we get a torus link instead of a knot, and its HOMFLY invariant is an average of a product of characters, as many as there are components in the link (the number equal to the common divisor of a and b). When JHEP08(2018)163 a = b = 2, this is actually the Hopf link, and its HOMFLY invariant is again a character, see [102] for a review and references. At the same time, the measure in this case is exactly that of the trigonometric model. In other words, we conclude that The framing factor at the l.h.s. of (9.6) has to be taken in degree 1 2 a b + b a in the generic torus knot/link case.
This strangely-looking shift of time variables (9.7) is in fact induced by action of the cut-and-join operator on the "topological locus" p * k = q N k −q −N k q k −q −k , which is the trigonometricmodel substitute of the locus p k = N in the Hermitian models: In the case of torus measure (9.2) with arbitrary a and b, the bi-trace correction to the action (8.4) is substituted by with ν = 0 and η k = 1 2 1 a 2k + 1 b 2k . Then

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sinh g 2 2ab (9.10) A generalization of the torus matrix model to non-torus knots is an open problem: for N = 2 it is nicely solved in [101], but the matrix model lifting to arbitrary N remains a challenge.
In the generic case,

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Only R and S of even sizes contribute to the sum. The integral of α is very immediate, so that finally but the individual coefficients contain skew characters and thus are a little more involved. For example, For the trigonometric models including the MacMahon one, the restriction to the traceless matrices works much simpler: since χ R [e M +iα ] = e iα|R| χ R [e M ], the α-dependence factors out, and its only effect is the additional factor N 2π in the average: In this formula, the average can be taken at any model: Hermitian, trigonometric, toric, since the generalized Vandermonde factors in the measure do not depend on α.

Conclusion
The central formula of this paper, looks like a statement that integration over M is reduced to the substitution of the "mean field" M = Id. This would look mysterious, but in fact this is not quite true: for an arbitrary function F {p k } the property is true only for the characters, and it is another kind of a mystery, more similar to a Duistermaat-Heckman (localization) trick with group theory origins rather than to any kind of an ordinary mean field calculation in quantum field theory.

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In the trigonometric case, the situation is different: It does not look like a mean-field formula: the exponentials at the l.h.s. turn into a somewhat different structure, the ratio of sinh's at the r.h.s. Instead, it is nicely consistent with the quasiclassical approximation µ → ∞ in (1.2): then dominating in the integral over M is the vicinity of M = 0 where Tr e kM = N . Note that, in the Hermitian case, taking the limit µ → ∞ makes no sense: the µ-dependence is fixed by dimensions of the operators: there is no any weak coupling regime at all, and formulas like (11.1) are exact. For straightforward q, t-deformation of (11.1) see [19], the clever thing to do in this case is just to take (11.1) as a definition of the model, which is much simpler and more practical than to proceed through multiple Jackson integrals and Pochhammer symbols.
Challenging are generalizations of (11.1) in at least five directions: • to non-Gaussian phases, where changing is only the coefficient in front of the characters at the r.h.s., see [17] for simplest examples, • to generic knots, not only torus ones, for a more detailed description of the problem see [101], • to the MacMahon matrix model (8.3), where the Vandermonde determinant is substituted by a product of the Barnes double Γ-functions, • to the network models [32,[103][104][105] describing contractions of multiple topological vertices, usual and refined, • to the Aristotelian tensor models of [34][35][36][37][38][39]108].
The challenge is well illustrated already by the operator counting rules. The ordinary characters (Schur functions) and their MacDonald deformations are labeled by the ordinary Young diagrams, and their abundance is described by the generating function n # Young · q n = n=1 1 1 − q n (11.4) For the plain partitions, which are labeling representations of the DIM algebra and the affine Yangian [106,107], it becomes n # plain · q n = (1−q n ) βn (11.6) where β n is the number of unlabeled dessins d'enfants with n edges [109][110][111][112].
Our main purpose in this paper was to demonstrate a technical possibility to attack all these problems in a systematic way. Our main emphasize was on the way the characterpreservation property unifies highly non-trivial and even previously unnoticed identities (sum rules) between characters. Usually such non-linear relations are described in terms of "integrability", but physically relevant quantities (non-perturbative partition functions) are long known to be more than integrable, the word superintegrable seems most adequate to describe the situation. In the standard language of matrix models, the story is that matrix model τ -functions are not just tau-functions, but satisfy an additional string equation (and, in result, the whole set of Virasoro or W -constraints), which altogether makes the model not just integrable, but explicitly solvable like additional integrals of motion do for superintegrable mechanical systems. However, the higher symmetry behind the superintegrable models, more complicated than the Coulomb force with its hidden O(d+1) symmetry is still under investigated and is not straightforward to reveal, because it is nonlinearly realized. This paper can be considered as a substantial step in this direction, which is based on the technique of character decompositions [1, 33-39, 108, 113, 114], see earlier reviews in [95,96]. One should now study character decompositions of various harmonics of higher W -operators, including the higher cut-and-join operators W R of [23,24], which are their zero-harmonics. These will again involve new summation rules for characters, which can, however, be more comprehensive than those implied by the genus expansions. One of the issues is to study the emerging hook structure of the sum rules (hook formulas) and its dependence on the shape of the diagram R. Exponentiation of cutand-join operators, which is a kind of trivial since all the eigenfunctions and eigenvalues are explicitly known from [23,24] is, however, also a source of highly non-trivial sum rules for characters. To conclude, the character-preservation property of matrix models reveals an entire new world of non-linear relations between the characters of linear and symmetric groups, which requires an understanding from the point of view of the basic group theory. This is especially important, because the combinatorial solution of matrix models survives various deformations: from Young diagrams to plain partitions, from matrices to tensors, from the Gaussian to higher Airy measures, from the Hermitian to trigonometric model and, probably, further, while the corresponding deformations of Lie algebra theory are yet unknown or, at best, extremely complicated. As usual, the matrix model approach provides a unifying view on the full set of problems and provides an efficient method to solve them.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.