Random Unitary Representations of Surface Groups I: Asymptotic Expansions

In this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _{g}$$\end{document}Σg denote a topological surface of genus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\ge 2$$\end{document}g≥2. We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{1}(\Sigma _{g})$$\end{document}π1(Σg) under a random representation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{1}(\Sigma _{g})$$\end{document}π1(Σg) into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {SU}(n)$$\end{document}SU(n). Each such expected value involves a contribution from all irreducible representations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {SU}(n)$$\end{document}SU(n). The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.


Introduction
Let g ∈ N with g ≥ 2 and let g denote a closed topological surface of genus g. If x 0 is a point in g , then we have The group g is called the surface group of genus g. For n ∈ N, the group U(n) is the group of n × n complex unitary matrices with respect to the standard Hermitian inner product on C n . Then SU(n) is the subgroup of U(n) consisting of matrices with unit determinant.
(1.1) This embedding shows that Hom( g , SU(n)) is an algebraic variety, but it is a variety with singularities [Gol84,pg. 204 Prop.]. We let Hom( g , SU(n)) irr denote the collection of homomorphisms φ such that φ is irreducible as a linear representation of g . The space Hom( g , SU(n)) irr then inherits the structure of a smooth non-complete manifold from (1.1) (ibid.).
There is an action of SU(n) on Hom( g , SU(n)) by postcomposition with inner automorphisms; from the point of view of (1.1), this is just the diagonal action of SU(n) by conjugation. This action factors through an action of PSU(n), that is, SU(n) modulo its finite center. The quotient by this action is denoted by Hom( g , SU(n))/PSU(n). It is shown by Goldman in (ibid.) that the action of PSU(n) on Hom( g , SU(n)) irr is free and the moduli space M g,n def = Hom( g , SU(n)) irr /PSU(n) is a smooth real manifold. This moduli space is the underlying set of random representations of g discussed in this paper. By a theorem of Narasimhan and Seshadri [NS65], if a complex structure on g is fixed, M g,n corresponds via a natural map to a moduli space of stable holomorphic rank-n vector bundles on g .
To describe the law of the random representation, we need to recall some further results of Goldman. In (ibid.), Goldman shows that there is a natural symplectic form ω g,n on M g,n , defined up to a scalar normalization that we fix in § §2.7. This symplectic form arose previously in the work of Atiyah and Bott [AB83]. It is analogous to the Weil-Petersson form on the Teichmüller space of complex structures on g and is defined precisely in § §2.7 of this paper. The symplectic form ω g,n yields a volume form The random representations in this paper are sampled according to this volume form, normalized to be a probability measure. We call the normalized measure the Atiyah-Bott-Goldman measure. The statistics of random representations we are interested in come from functions on Hom( g , SU(n)) that are invariant under conjugation by SU(n). The natural functions to integrate on moduli spaces like M g,n are geometric functions (as studied, e.g., by Mirzakhani [Mir07] in the Weil-Petersson context). These functions are also called Wilson loops in the theoretical physics literature.
We now fix a concrete instance of a family of geometric functions. For g ∈ U(n), let tr(g) denote the trace of g as an n × n matrix. Given any element γ ∈ g , we obtain a continuous function tr γ : Hom( g , SU(n)) → C, tr γ (φ) def = tr(φ(γ )).
Clearly tr γ is invariant under conjugation by SU(n) and hence yields a continuous bounded function that we give the same name tr γ : M g,n → C.
In this paper, we instigate a study of the expected value of tr γ , that is, M g,n tr γ dVol M g,n M g,n dVol M g,n . (1.2) For fixed γ , we are interested in the large n behavior of this expected value. One has the simple bound The significance of this prediction is that it extends a celebrated result of Voiculescu [Voi91,Theorem 3.8] on the asymptotic * -freeness of Haar unitary matrices, suitably interpreted, from free groups to surface groups. The current paper lays the groundwork for the proof of (1.3) in the next paper in the series [Mag21]. Not only that, but here we will expose a separate phenomenon for the values E g,n [tr γ ]: they can be approximated to any order O(n −M ) as n → ∞ by a Laurent polynomial in n depending on γ . The formal theorem is the following: Theorem 1.1. For any g ≥ 2 and γ ∈ g , there is an infinite sequence of rational numbers a −1 (γ ), a 0 (γ ), a 1 (γ ), a 2 (γ ), . . . such that for any M ∈ N, as n → ∞ E g,n [tr γ ] = a −1 (γ )n + a 0 (γ ) + a 1 (γ ) n + · · · + a M−1 (γ ) The main technical result we prove in order to establish Theorem 1.1 is interesting in its own right so we discuss this now. The quantity in the denominator of (1.2), i.e., the symplectic volume of M g,n , was calculated non-rigorously by Witten in [Wit91]. Witten's result is in terms of Witten zeta functions, so named by Zagier in [Zag94]. The Witten zeta function of SU(n) is defined by the series (1.5) Here SU(n) is the set of equivalence class of irreducible representations of SU(n). The sum in (1.5) converges absolutely for Re(s) > 2 n by a result of Larsen and Lubotzky [LL08, Thm. 5.1] (see also [HS19,§2] for an alternative proof of this fact). The following formula that Witten obtained was rigorously established to hold by Sengupta in [Sen03]. Theorem 1.5 (Witten's formula). With the normalization of Vol M g,n fixed as in § §2.7, we have M g,n dVol M g,n = nζ(2g − 2; n).
In fact, Sengupta also provided a method to compute the integral of any continuous function on M g,n with respect to Vol M g,n and this is the starting point of our work. We let F 2g def = a 1 , b 1 , . . . , a g , b g be the free group on the generators a 1 , b 1 , . . . , a g , b g .
Therefore, we have a surjective homomorphism F 2g q g − → g obtained from quotient by the normal subgroup generated by R g . We say that w ∈ F 2g represents the conjugacy class of γ ∈ g if q g (w) is conjugate to γ in g . For any w ∈ F 2g , there is a word map w : SU(n) 2g → SU(n) obtained by substituting elements of SU(n) into the letters of w. We write dμ Haar SU(n) 2g (x) for the probability Haar measure on SU(n) 2g , and this is the product of the probability Haar measures on the 2g factors.
Corollary 1.6. Let g ≥ 2 and γ ∈ g . Suppose that w ∈ F 2g is an element representing the conjugacy class of γ . Then if the sum on the right-hand side of (1.6) is absolutely convergent.
We explain how to obtain Corollary 1.6 from [Sen03, Thm.1] in § §2.7 using ideas already presented in [Sen03]. Let [ g , g ] denote the commutator subgroup of g . Using Corollary 1.6, it is not hard to show: , then there exists n 0 = n 0 (γ ) such that for n ≥ n 0 Proposition 1.7 is proved in § §3.2. This proves Theorem 1.1 in the case that γ / ∈ [ g , g ].
We now explain how we prove Theorem 1.1 in general by using Corollary 1.6. We first discuss the zeta function factor in (1.6). One has the following theorem due to Guralnick, Larsen and Manack. The limiting value arises from the trivial representation in (1.5); it is possible to boost the methods of [GLM12] to show that ζ(2g − 2; n) can be approximated to any order O(n −M ) by a Laurent polynomial in n. In fact, the results of this paper can be viewed as a far generalization of this result and accordingly, the just-mentioned result is established as a byproduct of our proofs in § §5.4. Thus, the proof of Theorem 1.1 amounts to showing that is absolutely convergent and can be approximated to any order by some Laurent polynomial. The obvious bad feature of the sum (1.8) is that it runs over infinitely many representations of SU(n), and moreover, there are more and more of these as n increases. We aim to approximate (1.8) by finitely many of its terms and this requires an ordering of the representations of SU(n).
The correct way to do this is as follows. For every k, ∈ N and pair of Young diagrams μ k, ν , with number of rows given by (μ), (ν), for every n ≥ (μ) + (ν) there is a rational family of irreducible representations 2 denoted by (ρ (1.9) then (B; n) is in one-to-one correspondence with the (μ, ν) such that (μ), (ν) ≤ B, μ 1 , ν 1 ≤ B 2 when n is sufficiently large. This specific choice of (B; n) is for technical convenience, becoming useful in § §5.4. We prove bounds on the I(w, ρ) in Theorem 4.1. The main challenges are that not only that estimates for I(w, ρ) must overcome the weights dim W in (1.8) but also that these bounds must remain effective for dim W much larger than n. It is quite well understood that matrix integrals such as I(w, ρ) are challenging in this regime as the main method of performing such integrals, known as the Weingarten calculus, often fails to produce understandable answers there. This is because the Weingarten function Wg n,k defined in (2.12) becomes increasingly complicated for k n, drawing on more and more different representations of large symmetric groups. We overcome this inherent difficulty as follows.
Firstly there is a minor observation that in all cases of interest, SU(n) can be replaced by U(n) in (1.7) (Proposition 3.1). The main idea is then that after some splitting up, parts of I(w, ρ) can be evaluated by integrating first tr(ρ(R g (x))) over all double cosets for a very large subgroup U(n − D) ≤ U(n) where D is bounded depending only on w, which is fixed. During this first integration, the structure of the word R g can be exploited to produce a lot of cancelation.
After this initial integral, we then apply the Weingarten calculus through a novel strategy (cf. § §4.3) making heavy use of representation theory of both symmetric groups and U(n). What we achieve is the following technical result. Theorem 1.9. Suppose that g ≥ 2, w ∈ F 2g and B ∈ N.
1. For n ≥ n 0 (w, g), the sum in (1.6) is absolutely convergent. 2. As n → ∞ where |w| denotes the length of w as a reduced word.
The point of (1.10) is not the exact form of the right-hand side, but rather, it gives effective control of the tail. Theorem 1.9 shows that by taking B sufficiently large and fixed depending on w, the contribution to E g,n [tr γ ] from (ρ, W ) ∈ SU(n)\ (B; n) can be made to decay as fast as any n −M , for M ∈ N. We have the following direct corollary of Theorem 1.9, Theorem 1.8, and Corollary 1.6. Corollary 1.10. Suppose that g ≥ 2, γ ∈ g , and w ∈ F 2g represents the conjugacy class of γ . For any B ∈ N, we have as n → ∞ 1.1. Related works I: Spaces of representations. The existence of an asymptotic expansion of E g,n [tr γ ] as in Theorem 1.1 follows a long line of related results. The most closely related of these is the analog of Theorem 1.1 when SU(n) is replaced by the family of symmetric groups S n . For π ∈ S n , let fix(ρ) denote the number of fixed points of π , and for γ ∈ g , let fix γ : Hom( g , S n ) → N be the function fix γ (φ) def = fix(φ(γ )). The representation space Hom( g , S n ) is finite and we let E g,S n [fix γ ] denote the expected value of fix γ with respect to the uniform probability measure on Hom( g , S n ). An exactly analogous result to Theorem 1.1 for E g,S n [fix γ ] was established by the author and Puder in [MP20, Thm. 1.1].
Similarly, if instead of using a surface group g , we consider a free group F r with r ≥ 2, for any compact Lie group G the representation space Hom(F r , G) can be identified with G r and hence can be given the corresponding probability Haar measure. If G is finite, this is simply the uniform probability measure. For any character χ of G and w ∈ F r , we obtain a function Then we can define E F r ,G [χ w ] to be the expected value of χ w with respect to the Haar probability measure. Not only is the analog of Theorem 1.1 true for many natural families of (G(n), χ (n)), but actually, in the case of free groups, E F r ,G(n) [χ(n) γ ] is a rational function of n for n sufficiently large. Indeed, for fixed w ∈ F r , E F r ,G(n) [χ(n) γ ] agrees with a rational function of n for n w 1 when • G(n) = S n and χ(n) = fix [Nic94,LP10] • G(n) is a family of generalized symmetric groups, e.g., hyperoctahedral groups, and χ(n) is the trace in a natural defining representation [MP19c] • G(n) = U(n) and χ(n) = tr [MP19a] • G(n) = O(n) or Sp(n) and χ(n) = tr [MP19b].
is, in the context of a quantum SU(n) Yang-Mills theory on g , a heuristic definition of the expected value of the Wilson loop measuring the trace of the holonomy around the loop in g that w ∈ F 2g represents. Here C(ρ) is the Casimir eigenvalue of the representation (ρ, W ) and t is a coupling constant. (We have inserted the factor 1 n in the above exponent so that it matches with, e.g., [GT93].) It is worth pointing out that the methods of the current paper should also allow one to effectively and rigorously approximate (1.11) although this is not pursued here.
The emphasis in the physics literature is not on the rigorous analytic approximation of integrals such as (1.11), but rather, on the interpretation of (1.11) as a formal power series and then reordering the terms and truncating in a formal way. These manipulations are not intended as having rigorous mathematical consequences. For example, as we understand, none of the ' 1 n -expansions' of (1.11) obtained before in any sense rigorously approximate (1.11); nonetheless, they are significant to physicists. Since the values E g,n [tr γ ] that we focus on here correspond to the t = 0 case of (1.11) via Corollary 1.6, we briefly survey what is known to physicists for general t, with the disclaimer that the author is by no means an expert in the concepts of theoretical physics.
In physics literature, the chiral expansion means that the (ρ, W ) are parameterized by (ρ λ n , W λ n ) where λ runs over Young diagrams. For this parameterization to work (cf. § §2.4), one should restrict to Young diagrams with less than n rows. This is referred to as the finite-n expansion. However, in some cases this restriction is lifted and (1.11) is interpreted as a sum over all Young diagrams. This is called the large-n expansion. In the chiral expansion, the Young diagrams are ordered by the number of boxes that they contain.
The partition function of the quantum Yang-Mills theory corresponds to (1.11) in the case w = id, up to a factor 1 n , and is given by [GT93,2.4] A large-n chiral expansion of the partition function was obtained by Gross and Taylor in [GT93]. The coefficients of this expansion are interpreted in terms of branched covers of surfaces, and from this, Gross and Taylor deduce their titular statement that 'Two dimensional QCD is a String Theory.' A finite-n chiral expansion of the partition function in terms of branched covers with some extra data was obtained by Baez and Taylor in [BT94]. In [Ram96], Ramgoolam gives a large-n chiral expansion of (1.11) in terms of branched covers of surfaces.
In the language of these papers, the expansion we obtain in Theorem 1.1 is a finite-n, fully non-chiral expansion of the expected value of a Wilson loop, when the coupling constant is set to zero. The main point is that this asymptotic expansion is established rigorously through Theorem 1.9. To We let e(θ ) def = exp(2πiθ) for θ ∈ R. If G is a group, and g 1 , g 2 ∈ G, we let [g 1 , g 2 ] def = g 1 g 2 g −1 1 g −1 2 . We write [G, G] for the subgroup of G generated by elements of the form [g 1 , g 2 ] for g 1 , g 2 ∈ G. If V is a complex vector space and q ∈ N, we let in general, if we write a tensor product without explicit subscript it is over C. We write Q(t) for the ring of rational functions in an indeterminate t, i.e., ratios of polynomials. We use Vinogradov notation as follows. If f and h are functions of n ∈ N, we write f h to mean that there are constants n 0 ≥ 0 and C 0 ≥ 0 such that for n ≥ n 0 , h and h f . If in any of these statements the implied constants depend on additional parameters, we add these parameters as subscript to , O, or . Throughout the paper, we view the genus g as fixed and so any implied constant may depend on g.

Young diagrams and tableaux.
Young diagrams A Young diagram (YD) is a collection of left-aligned rows of identical square boxes in the plane, where the number of boxes in each row is non-increasing from top to bottom. Hence, we use the English convention for Young diagrams throughout the text. Any YD λ also gives a non-increasing sequence of natural numbers (λ 1 , λ 2 , . . . , λ (λ) ) where λ i is the number of boxes in the i th (from top to bottom) row of λ, and (λ) is the number of rows of λ. A finite non-increasing sequence of natural numbers is called a partition. We think of partitions and YDs interchangeably in this paper via the above correspondence. For example, the partition (k) corresponds to the Young diagram with one row consisting of k boxes. The empty YD with no boxes is denoted by ∅. The size of a YD λ is the number of boxes that it contains, or The size of λ is denoted by |λ|, and the statement |λ| = k is sometimes written λ k.
Given two Young diagrams λ and μ, we say μ ⊂ λ if every box of μ is a box of λ. A skew Young diagram (SYD) is a pair λ, μ of Young diagram such that μ ⊂ λ. This is usually written as λ/μ and λ/μ is thought of as the collection of boxes of λ that are not boxes of μ. A Young diagram λ is identified with λ/∅, and in this way, YDs are special cases of SYDs.
Semistandard Young tableaux Given a SYD λ/μ (which may in fact be a YD) and a subset S ⊂ N, a semistandard tableau of shape λ/μ with entries in S is a filling of the boxes of λ/μ with the numbers of S such that the numbers in the boxes are strictly increasing along columns from top to bottom and non-strictly increasing along rows from left to right. If λ/μ is a SYD, we write for the semistandard tableaux of shape λ/μ with entries in [k, ]. We also use all obvious variants of this notation, e.g., for YD λ and n ∈ N, SST [n] (λ) is the collection of semistandard tableaux of shape λ with entries in [n]. There is by convention a unique semistandard tableau (with any entry set) of shape the empty YD (or an empty SYD). If λ/μ is a SYD, n > m, T 1 ∈ SST [m] (μ) and T 2 ∈ SST [m+1,n] (λ/μ), we write T 1 T 2 for the semistandard tableau obtained by adjoining the numbers-in-boxes of T 2 to those of T 1 . It is easy to see this is always a valid semistandard tableau of shape λ.
Given T ∈ SST [k,m] (λ/μ), the weight of T is the function where ω T ( j) is the number of occurrences of j in T . Given T ∈ SST [n] (λ), λ a YD, and m ∈ N 0 with m ≤ n, we write T | >m for the skew semistandard tableau formed by the numbers-in-boxes of T with numbers > m, and similarly write T | ≤m for the semistandard tableau formed by the numbers-in-boxes of T with numbers ≤ m.

General representation theory.
Here we clarify the language of representation theory that will appear throughout the paper.
A unitary representation of a compact Lie group 3 G consists of a Hilbert space 4 H and a homomorphism ρ 1 , G → U(H) where U(H) is the group of unitary operators on H. Any compact Lie group has a trivial unitary representation (triv G , C) with inner product on C given by z 1 , z 2 = z 1 z 2 and triv G (g) = 1 for all g ∈ G.
In many cases, the maps ρ 1 and ρ 2 will be tacitly inferred from the group G and the spaces H 1 and H 2 . When this is the case, we write for the vector space of linear intertwiners between (ρ 1 , H 1 ) and (ρ 2 , H 2 ).
For (ρ, W ) a unitary representation of a compact Lie group G, and H ≤ G a compact Lie subgroup with unitary representation (π, V ), we define the (π, V )-isotypic subspace of W for the subgroup H to be the subspace spanned by the images of all elements of where Res G H W is the restriction of ρ to H . Any such isotypic space is itself a unitary subrepresentation of Res G H W (for the group H ). If V is an irreducible representation of H , the dimension of Hom H (V, W ) is the multiplicity with which V appears in Res G H W . Let (π, W ), (π 1 , W 1 ) and (π 2 , W 2 ) be finite-dimensional unitary representations of a compact Lie group. We explain some basic constructions.
There is a dual unitary representation on the space of complex linear functionals on W with inner product induced by that on W ; this representation is also irreducible if We denote the dual representation by (π,W ) or simplyW . Given two finite-dimensional unitary representations (π 1 , W 1 ) and (π 2 , W 2 ) of a compact Lie group G, the tensor product W 1 ⊗ W 2 def = W 1 ⊗ C W 2 has an action of G that is 'diagonal' by π 1 on the first factor and π 2 on the second factor. The tensor product inherits a Hermitian inner product from that on W 1 and W 2 where w 1 ⊗ w 2 , w 1 ⊗ w 2 def = w 1 , w 1 w 2 , w 2 that makes W 1 ⊗ W 2 a unitary representation of G under the diagonal action. This extends to tensor powers of W ; any W ⊗k for k ∈ N is in this way a unitary representation of G under the diagonal action.
There is a canonical isomorphism of linear representations, where the right-hand side is the vector space of linear maps from W 2 to W 1 , where G acts 'diagonally' on the left-hand side, and by conjugation (g : A → π 1 (g)Aπ 2 (g) −1 ) on the right-hand side. If W 1 , W 2 are subrepresentations of W 1 , W 2 , then we view Hom(W 2 , W 1 ) as a subrepresentation of Hom(W 2 , W 1 ) via (2.1). This corresponds to extending linear maps by 0 on the orthogonal complement of W 2 in W 2 .
In the case W 1 = W 2 = W (2.1) is moreover an isomorphism of unitary representations W ⊗W ∼ = End(W ) if End(W ) is given the Hilbert-Schmidt inner product A, B def = tr(AB * ), B * standing for the Hermitian transpose of B.

Representation theory of symmetric groups.
Although the problems of this paper are not initially posed in a way that involves symmetric groups, the representation theory of unitary groups is intimately connected via Schur-Weyl duality (see §2.4) to the representation theory of symmetric groups, so it plays a large part in this paper.
We write S k for the symmetric group of permutations of the set [k], and C[S k ] for the group algebra of S k . As a technicality, the group S 0 is the group with one element.
The equivalence classes of irreducible representations of S k are in one-to-one correspondence with YDs λ k [FH91, §4.2]. The irreducible unitary representation of S k corresponding to λ k will be denoted by (π λ , V λ ) and simply referred to as V λ . We write χ λ for the character of V λ , i.e., All characters of irreducible representations of symmetric groups are real-valued (e.g., by [FH91,Frobenius Formula 4.10]).
The dimension of V λ is given by the Frame-Robinson-Thrall hook length formula as follows. The hook of a box in λ k is the collection of boxes either to the right of, or below, λ, including the box itself. We write h λ ( ) for the number of boxes in the hook of . The hook length formula [FRT54] states Before proceeding, we fix some notation. If we refer to S ≤ S k with ≤ k we always view S as the subgroup of permutations that fix every element of [ + 1, k]. As a consequence, we obtain fixed inclusions C[S ] ⊂ C[S k ] for and k as above. When we write S × S k− ≤ S k , the first factor is S as defined above and the second factor is S k− which is our notation for the subgroup of permutations that fix every element of [ ].
Given λ , the element is a central idempotent in C[S ] with the following important property. If (π, V ) is any unitary irreducible representation of S k with k ≥ , then by linear extension π : The branching rules for S k imply that d λ/μ = 0 unless μ ⊂ λ. Applying Frobenius reciprocity to the pair (V μ , V λ ) then taking the dimension of the induced representation Ind S k S V μ gives the formula, for μ fixed and b = k − , Suppose that 1 , 2 ∈ N 0 and 1 + 2 = k. The irreducible representations of S 1 × S 2 are of the form V μ 1 ⊗ V μ 2 with μ i i for i = 1, 2; S 1 acts on the first factor and S 2 acts on the second factor. The numbers are called Littlewood-Richardson coefficients. They are notoriously difficult to work with, but thankfully, in this paper the most detail we need about them is the following: Proof. This is a standard fact but normally presented slightly differently. To obtain this version, one needs to know that LR λ μ,( 2 ) is also the coefficient of the Schur polynomial s μ in the expansion of s λ s ( 2 ) in Schur polynomials (see [FH91, eq. (A.8), Ex. 4.43] for the equivalence of these definitions of Littlewood-Richardson coefficients). Then the lemma follows from the version of Pieri's formula for s λ s ( 2 ) given in [FH91, eq. (A.7)].

Representation theory of U(n) and SU(n).
Here we give a brief account of the representation theory of U(n) and SU(n). The details as well as more background can be found in [FH91].
The equivalence classes of irreducible representations of U(n) are in one-to-one correspondence with their highest weights by the theorem of the highest weight. These highest weights are given by linear combinations λ 1ω1 + · · · + λ nωn where (λ 1 , . . . , λ n ) ∈ Z n is a non-increasing sequence of integers andω 1 , . . . ,ω n is a system of fundamental weights for U(n). The sequence (λ 1 , . . . , λ n ) is called the signature of the representation. If all λ i ∈ Z ≥0 , then the signature corresponds to a Young diagram λ with (λ) ≤ n, and conversely, any YD λ with (λ) ≤ n gives rise to an equivalence class of irreducible representation of U(n) denoted by (ρ λ n , W λ n ). Every irreducible representation of U(n) restricts to an irreducible representation of SU(n), and every irreducible representation of SU(n) arises by restriction from U(n). Two irreducible representations of U(n) restrict to equivalent representations of SU(n) if and only if their signatures differ by an integer multiple of (1, 1, . . . , 1). This immediately shows that the equivalence classes of irreducible representations of SU(n) are in one-to-one correspondence with YDs λ such that (λ) ≤ n − 1, since there is a unique signature (λ 1 , . . . , λ n ) whose representation restricts to the given one of SU(n) with λ n = 0 and this corresponds to a YD by deleting trailing zeros. For such a λ, we write (ρ λ n , W λ n ) for the restriction of this representation of SU(n). To be less verbose, in the rest of the paper we will tend to refer to representations simply by the vector space whenever the module structure can be inferred.
The trace of g ∈ U(n) on W λ n is given by s λ (e(θ 1 ), . . . , e(θ n )) where s λ is the Schur polynomial associated with the YD λ and e(θ 1 ), . . . , e(θ n ) are the eigenvalues of g. As such, the dimension of W λ n is given by the specialization of the Schur polynomial, letting There is a formula for s λ (1 n ) due to Stanley called the hook content formula. Given a YD λ, for any box of λ, we define the content of the box to be (2.5) By a slight abuse of notation, if g is any element of U(n), we also write s λ (g) for tr W λ n (ρ λ (g)); strictly speaking this is the Schur polynomial evaluated at the eigenvalues of g. The hook content formula (2.5) implies that for a fixed YD λ, D λ (n) is a polynomial function of n with coefficients in Q.
There is an important formula expressing s λ (g) in terms of power sum symmetric functions. Given a partition λ = (λ 1 , . . . , λ r ), and g ∈ U(n), we define where μ(π) k is the partition given by the cycle type of π ∈ S k . Besides the polynomial families of representations of U(n), as n varies, obtained by fixing λ and varying n, there are similar rational families that we explain now. Let μ k and ν be fixed Young diagrams. Then for every n ≥ (μ) + (ν), there is an irreducible unitary representation of U(n) with signature for the character of this representation at g ∈ U(n). The representations W [μ,ν] n directly generalize the case of W λ n with λ fixed by taking μ = λ and ν to be the empty Young diagram. The character s [λ,μ] (g) can be written in terms of Schur polynomials by a result of Koike.
whereν 1 denotes the transposed YD of ν 1 (obtained from ν 1 by switching rows and columns). Inspection of Theorem 2.2 shows that D [μ,ν] (n) = s [μ,ν] (1 n ) is a finite linear combination, with integer coefficients, of (2.7) The linear combination itself does not depend on n, and by the hook content formula (2.5), each of the terms in (2.7) is a polynomial function of n with coefficients in Q for n ≥ (μ) + (ν). Moreover, the term in (2.7) is n p 2 + p 3 so the contribution to s [λ,μ] (1 n ) of maximal growth corresponds to p 2 = k and p 3 = hence is a unique term with nonzero (in fact, unity) coefficient. These arguments yield the following corollary to Theorem 2.2. with μ, ν fixed should be considered alongside the representations W λ n with λ fixed.

Schur-Weyl duality
We always understand that C n has the standard Hermitian inner product and U(n) acts unitarily on C n in its defining representation. This representation coincides with (ρ λ , W λ n ) where λ consists of one box in the formalism of §2.4. We write {e 1 , . . . , e n } for the standard basis of C n . The tensor power (C n ) ⊗k def = C n ⊗ · · · ⊗ C n k has an induced inner product that makes it into a unitary representation of U(n), where U(n) acts diagonally. The space (C n ) ⊗k is also a unitary representation of S k where S k acts by permuting indices. The actions of U(n) and S k commute, and hence, (C n ) ⊗k is a unitary representation of U(n) × S k . We write π k n : U(n) → End((C n ) ⊗k ) for the diagonal action of U(n) and ρ k n : S k → End((C n ) ⊗k ) for the action that permutes coordinates. Schur-Weyl duality gives the following full description of the decomposition of (C n ) ⊗k into irreducible representations of U(n) × S k , which are a priori tensor products Branching rules For any n > r ≥ 0 we view U(n − r ) as the subgroup of U(n) of elements that fix pointwise the standard basis elements e n−r +1 , . . . , e n of C n in the defining representation of U(n). We have [FH91, Ex. 6.12] for (λ) ≤ n (2.8) By iterating (2.8), we obtain a orthogonal direct sum decomposition We make the observation that a sequence gives a semistandard tableau of shape λ by filling in the boxes of μ i+1 /μ i with the number i. This gives a one-to-one correspondence (μ 1 , . . . , μ n−1 ) → T (μ 1 , . . . , μ n−1 ) between such sequences of YDs as in (2.9) and SST [n] (λ); for and pick a unit-norm vector w T in each W λ n,T (this is unique up to a unit complex number). We do this now once and for all for all λ and T ∈ SST [n] (λ). The resulting orthonormal basis (2.10) We have use for the following fact, analogous to [MP20, Lemma 2.4].
has an orthonormal basis given by Proof. The proof is essentially the same as that of [MP20, Lemma 2.4].

The Weingarten calculus. The Weingarten calculus is a method of calculating integrals of the form
where dμ Haar U(n) is the probability Haar measure on U(n). A large-n asymptotic estimate for (2.11) was first obtained by Weingarten in [Wei78] and this was expanded upon by Xu in [Xu97] where a full large-n asymptotic expansion was given. A new method of evaluating (2.11) in terms of characters of symmetric groups was developed by Collins [Col03] and Collins andŚniady [CŚ06] and this is what we refer to here as the Weingarten calculus. The work [CŚ06] involves reinterpreting (2.11) as a problem in calculating the orthogonal projection onto the the diagonal action. This is actually the point of view that will be relevant to this paper.
The Weingarten function, depending on parameters n, k is the following element of (2.12) Write P n,k for the orthogonal projection in End((C n ) ⊗k ) onto the U(n)-invariant vectors.
Henceforth, tr will always denote the usual trace of a linear map from a finite-dimensional vector space to itself. We have the following proposition of Collins andŚniady [CŚ06, Proposition 2.6 (Collins-Śniady). Let n, k ∈ N. Suppose A ∈ End((C n ) ⊗k ). Then 2.6. Free groups and surface groups. Recall that g is a closed topological surface of genus g with base point x 0 and we have identified Given w ∈ F 2g , we view w as a combinatorial word in a 1 , a −1 The abelianization of g coincides with the first singular homology group of g and is isomorphic to Z 2g , generated by the images of a 1 , b 1 , . . . , a g , b g . As such, the map induced by q w from F 2g to the abelianization of g has kernel 2.7. The Atiyah-Bott-Goldman measure. We first follow Goldman [Gol84] to define the Atiyah-Bott-Goldman measure on M g,n . Consider the word map We view Hom( g , SU(n)) as R −1 g (id) ⊂ SU(n) 2g via the embedding (1.1). The diagonal action of SU(n) on SU(n) 2g by conjugation factors through an action of PSU(n). This action preserves Hom( g , SU(n)).
Let Hom( g , SU(n)) irr denote the set of φ ∈ Hom( g , SU(n)) that are irreducible as linear representations. The conjugation action of PSU(n) on Hom( g , SU(n)) irr is clearly free and the action is also proper [Gol84,pg. 205]. Hence, The complement of M g,n in Hom( g , SU(n))/PSU(n) is the union of finitely many manifolds of strictly lower dimension than M g,n .
By We normalize the Killing form on su(n) so that the induced Riemannian volume on SU(n) has unit total mass, i.e., it gives the probability Haar measure on SU(n). Cup product together with the Killing form on su(n) and Poincaré duality induces an alternating non-degenerate bilinear form where the last isomorphism used that φ is irreducible (see [Gol84,§1.4] for more details on this argument). This yields via (2.13) a differential two-form ω ABG g,n on M g,n that turns out to be closed and hence symplectic ([Gol84, pg. 208]; the closedness of the form was originally proved by Atiyah and Bott [AB83]).
The Atiyah-Bott-Goldman (ABG) measure μ ABG g,n is the probability measure on M g,n induced by the symplectic volume form: If f is a continuous function on M g,n then the expected value of f with respect to μ ABG g,n is given by (2.14) Having defined the ABG measure, we now explain how to deduce Corollary 1.6 from Sengupta's work [Sen03]. For mostly technical reasons, this involves the introduction of the heat kernel 5 on SU(n). The heat kernel is a one parameter family of functions Q t : SU(n) → R for t ∈ (0, ∞) that, as a function on (0, ∞)×SU(n), is a fundamental solution to the heat equation on SU(n) (cf. [Sen03, § §1.2] for the precise definition). What is important here is that the heat kernel has the expansion The notation lim t→0 + means the limit is taken along positive values of t.
Proof. (Deduction of Corollary 1.6 from Theorem 2.7) Under the same assumptions as Theorem 2.7, using (2.15), one may write for fixed where the interchange of sum and integral is valid by uniform convergence of the heat kernel expansion. We are given γ ∈ g and w ∈ F 2g representing the conjugacy class of γ . We take f = tr • w (w denoting the word map of w) so thatf = tr γ . Recall the notation I(w, ρ) from 1.7, in this case, Theorem 2.7 together with (2.16) gives (2.17) Since we assume the hypothesis that (ρ,W )∈ SU(n) dim W I(w, ρ) is absolutely convergent, using dominated convergence in (2.17) proves Combining this with Witten's formula (Theorem 1.5) we obtain Corollary 1.6.

Models of representations.
In light of Corollary 1.6, we must evaluate I(w, ρ) for (ρ, W ) ∈ SU(n). We use different models of (ρ, W ) throughout the paper: recalling the definition of (B; n) from (1.9), viewing B as fixed and assuming n is large enough depending on B ) for some μ, ν with |μ|, |ν| ≤ B 3 uniquely determined by (ρ, W ). Hence, • If (ρ, W ) ∈ SU(n)\ (B; n), then we identify (ρ, W ) ∼ = (ρ λ n , W λ n ) for some λ with (λ) ≤ n − 1 uniquely determined by (ρ, W ). Hence, 3.2. Integrating over SU(n) 2g vs U(n) 2g . At various points, it is convenient to execute the integral defining I(w, λ) or I(w, [μ, ν]) as an integral over U(n) 2g rather than SU(n) 2g . This is indeed possible in all cases that the large-n behavior of E g,n [tr γ ] is interesting. Recall that [F 2g , F 2g ] is the commutator subgroup of F 2g .
In other words, the integrals can be computed using U(n) instead of SU(n).

On the other hand
, then there is n 0 = n 0 (w) such that for n ≥ n 0 , for any (ρ, W ) ∈ SU(n), Proof. By uniqueness of the Haar measure on U(n) 2g , we have where * denotes convolution of measures. Therefore, The first equality used (3.2), the second used R g (zx) = R g (x) for all x ∈ U(n) 2g and z ∈ Z (U(n) 2g ), and the last used Fubini's theorem. If w ∈ [F 2g , F 2g ], then by the discussion in § §2.6 the letters of w are 'balanced' and the inner integrand is tr(w(zx)) = tr(w(x)). Thus, we obtain In other words, the Fourier coefficient can be computed using U(n) instead of SU(n where all m i ∈ Z and at least one m i = 0. On the other hand, R g (zx) = R g (x) as before. Suppose without loss of generality in the following that m 1 = 0. Let z 0 = e 1 n id, . . . , id . We have by left invariance of Haar measure Hence, for n > m 1 (w) we obtain I(w, ρ) = 0 for any (ρ, W ) ∈ SU(n).
This means the main theorem is proved when γ / ∈ [ g , g ].
As Theorem 1.1 has now been proved for γ / ∈ [ g , g ], in the rest of the paper, we may assume γ ∈ [ g , g ], and hence, any w representing the conjugacy class of γ is

Rationality of the contribution from (B; n).
Here we prove the following theorem. Let Q(t) denote the ring of rational functions in an indeterminate t with coefficients in Q. For f ∈ Q(t) and t 0 ∈ Q we write f (t 0 ) for the evaluation of the rational function at t 0 , provided that t 0 is not a pole of f . Let w ∈ [F 2g , F 2g ] represent the conjugacy class of γ ∈ [ g , g ].

Theorem 3.2.
There is a rational function Q B,w ∈ Q(t) such that for n ≥ |w| + 2B 3 Since the index set of the sum is finite and does not depend on n, it now suffices to prove that each D [μ,ν] (n)I(w, [μ, ν]) agrees with a rational function of n for fixed μ, ν as above. By Corollary 2.3, each D [μ,ν] (n) agrees with a polynomial function of n with coefficients in Q when n ≥ 2B. Therefore, the proof of the theorem is reduced to the following: Claim: For each (μ, ν) ∈ (B), when n ≥ |w| + 2B 3 , I(w, [μ, ν]) agrees with a rational function of n with coefficients in Q.

Statement of main sectional result and setup. For n, ∈ N, let [n]
def = [n] × · · · × [n] ; this set has a diagonal action of S n and we write S n \[n] for the quotient set. The main theorem of this §4 is the following key estimate that will be used for (ρ λ n , W λ n ) ∈ SU(n)\ (B; n).  In the rest of this §4, we assume g = 2 for simplicity of exposition. The proofs extend in a straightforward way to g ≥ 3. We write {a, b, c, d} for the generators of F 4 and R = [a, b][c, d]. We write w in reduced form: The expression (4.1) implies that for h After interchanging summation and integration in (3.1), we obtain Since we have an inclusion S n ⊂ U(n) via 0-1 matrices, for σ ∈ S n we can change variables The measure dμ Haar U(n) 4 is invariant by this change of variables and (R(h a , h b , h c , h d )).
For I = (i 1 , i 2 , . . . , i |w| ) and σ ∈ S n , we define σ (I ) = (σ (i 1 ), σ (i 2 ), . . . , σ (i |w| )) and we have so we obtain in total We can therefore rewrite (4.2) as where D(I ) is the number of distinct entries in I . Most of the rest of the section will be devoted to estimating I * (w I , λ); the point of the previous calculations is that we can assume I ∈ [n − D(I ) + 1, n] |w| and this will be exploited in § §4.2.

First integrating over a large subgroup.
We keep the assumptions of the previous section and also assume I ∈ [n − D(I ) + 1, n] |w| .
We fix I and hence write D = D(I ). This assumption means that the function w I : To simplify notation, all integrals over groups are done with respect to the probability Haar measure and this will be denoted by dg where g is the group element. Our goal is to calculate I * (w I , λ). The bi-invariance of dμ Haar U(n) 4 and the U(m) 4 -biinvariance of w I means we can write The first cancelation we will obtain comes from the integral h 1 ,h 2 ∈U(m) 4 s λ (R(h 1 hh 2 ))dh 1 dh 2 , h ∈ U(n) 4 . (4.5) Our approach to this integral follows the same lines as [MP20, §4.4]. We consider the vector space which is a unitary representation of U(n) 4 , the subscripts indicating which elements of (h a , h b , h c , h d ) act on which factor, so each h f acts diagonally on two factors. Let B λ ∈ End(W λ R ) be defined via matrix coefficients by the formula We have the following lemma analogous to [MP20, Lemma 4.7].

Lemma 4.2. For any h
The purpose of Lemma 4.2 is that it turns the integral in (4.5) into a projection operator. Indeed, let Q denote the orthogonal projection in W λ R onto the U(m) 4 -invariant vectors.

Lemma 4.3. We have
Recalling the definition of E λ,μ,m R 1 ,R 2 from (2.10), we are able to calculate tr W λ R (h Q B λ Q) as follows.

Proposition 4.4. We have
Proof. An orthonormal basis for the U(m) 4 -invariant vectors in W λ R is given by where the μ i range over all μ i ⊂ D λ with (μ i ) ≤ m and each S i , T i ∈ SST [m+1,n] (λ/μ i ). This can be extended to a full orthonormal basis of W λ R , and hence, (4.8) The matrix coefficient will now be calculated in stages. Firstly We have Combining the previous two calculations, we obtain Therefore, from (4.8) This is progress because now the integrand is U(m) 4 -bi-invariant, and hence, the integral is now essentially over U(m) 4 \U(n) 4 /U(m) 4 rather than the full U(n) 4 . In the process of integrating over U(m) 4 , by exploiting the structure of the relator R of g we picked up a helpful factor of 1 D μ (m) 3 and forced μ to be the same in each tensor factor of the equation stated in Proposition 4.5. We show how to proceed further in the next section.

Second integration: strategy.
The only method we really have to proceed from Proposition 4.5 is to use the Weingarten calculus from § §2.5. The caveat is that this works well when integrating functions that are finite products of matrix coefficients in the standard representation of U(n). Here we are concerned with the integral appearing in Proposition 4.5. So to use the Weingarten calculus we must write the integrand of (4.9) as a product of matrix coefficients in some End((C n ) ⊗K ).
Since we assume w ∈ [F 4 , F 4 ], by the observation about the exponents of letters in the reduced word of w from § §2.6, (4.9) is equal to the product of four independent integrals over U(n) of the form So we will estimate (4.10) under the assumptions (4.11) and later take the product of the four resulting bounds to estimate (4.9). We now outline the strategy that we will follow to estimate the integral (4.10). This means that

We can write
The implication for this to our key integral (4.10) is that where P is the self-adjoint orthogonal projection on End We are able to do this using the Weingarten calculus; this is the reason for the construction of A λ,μ s,t . This calculation takes place in § §4.6. 5. We estimate P [A λ,μ s,t ⊗ X ] in Proposition 4.9. 6. On the other hand, after suitable identifications, we have and hence, ⊗ X ] , and putting these estimates into (4.12), we obtain an upper bound on |(4.10) |, as desired.
Remark 4.6. It does not seem possible to exactly evaluate (4.9) using this method. Indeed one might hope that The components of this strategy will be carried out in the following sections.

Construction of A
From these data, we will construct a special vector in (C n ) ⊗B ⊗(Č n ) ⊗B ∼ = End((C n ) ⊗B ). This will be done in stages.
Recall the weight functions ω s , ω t : [m + 1, n] → N 0 from §2.1. For z ∈ C[S k ], we will write ρ k n (z) for the resulting element of End((C n ) ⊗k ) according to the representation of S k on (C n ) ⊗k . We also have fixed linear embeddings for M ≤ n C M = e 1 , . . . , e M ⊂ C n = e 1 , . . . , e n ; similarly,Č M ⊂Č n , hence using the type of isomorphism as in (2.1) for M 1 , M 2 ≤ k we obtain a fixed linear embedding The skew tableaux s yields a sequence of tableaux where ν j (s) is the YD given by μ along with the boxes of s containing numbers ≤ m + j.
Recall the projection operators p ν from § §2.3. We use the natural inclusions, e.g., We have the following proposition. Recalling the intertwiner F from Schur-Weyl duality (Proposition 2.4), we obtain a map (4.14) this map is an intertwiner for Proof. The proof relies on the fact that up to scalar multiplication, E λ,μ,m s,t ∈ End(W λ n ) is uniquely characterized by the following properties: s,t commutes with U(m), or in other words, is invariant under the diagonal subgroup U(m) ≤ U(m) × U(m). Consider the linear map by which A i+1 is obtained from A i for i ≥ 0, that is, This is a linear intertwiner for the group U(m Proof of Claim A. We prove by induction on j the following statement: Consider the base cases j = i. By Schur-Weyl duality, and in particular (4.14), the fact that A i has the form Proof of Claim B. Using the intertwining properties of the maps f i again, we have for u ∈ U(m)

This completes the proof of Claim B.
Now by Claim A with i = D we have that, in reference to (4.14), and the analogs of Claim A and Claim B hold for this element as F ⊗F is a U(n) × U(n) intertwiner. Therefore, by the fact that E λ,μ,m s,t is uniquely characterized by a and b, we must have This proves Proposition 4.7.

Normalization of A
λ,μ s,t . The goal of this section is to calculate A λ,μ s,t , where the norm is the standard norm on End((C n ) ⊗B ).
We identify End((C n ) ⊗B ) ⊗ End((C n ) ⊗ p ) ∼ = End((C n ) ⊗(B+ p) ) via canonical isomorphisms as in (2.1) and the map e i 1 ⊗ · · · ⊗ e i B ⊗ě j 1 ⊗ · · · ⊗ě j B ⊗ e i B+1 ⊗ · · · ⊗ e i B+ p ⊗ě j B+1 ⊗ · · · ⊗ě j B+ p → e i 1 ⊗ · · · ⊗ e i B ⊗ e i B+1 ⊗ · · · ⊗ e i B+ p ⊗ě j 1 ⊗ · · · ⊗ě j B ⊗ě j B+1 ⊗ · · · ⊗ě j B+ p . (4.17) Using this identification, we view End((C n ) ⊗B ) ⊗ End((C n ) ⊗ p ) as a unitary representation of U(n) × S B+ p × U(n) × S B+ p ; the mapP is a C[S B+ p ]-bimodule morphism. Hence, where X 1 def = e ⊗ω t (m+D) m+D ⊗ě ⊗ω s (m+D) m+D ⊗ X . Repeating this argument giveŝ So it remains to calculateP[ρ b m (p μ ) ⊗ X D ]. The Weingarten calculus (Proposition 2.6) yieldŝ where and X D ∈ End( e m+1 , . . . , e n ⊗B−b+ p ) is the element corresponding to X D under the fixed isomorphism that preserves the order of the factors in both so that We make an observation that if q r denotes the number of ∈ [p] such that j = m +r , and p r denotes the number of ∈ [p] such that i = m + r , if we do not have Therefore, (4.23) Here we write (σ 1 , σ 2 ) for the element of S B+ p corresponding to (σ 1 , σ 2 ) ∈ S b × S B−b+ p . Given From (4.23), we now have is an element which in any unitary representation of S B+ p gives the orthogonal projection onto the V μ ⊗ triv S s -isotypic subspace for S b × S s . Combining (4.20) and (4.24) and recalling the definition of Wg n,B+ p from (2.12) givê Combined with (4.19) this giveŝ that is the main result of this section.

The norm of the projection of A
λ,μ s,t ⊗ X. We now turn to the calculation of P [A λ,μ s,t ⊗ X ] where the norm is the natural one induced by the standard inner product on C n . We identify End((C n ) ⊗B+ p ) ∼ = End((C n ) ⊗B ) ⊗ End((C n ) ⊗ p ) using the isomorphism (4.17); after this identification the norm on End((C n ) ⊗B ) ⊗ End((C n ) ⊗ p ) coincides with the Hilbert-Schmidt norm on End((C n ) ⊗B+ p ). The main result of this section is: Proposition 4.9. If (4.22) holds, we have .
We view (C n ) ⊗B+ p with its given inner product as a unitary representation of U(n) × S B+ p in the usual way involved in Schur-Weyl duality. For λ m + p with (λ ) ≤ n, we let Z λ denote the W λ n × V λ -isotypic subspace of (C n ) ⊗B+ p for U(n) × S B+ p . By Schur-Weyl duality (Proposition 2.4), this space is itself an irreducible representation We may assume (4.22) holds, otherwiseP[A λ,μ s,t ⊗ X ] = 0 and Proposition 4.9 is proved. In this case, using that p λ is central in C[S B+ p ], inspection of (4.25) gives that where for z ∈ C[S B+ p ] we write z V λ for the Hilbert-Schmidt norm of z acting on V λ . Notice that we obtained a factor √ D λ (n) from the multiplicity of V λ in Z λ . Due to the presence of p ν D (s) = p λ in (4.27), the right-hand side of (4.27) is zero unless λ ⊂ p λ , and hence, (4.28) To calculate the Hilbert-Schmidt norm in (4.27), we study in detail the operator Let π λ denote the representation of S B+ p on V λ .
Lemma 4.10. With notation as above, π λ (Q) is a self-adjoint idempotent element of Proof. Recall that q s is the orthogonal projection onto the V μ ⊗ triv S s -isotypic subspace for One and hence commutes with all elements p ν j (s) with j ≤ i. Thus, π λ (Q) is a product of commuting self-adjoint idempotents and so is itself a selfadjoint idempotent. The same arguments show that π λ (p ν 1 (t) · · · p ν D (t) ) is a self-adjoint idempotent.
We now calculate the dimension of π λ (Q)V λ . This is the eigenspace of π λ (Q) with eigenvalue 1 and also the orthogonal complement to the kernel of π λ (Q).
Lemma 4.11. With notation as above, if Proof. Firstly, as Q ∈ C[S B ] and contains the factor (4.29) Given this, we now consider Q acting on V λ . This is the same as We have π λ (Q 1 ) is a self-adjoint idempotent by the same proof as that of Lemma 4.10.
Then Q 1 is supported on, and preserves, the V ν D−1 (s) ⊗ triv-isotypic component for By Pieri's formula (Lemma 2.1), as ν D−1 ⊂ 1 ν D = λ, this isotypic component consists of a unique isomorphic copy of V ν D−1 (s) ⊗ triv in V λ . Moreover, when identifying this space with V ν D−1 (s) , the action of π λ (Q 1 ) is given by where q 2 is the projection onto the V μ ⊗ triv-isotypic subspace for Iterating this argument, using Pieri's formula each time, gives eventually that Thus, in total, going back to (4.29), we obtain the required Proof of Proposition 4.9. Pick an orthonormal basis {v 1 , . . . , v D } for π λ (Q)V λ with D = d λ /λ d μ and extend this to an orthonormal basis of V λ . In regard to (4.27), we have (4.30) Now, each π λ (id S b , π 0 τ ) is unitary, and by the second statement of Lemma 4.10, π λ (p ν D (t) · · · p ν 1 (t) ) is a self-adjoint idempotent and hence has operator norm bounded by one. Hence, for i ∈ [D] π λ (p ν D (t) · · · p ν 1 (t) ) Combining this with (4.30), we obtain Using this in (4.27) gives Finally incorporating this estimate into (4.26) gives

Completing the outlined strategy.
Proposition 4.12. With notation as in the previous sections, (4.32) Recalling that the block sizes of are given by either side of (4.22), we have Next, by using twice both the hook length formula (2.2) and the hook content formula (2.5) we obtain where the last inequality used for each f ∈ {a, b, c, d}. (For general g the exponents 8|w| and 4 in (4.35) are 4g|w| and 2g, respectively.) Next using Proposition 4.5, we obtain for m(I ) = n − D(I )

Statement of main sectional results. The main task of this section is to estimate the sum
appearing in the left-hand side of (1.10). Let (B; n) denote the collection of Young diagrams of length at most n − 1 such that λ ∈ (B; n) → (ρ λ n , W λ n ) ∈ SU(n)\ (B; n) is a one-to-one parametrization. The goal of this section is to prove Theorem 1.9, which amounts to establishing a bound for 2 (w, B, n) def = λ∈ (B;n) D λ (n)I(w, λ) and proving its absolute convergence. Theorem 1.9 has the following corollary that will be useful later.
Corollary 5.1. Let g ≥ 2 and B ∈ N be fixed. We have as n → ∞.

Preliminary estimates.
Even in the simple case of γ = id (5.2) shows that the problem of estimating 2 is related to the large-n convergence of the Witten zeta function as in Theorem 1.8. The techniques used in [GLM12] can be adapted to deal with (5.2), and indeed, are also useful in dealing with general γ (or w). The key estimate we take from (ibid.) is the following. Given a YD λ with (λ) < n, we define x i (λ) def = λ i − λ i+1 , setting λ i = 0 for i > (λ). These are the coefficients of the highest weight of (ρ λ n , W λ n ) with respect to a system of fundamental weights for SU(n).

C(w,g)
where C(w, g) > 0. Let C = C(w, g). We need a key observation relating the weight coefficients x(λ) to the set (B; n) and hence (B; n). It is not hard to check that if λ ∈ (B; n) then either and given x ∈ Z n−1 ≥0 satisfying these conditions there is at most one corresponding λ ∈ (B; n). (5.5) Here v j are the constants from Lemma 5.2 satisfying for 0 ≤ j ≤ n 2 v j = v n− j ≥ j max(1, log(n − 1) − log j).
Notice that if j ∈ [ n−1 e , n − n−1 e ] we have v j ≥ n−1 e . The function x → x log n−1 x has nonnegative derivative on [1, n−1 e ], so the minimum value of v j for j ∈ [1, n−1 e ] ∪ [n − 1 − n−1 e , n − 1] is v 1 ≥ log(n − 1). We have n−1 e ≥ log(n − 1) for n 1 so for n 1 we have is an absolutely convergent sum defining Riemann's zeta function. Notice that for n w,g 1, 2v j − C ≥ 2 log(n − 1) − C > 1, so (5.7) shows that 2 (w, B, n) is defined by an absolutely convergent sum. This proves the first statement of Theorem 1.9. We now turn to finer estimates for 2 . Assume n ≥ 2B max(2B, C +1) and 2 log(n − 1) − C > 2, so that 2v j − C ≥ 3 .
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