2. Preliminaries

2.1 Some facts in representation theory

For a compact group G, we denote the set of irreducible complex characters of G by $\mathrm{Irr}(G)$ . Given a subgroup $H\leq G$ and a character $\chi\in\mathrm{Irr}(H)$ , we denote the induction of $\chi$ to G by $\mathrm{Ind}_{H}^{G}\chi$ . We normalize the Haar measure to be a probability measure and denote the expectation with respect to the Haar measure by $\mathbb{E}$ . The standard inner product on functions on G is $\langle f_{1},f_{2}\rangle_{G}=\mathbb{E}f_{1}\overline{f_{2}}$ .

2.1.1 Representation theory of the symmetric group

Given $m\in\mathbb{N}$ , a partition of m is a non-increasing sequence $\lambda=(\lambda_{1},\ldots,\lambda_{k})$ of non-negative integers that sum to m. In this case, we write $\lambda\vdash m$ . Two partitions are equivalent if they differ only by a string of zeros at the end. A partition $\lambda=(\lambda_{1},\ldots,\lambda_{k})$ , with $\lambda_{k}>0$ , is graphically encoded by a Young diagram, which is a finite collection of boxes (or cells) arranged in k left-justified rows, where the jth row has $\lambda_{j}$ boxes. The length $\ell(\lambda)$ of a partition $\lambda\vdash m$ is the number of non-zero parts $\lambda_{i}$ or, equivalently, the number of rows in the corresponding Young diagram.

The irreducible representations of $S_{m}$ are in bijection with partitions $\lambda\vdash m$ . We write $\chi_{\lambda}\in\mathrm{Irr}(S_{m})$ for the corresponding character. For each cell (i,j) in the Young diagram of $\lambda$ , the hook length $h_{\lambda}(i,j)$ is the number of cells (a,b) in the Young diagram of $\lambda$ such that either $a=i$ and $b\geq j$ , or $a\geq i$ and $b=j$ . The hook-length formula states that

(2.1) \begin{equation}\chi_{\lambda}(1)=\frac{m!}{\prod_{(i,j)\in\lambda}h_{\lambda}(i,j)}.\end{equation}

Definition 2.1.

1. (1) Fix a Young diagram $\lambda$ and let $n\in\mathbb{N}$ . An n-expansion of $\lambda$ is any Young diagram obtained by adding n boxes to $\lambda$ in such a way that no two boxes are added in the same column.

2. (2) Given a partition $\lambda=(\lambda_{1},\ldots,\lambda_{l_{1}})\vdash k$ and a partition $\mu=(\mu_{1},\ldots,\mu_{l_{2}})\vdash l$ , a $\mu$ -expansion of $\,\lambda$ is defined to be a $\mu_{l_{2}}$ -expansion of a $\mu_{l_{2}-1}$ -expansion of a $\cdots$ of a $\mu_{1}$ -expansion of the Young diagram of $\lambda$ . For a $\mu$ -expansion of $\lambda$ , we label the boxes added in the $\mu_{l_{j}}$ -expansion by the number j and order the boxes lexicographically by their position, first from top to bottom and then from right to left. We say that a $\mu$ -expansion of $\lambda$ is strict if, for every $p\in\{1,\ldots,l_{2}-1\}$ and every box t, the number of boxes coming before t that are labeled p is greater than or equal to the number of boxes coming before t that are labeled $(p+1)$ .

Theorem 2.2 (Littlewood–Richardson rule [Reference MacdonaldMac95, I.9]). Let $\lambda\vdash k$ and $\mu\vdash m$ . Then,

\[\mathrm{Ind}_{\mathrm{S}_{k}\times S_{m}}^{S_{k+m}}(\chi_{\lambda}\otimes\chi_{\mu})=\bigoplus_{\nu\vdash k+m}N_{\lambda\mu\nu}\chi_{\nu},\]

where $N_{\lambda\mu\nu}$ is the number of strict $\mu$ -expansions of $\lambda$ that are a Young diagram of the partition $\nu$ .

We need the following consequence of Theorem 2.2.

Lemma 2.3. Let $l\in\mathbb{Z}_{\geq2}$ and identify $S_{m}^{l}$ with its image in the standard embedding $S_{m}^{l}\hookrightarrow S_{ml}$ . Then, each $\chi_{\nu}\in\mathrm{Irr}(S_{ml})$ appearing in $\mathrm{Ind}_{\mathrm{S}_{m}^{l}}^{S_{ml}}(1)$ (respectively, $\mathrm{Ind}_{\mathrm{S}_{m}^{l}}^{S_{ml}}(\mathrm{sgn})$ ) corresponds to a partition $\nu\vdash ml$ with at most l rows (respectively, l columns).

Proof. We prove the statement for the trivial representation 1 by induction on l. The proof for sgn is similar. The character 1 of $S_{m}$ corresponds to the partition $\lambda$ consisting of one row of length m. By the induction hypothesis, we may assume that $\mathrm{Ind}_{\mathrm{S}_{m}^{j}}^{S_{mj}}(1)=\bigoplus_{\mu\vdash mj}m_{\mu}\chi_{\mu}$ , with $m_{\mu}>0$ only if $\mu$ has at most j rows, for all $j<l$ . Hence, we can write

(2.2) \begin{equation}\mathrm{Ind}_{\mathrm{S}_{m}^{l}}^{S_{ml}}(1)=\mathrm{Ind}_{\mathrm{S}_{m(l-1)}\times S_{m}}^{S_{ml}}\big(\mathrm{Ind}_{\mathrm{S}_{m}^{l-1}}^{\mathrm{S}_{m(l-1)}}(1)\otimes1\big)=\bigoplus_{\mu\vdash m(l-1)}m_{\mu}\,\mathrm{Ind}_{\mathrm{S}_{m(l-1)}\times S_{m}}^{S_{ml}}(\chi_{\mu}\otimes1).\end{equation}

By Theorem 2.2 and since a strict $\lambda$ -expansion of $\mu$ increases the number of rows by at most one, the lemma follows.

2.1.2 Representation theory of the unitary group

The irreducible representations of $\mathrm{U}_{d}$ can be identified with the irreducible rational representations of $\mathrm{GL}_{d}(\mathbb{C})$ . More precisely, the restriction map $\rho\mapsto\rho|_{\mathrm{U}_{d}}$ induces a bijection $\mathrm{Irr}(\mathrm{GL}_{d}(\mathbb{C}))\rightarrow\mathrm{Irr}(\mathrm{U}_{d})$ . Moreover, the set $\mathrm{Irr}(\mathrm{U}_{d})$ is in bijection with the set $\Lambda$ of dominant weights,

\[\Lambda:=\{(\lambda_{1},\ldots,\lambda_{d}):\lambda_{1}\geq\cdots\geq\lambda_{d},\ \lambda_{i}\in\mathbb{Z}\}.\]

We denote the representation corresponding to $\lambda\in\Lambda$ by $(\rho_{\lambda},V_{\lambda})$ . The irreducible representations

\[\mathbb{C}^{d},\bigwedge\nolimits^{\!2}\mathbb{C}^{d},\ldots,\bigwedge\nolimits^{\!d}\mathbb{C}^{d},\]

are called the fundamental representations, and we have $\bigwedge\nolimits^{\!m}\mathbb{C}^{d}\simeq V_{(1,\ldots,1,0,\ldots,0)}$ , with 1 appearing m times. In particular, the standard representation $\mathbb{C}^{d}$ is $V_{(1,0,\ldots,0)}$ . Note that $\bigwedge^{\!d}\mathbb{C}^{d}$ is the determinant representation $\chi_{\det}$ . We identify a weight $\lambda\in\Lambda$ such that $\lambda_{d}\geq0$ with a partition $(\lambda_{1},\ldots,\lambda_{d})$ .

Remark 2.4 [Reference Fulton and HarrisFH91, I.6, Exc. 6.4]. For each $\lambda=(\lambda_{1},\ldots,\lambda_{d})\vdash m$ ,

(2.3) \begin{equation}\rho_{\lambda}(1)=\frac{\chi_{\lambda}(1)\cdot\prod_{(i,j)\in\lambda}(d+j-i)}{m!},\end{equation}

where (i,j) are the coordinates of the cells in the Young diagram with shape $\lambda$ .

Given $\lambda,\mu\in\Lambda$ , the irreducible subrepresentations of $\rho_{\lambda}\otimes\rho_{\mu}$ are determined by the Littlewood–Richardson rule as follows.

Theorem 2.5 (Littlewood–Richardson rule; see, e.g., [Reference Fulton and HarrisFH91, I.6, Equation (6.7)]). Let $\lambda,\mu\in\Lambda$ and suppose that $\lambda_{d},\mu_{d}\geq0$ . Let $N_{\lambda\mu\nu}$ be the coefficients from Theorem 2.2. Then,

\[\rho_{\lambda}\otimes\rho_{\mu}=\bigoplus_{\nu:\nu_{d}\geq0}N_{\lambda\mu\nu}\rho_{\nu}.\]

Remark 2.6. For $\lambda,\mu\in\Lambda$ , set $\widetilde{\lambda}:=\lambda-(\lambda_{d},\ldots,\lambda_{d})$ and $\widetilde{\mu}:=\mu-(\mu_{d},\ldots,\mu_{d})$ . Then $\rho_{\lambda}=\chi_{\det}^{\lambda_{d}}\cdot\rho_{\widetilde{\lambda}}$ and $\rho_{\mu}=\chi_{\det}^{\mu_{d}}\cdot\rho_{\widetilde{\mu}}$ , and hence by Theorem 2.5, one has

(2.4) \begin{equation}\rho_{\lambda}\otimes\rho_{\mu}=\chi_{\det}^{\lambda_{d}+\mu_{d}}\rho_{\widetilde{\lambda}}\otimes\rho_{\widetilde{\mu}}=\chi_{\det}^{\lambda_{d}+\mu_{d}}\bigoplus_{\nu}N_{\widetilde{\lambda}\widetilde{\mu}\nu}\rho_{\nu}.\end{equation}

2.1.3 Averaging characters over cosets

Lemma 2.7. Let G be a finite group, let $(\pi,V)$ be an irreducible representation of G, let $H\leq G$ be a subgroup, and let $\lambda$ be any one-dimensional character of H. Then, for every $g\in G$ ,

\[\bigg|\frac{1}{|H|}\sum_{h\in H}\lambda^{-1}(h)\chi_{\pi}(gh)\bigg|\leq\langle \chi_{\pi}|_{H},\lambda\rangle_{H}.\]

In particular, if $\langle \chi_{\pi}|_{H},\lambda\rangle_{H}=0$ , then $\sum_{h\in H}\lambda^{-1}(h)\chi_{\pi}(gh)=0$ .

Proof. Write $\pi|_{H}=\bigoplus_{i=1}^{\widetilde{N}}\pi_{i}$ with each $(\pi_{i},V_{i})$ an irreducible representation of H. For each i and $h'\in H$ ,

\begin{align*}\bigg(\sum_{h\in H}\lambda^{-1}(h)\pi_{i}(h)\bigg)\pi_{i}(h') & =\sum_{h\in H}\lambda^{-1}(h)\pi_{i}(hh')=\sum_{h\in H}\lambda^{-1}(hh'^{-1})\pi_{i}(h)\\& =\sum_{h\in H}\lambda^{-1}(h'^{-1}h)\pi_{i}(h)=\sum_{h\in H}\lambda^{-1}(h)\pi_{i}(h'h)\\&=\pi_{i}(h')\bigg(\sum_{h\in H}\lambda^{-1}(h)\pi_{i}(h)\bigg).\end{align*}

By Schur’s lemma, $\sum_{h\in H}\lambda^{-1}(h)\pi_{i}(h)$ is a scalar matrix $\alpha\cdot I_{V_{i}}$ , for some $\alpha\in\mathbb{C}$ . Hence,

(2.5) \begin{equation}\alpha\cdot\chi_{\pi_{i}}(1)=\mathrm{tr}\bigg(\sum_{h\in H}\lambda^{-1}(h)\pi_{i}(h)\bigg)=\sum_{h}\lambda^{-1}(h)\chi_{\pi_{i}}(h)=\begin{cases}|H| & \text{if }\chi_{\pi_{i}}=\lambda,\\0 & \text{otherwise.}\end{cases}\end{equation}

Let $L:=\{ v\in V:\pi(h)v=\lambda(h)\cdot v,\ \forall h\in H\} $ be the subspace of $(H,\lambda)$ -equivariant vectors in V and let $L^{\bot}$ be an H-invariant subspace of V with $V=L\oplus L^{\bot}$ . By (2.5), the map $A:=\sum_{h\in H}\lambda^{-1}(h)\pi(h)\in\mathrm{End}(V)$ satisfies $A|_{L^{\bot}}=0$ and $A|_{L}=|H|\cdot I_{L}$ . Take an orthonormal basis $v_{1},\ldots,v_{N}$ for V with $L=\langle v_{1},\ldots,v_{M}\rangle$ , $L^{\bot}=\langle v_{M+1},\ldots,v_{N}\rangle$ . Then,

\[\bigg|\!\sum_{h\in H}\lambda^{-1}(h)\chi_{\pi}(gh)\!\bigg|=\bigg|\!\sum_{i=1}^{N}\bigg\langle\pi(g)\bigg(\sum_{h\in H}\lambda^{-1}(h)\pi(h)\bigg)v_{i},v_{i}\bigg\rangle\!\bigg|=|H|\bigg|\!\sum_{i=1}^{M}\langle\pi(g)v_{i},v_{i}\rangle\!\bigg|\leq M|H|,\]

and the lemma follows.

The following lemma gives a different estimate on the average of a character over a coset, and this estimate is sharper when the double coset HgH is large. We will not need these alternative estimates, but we thought it could be useful to state them.

Lemma 2.8. Let G be a finite group, and let $H\leq G$ be a subgroup. Then, for each $\chi\in\mathrm{Irr}(G)$ and each $g\in G$ ,

\[\bigg|\frac{1}{|H|}\sum_{h\in H}\chi(hg)\bigg|\leq\frac{\langle\chi,1\rangle_{H}^{1/2}\cdot|G|^{1/2}}{|HgH|^{1/2}\chi(1)^{1/2}}.\]

Proof. Let G be a finite group. For each $\chi\in\mathrm{Irr}(G)$ , we denote by $(\pi_{\chi},V_{\chi})$ the representation corresponding to $\chi$ . The non-commutative Fourier transform (see, e.g., [Reference ApplebaumApp14, Section 2.3]) is the map ${\mathcal F}:\mathbb{C}[G]\rightarrow\bigoplus_{\chi\in\mathrm{Irr}(G)}\mathrm{End}(V_{\chi})$ defined by $f\mapsto\widehat{f}:=(\widehat{f}(\chi))_{\chi\in\mathrm{Irr}(G)}$ , where $\widehat{f}(\chi)=(\frac{1}{|G|})\sum_{g'\in G}f(g')\pi_{\chi}(g'^{-1})$ . We denote by $\Vert f\Vert_{2}:=\big((\frac{1}{|G|})\sum_{g'\in G}|f(g')|^{2}\big)^{{1}/{2}}$ . Similarly, for a collection of endomorphisms $(A_{\chi})_{\chi\in\mathrm{Irr}(G)}\in\bigoplus_{\chi\in\mathrm{Irr}(G)}\,\mathrm{End}(V_{\chi})$ , with $A_{\chi}\in\mathrm{End}(V_{\chi})$ , we define

\[\Vert\! (A_{\chi})_{\chi\in\mathrm{Irr}(G)}\!\Vert_{2}:=\bigg(\sum_{\chi\in\mathrm{Irr}(G)}\chi(1)\cdot\Vert A_{\chi}\Vert_{\mathrm{HS}}^{2}\bigg)^{\!\frac{1}{2}},\]

where $\Vert A_{\chi}\Vert_{\mathrm{HS}}:=\mathrm{tr}(A_{\chi}\cdot A_{\chi}^{*})^{{1}/{2}}$ is the Hilbert–Schmidt norm on $\mathrm{End}(V_{\chi})$ . The Plancherel theorem (see, e.g., [Reference ApplebaumApp14, Theorem 2.3.1(2)]), states that

(2.6) \begin{equation}\Vert f\Vert_{2}=\Vert \widehat{f}\Vert_{2}.\end{equation}

Let $\psi_{HgH}:=(\frac{1}{|HgH|})1_{HgH}$ . For each $\chi\in\mathrm{Irr}(G)$ , one has

\[\widehat{\psi_{HgH}}(\chi)=\frac{1}{|G|}\sum_{g'\in G}\psi_{HgH}(g')\pi_{\chi}(g'^{-1})=\frac{1}{|HgH||G|}\sum_{g'\in HgH}\pi_{\chi}(g'^{-1}).\]

The square of the $L^{2}$ -norm of $\psi_{HgH}$ is given by

(2.7) \begin{equation}\Vert \psi_{HgH}\Vert_{2}^{2}=\frac{1}{|G|}\sum_{g'\in G}(\psi_{HgH}(g'))^{2}=\frac{1}{|G|}\sum_{g'\in HgH}\frac{1}{|HgH|^{2}}=\frac{1}{|HgH||G|}.\end{equation}

Let $v_{1},\ldots,v_{M}$ be an orthonormal basis of $V_{\chi}^{H}:=\{ v\in V_{\chi}:\pi_{\chi}(h)\cdot v=v,\ \forall h\in H\}$ with respect to some G-invariant inner product $\langle\,,\,\rangle$ on $V_{\chi}$ , with $M=\langle\chi,1\rangle_{H}$ . Let $\big(V_{\chi}^{H}\big)^{\perp}$ be the orthogonal complement to $V_{\chi}^{H}$ in $V_{\chi}$ . In the proof of Lemma 2.7, in the case that $\lambda=1$ , we have seen that

(2.8) \begin{equation}\sum_{h\in H}\pi_{\chi}(h)\cdot v=\begin{cases}0 & \text{if }v\in(V_{\chi}^{H})^{\perp},\\|H|\cdot v & \text{if }v\in V_{\chi}^{H}.\end{cases}\end{equation}

In particular, we have

\begin{align*}&\bigg\langle\! \sum_{g'\in HgH}\pi_{\chi}(g'^{-1})\cdot v,v\bigg\rangle\\&\quad =\frac{|HgH|}{|H|^{2}}\bigg\langle\! \sum_{h',h\in H}\pi_{\chi}(h'g^{-1}h)\cdot v,v\bigg\rangle=\frac{|HgH|}{|H|^{2}}\bigg\langle\! \bigg(\sum_{h'\in H}\pi_{\chi}(h')\bigg)\cdot\bigg(\sum_{h\in H}\pi_{\chi}(g^{-1}h)\cdot v\bigg),v\bigg\rangle\\&\quad =\frac{|HgH|}{|H|^{2}}\bigg\langle\! \sum_{h\in H}\pi_{\chi}(g^{-1}h)\cdot v,\sum_{h'\in H}\pi_{\chi}(h')v\bigg\rangle =\begin{cases}0 & \text{if }v\in(V_{\chi}^{H})^{\perp}\\|HgH|\langle\pi_{\chi}(g^{-1})\cdot v,v\rangle & \text{if }v\in V_{\chi}^{H}.\end{cases}\end{align*}

Hence,

(2.9) \begin{align}\Vert \widehat{\psi_{HgH}}\Vert_{2}^{2}&=\sum_{\chi\in\mathrm{Irr}(G)}\chi(1)\bigg\Vert\frac{1}{|HgH||G|}\sum_{g'\in HgH}\pi_{\chi}(g'^{-1})\bigg\Vert_{\mathrm{HS}}^{2}\nonumber\\&=\sum_{\chi\in\mathrm{Irr}(G)}\frac{\chi(1)}{|G|^{2}}\sum_{i,j=1}^{M}|\!\langle\pi_{\chi}(g^{-1}).v_{i},v_{j}\rangle\!|^{2}.\end{align}

By (2.6), (2.7) is equal to (2.9), hence,

\begin{align*}\bigg|\frac{1}{|H|}\sum_{h\in H}\chi(hg^{-1})\!\bigg|^{2} &=\bigg|\!\sum_{i=1}^{M}\langle\pi_{\chi}(g^{-1})\cdot v_{i},v_{i}\rangle\!\bigg|^{2}\leq M\sum_{i=1}^{M}|\!\langle\pi_{\chi}(g^{-1})\cdot v_{i},v_{i}\!\rangle|^{2}\\& \leq M\sum_{i,j=1}^{M}|\!\langle\pi_{\chi}(g^{-1})\cdot v_{i},v_{j}\rangle\!|^{2}\leq\frac{M|G|}{\chi(1)|HgH|},\end{align*}

where the first equality follows from (2.8), and the first inequality follows from Cauchy–Schwarz inequality.

2.2 Weingarten calculus

In §§ 2.1.1 and 2.1.2 we stated that each partition $\lambda\vdash m$ with $\ell(\lambda)\leq d$ induces two different representations, $\rho_{\lambda}\in\mathrm{Irr}(\mathrm{U}_{d})$ and $\chi_{\lambda}\in\mathrm{Irr}(S_{m})$ . There is a deeper connection between $\rho_{\lambda}$ and $\chi_{\lambda}$ coming from the Schur–Weyl duality: the space $(\mathbb{C}^{d})^{\otimes m}$ carries a natural action of $\mathrm{U}_{d}\times S_{m}$ , where $A\in\mathrm{U}_{d}$ acts diagonally $A\cdot(v_{1}\otimes\cdots\otimes v_{m})=Av_{1}\otimes\cdots\otimes Av_{m}$ , and $\sigma\in S_{m}$ acts by $\sigma\cdot (v_{1}\otimes\cdots\otimes v_{m})=v_{\sigma(1)}\otimes\cdots\otimes v_{\sigma(m)}$ . The Schur–Weyl duality can be phrased as follows.

Theorem 2.9 (Schur–Weyl duality [Reference WeylWey39]). The space $(\mathbb{C}^{d})^{\otimes m}$ is a multiplicity-free representation of $\mathrm{U}_{d}\times S_{m}$ . The decomposition of $(\mathbb{C}^{d})^{\otimes m}$ into irreducible components is given by

(2.10) \begin{equation}(\mathbb{C}^{d})^{\otimes m}=\bigoplus_{\lambda\vdash m,\ell(\lambda)\leq d}\rho_{\lambda}\otimes\chi_{\lambda}.\end{equation}

There are two special functions on $S_{m}$ which come from (2.10). First, writing $\ell(\sigma)$ for the number of disjoint cycles in $\sigma\in S_{m}$ , the character of $(\mathbb{C}^{d})^{\otimes m}$ as a representation of $S_{m}$ is the function $\sigma\mapsto d^{\ell(\sigma)}$ .

Recall we have an isomorphism of algebras $\mathbb{C}[S_{m}]\simeq\bigoplus_{\lambda\vdash m}\mathrm{End}(V_{\chi_{\lambda}})$ , where the multiplication in $\mathbb{C}[S_{m}]$ is the convolution operation $f_{1}*f_{2}(y):=\sum_{x\in S_{m}}f(x)g(x^{-1}y)$ . We denote by $\mathbb{C}_{d}[S_{m}]$ the subalgebra corresponding to $\bigoplus_{\lambda\vdash m,\ell(\lambda)\leq d}\mathrm{End}(V_{\chi_{\lambda}})$ .

Definition 2.10 [Reference Collins and ŚniadyCS06, Proposition 2.3]. Let $d\in\mathbb{N}$ . The Weingarten function $\mathrm{Wg}_{d}:S_{m}\rightarrow\mathbb{C}$ is the inverse of the function $d^{\ell(\sigma)}$ in the ring $\mathbb{C}_{d}[S_{m}]$ . It has the following Fourier expansion:

(2.11) \begin{equation}\mathrm{Wg}_{d}(\sigma)=\frac{1}{m!^{2}}\sum_{\lambda\vdash m,\ell(\lambda)\leq d}\frac{\chi_{\lambda}(1)^{2}}{\rho_{\lambda}(1)}\chi_{\lambda}(\sigma).\end{equation}

Remark 2.11. Since in this paper we only consider $\mathrm{Wg}_{d'}(\sigma)$ for $d'=d$ , we write Wg instead of $\mathrm{Wg}_{d}$ .

The Weingarten calculus, developed in [Reference WeingartenWei78, Reference CollinsCol03, Reference Collins and ŚniadyCS06], utilizes the Schur–Weyl duality to express integrals on unitary groups as finite sums of Weingarten functions on symmetric groups. One formulation is the following theorem by Collins and Śniady.

Theorem 2.12 [Reference Collins and ŚniadyCS06, Corollary 2.4]. Let $(i_{1},\ldots,i_{m})$ , $(j_{1},\ldots,j_{m})$ , $(i'_{1},\ldots,i'_{m})$ , and $(j'_{1},\ldots,j'_{m})$ be tuples of integers in [d]. Then,

(2.12) \begin{align}& \mathbb{E}_{X\in\mathrm{U}_{d}}\big(X_{i_{1},j_{1}}\cdots X_{i_{m},j_{m}}\cdot\overline{X_{i'_{1},j'_{1}}\cdots X_{i'_{m},j'_{m}}}\big)\nonumber\\&\quad = \sum_{\sigma,\tau\in S_{m}}\delta_{i_{1},i'_{\sigma(1)}}\cdots\delta_{i_{m},i'_{\sigma(m)}}\delta_{j_{1},j'_{\tau(1)}}\cdots\delta_{j_{m},j'_{\tau(m)}}\cdot\mathrm{Wg}_{d}(\sigma^{-1}\tau).\end{align}

We will use a coordinate-free version of Theorem 2.12 which we proceed to state.

Definition 2.13. Let $\Omega$ be a set.

1. (1) A symmetric partition $\Phi$ of $\Omega$ is a partition $\Omega=\bigsqcup_{i=1}^{r}A_{i}\sqcup\bigsqcup_{i=1}^{r}B_{i}$ , where $|A_{i}|=|B_{i}|$ .

2. (2) Given a symmetric partition $\Phi=(A_{1},\ldots,A_{r},B_{1},\ldots,B_{r})$ , let

   \[S_{\Phi}=\{ \Sigma\in\mathrm{Sym}(\Omega):\Sigma(A_{i})=B_{i},\Sigma(B_{i})=A_{i}\} .\]

3. (3) If $\Sigma\in S_{\Phi}$ , then $\Sigma^{2}(A_{i})=A_{i}$ and we define $\widetilde{\mathrm{Wg}}(\Sigma^{2})=\prod_{i=1}^{r}\mathrm{Wg}(\Sigma^{2}|_{A_{i}})$ .

Proposition 2.14. Let $\Phi=(A,B)$ be a symmetric partition of $\Omega$ and let $F,H:\Omega\rightarrow[d]$ . Then

\begin{align*}\mathbb{E}_{X\in\mathrm{U}_{d}}\bigg(\prod_{x\in A}X_{F(x),H(x)}\prod_{y\in B}X_{F(y),H(y)}^{-1}\bigg)&=\mathbb{E}\bigg(\prod_{x\in A}X_{F(x),H(x)}\prod_{y\in B}\overline{X_{H(y),F(y)}}\bigg)\\&=\sum_{\Sigma\in S_{\Phi}:\,H=F\circ\Sigma}\widetilde{\mathrm{Wg}}(\Sigma^{2}).\end{align*}

Proof. Identify $A\cong\{ 1,\ldots,m\} $ and $B\cong\{ -1,\ldots,-m\}$ and let $\overrightarrow{\!i},\overrightarrow{\!j},\overrightarrow{\!i}',\overrightarrow{\!j}'\in[d]^{m}$ be

\[i_{k}=F(k),\quad j_{k}=H(k),\quad i'_{k}=H(-k),\quad j'_{k}=F(-k).\]

Then, by Theorem 2.12,

\begin{align*}&\mathbb{E}_{X\in\mathrm{\mathrm{U}}_{d}}\bigg(\prod_{x\in A}X_{F(x),H(x)}\prod_{y\in B}X_{F(y),H(y)}^{-1}\bigg) =\mathbb{E}_{X\in\mathrm{\mathrm{U}}_{d}}\Big(X_{i_{1},j_{1}}\cdots X_{i_{m},j_{m}}\overline{X_{i'_{1},j'_{1}}\cdots X_{i'_{m},j'_{m}}}\Big)\\&\quad =\sum_{\sigma,\tau\in S_{m}}\delta_{i_{1},i'_{\sigma(1)}}\cdots\delta_{i_{m},i'_{\sigma(m)}}\cdot\delta_{j_{1},j'_{\tau(1)}}\cdots\delta_{j_{m},j'_{\tau(m)}}\cdot\mathrm{Wg}(\sigma^{-1}\tau).\end{align*}

For $\sigma,\tau\in S_{m}$ , let $\Sigma_{(\sigma,\tau)}\in\mathrm{Sym}(A\sqcup B)\cong\mathrm{Sym}(\{-m,\ldots,-1,1,\ldots,m\} )$ be the permutation

\[\Sigma_{(\sigma,\tau)}(x)=\begin{cases}-\tau(x) & x\in\{ 1,\ldots,m\}, \\\sigma^{-1}(-x) & x\in\{ -1,\ldots,-m\} .\end{cases}\]

The map $(\sigma,\tau)\mapsto\Sigma_{(\sigma,\tau)}$ is a bijection $S_{m}^{2}\cong S_{\Phi}$ and the condition $\delta_{i_{1},i'_{\sigma(1)}}\cdots\delta_{i_{m},i'_{\sigma(m)}}\cdot\delta_{j_{1},j'_{\tau(1)}}\cdots\delta_{j_{m},j'_{\tau(m)}}=1$ is equivalent to $H=F\circ\Sigma_{(\sigma,\tau)}$ . Finally, the permutation $(\Sigma_{(\sigma,\tau)})^{2}$ acts on A as $\sigma^{-1}\tau$ , and the result follows.

Corollary 2.15. Let $\Phi=(A_{1},\ldots,A_{r},B_{1},\ldots,B_{r})$ be a symmetric partition of $\Omega$ and let $F,H:\Omega\rightarrow[d]$ . Then

\[\mathbb{E}\bigg(\prod_{i=1}^{r}\bigg(\prod_{x\in A_{i}}(X_{i})_{F(x),H(x)}\prod_{y\in B_{i}}(X_{i}^{-1})_{F(y),H(y)}\bigg)\bigg)=\sum_{\Sigma\in S_{\Phi}:\,H=F\circ\Sigma}\widetilde{\mathrm{Wg}}(\Sigma^{2}).\]

3. The Engel word as a model case

‘Those who run to long words are mainly the unskillful and tasteless; they confuse pomposity with dignity, flaccidity with ease, and bulk with force.’ [Reference FowlerFow65, p. 342]

In this section we prove the following simplified version of Theorem 1.3 for the Engel word. We chose the Engel word since it is short enough to make the proof easier to digest, while at same time complicated enough so that the proof contains most of the key ideas in the paper.

Theorem 3.1. Let X,Y be independent random variables with respect to the normalized Haar measure on $\mathrm{U}_{d}$ . For every $d\geq2m$ , one has

\[\mathbb{E}(c_{m}([[X,Y],Y]))<2^{17m}.\]

Let $w=[[x,y],y]=xyx^{-1}yxy^{-1}x^{-1}y^{-1}$ be the Engel word. We would like to compute $\mathbb{E}\big(\mathrm{tr}\bigwedge\nolimits^{\!m}w(X,Y)\big)$ . Denote ${\mathcal I}_{m,d}:=\{a_{1}<\cdots<a_{m}:a_{i}\in[d]\}$ , and note that

(3.1) \begin{equation}\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X,Y)\Big)=\sum_{\overrightarrow{\!a}\in{\mathcal I}_{m,d}}\sum_{\pi\in S_{m}}(-1)^{\pi}w(X,Y)_{a_{1}a_{\pi(1)}}\cdots w(X,Y)_{a_{m}a_{\pi(m)}}.\end{equation}

We have

(3.2) \begin{align}w(X,Y)_{a_{i}a_{\pi(i)}} &=\sum_{b_{i},c_{i},d_{i},A_{i},B_{i},C_{i},D_{i}\in[d]}X_{a_{i},D_{i}}Y_{D_{i},c_{i}}X_{c_{i},A_{i}}^{-1}Y_{A_{i},b_{i}}X_{b_{i},C_{i}}Y_{C_{i},d_{i}}^{-1}X_{d_{i},B_{i}}^{-1}Y_{B_{i},a_{\pi(i)}}^{-1}\nonumber\\& =\sum_{b_{i},c_{i},d_{i},A_{i},B_{i},C_{i},D_{i}\in[d]}X_{a_{i},D_{i}}X_{b_{i},C_{i}}\overline{X_{A_{i},c_{i}}X_{B_{i},d_{i}}}Y_{A_{i},b_{i}}Y_{D_{i},c_{i}}\overline{Y_{a_{\pi(i)},B_{i}}Y_{d_{i},C_{i}}}.\end{align}

The group $S_{m}$ acts on $[d]^{m}$ by $\sigma(\overrightarrow{\!v})_{i}=\overrightarrow{\!v}_{\sigma^{-1}(i)}$ for any $\sigma\in S_{m}$ and $\overrightarrow{\!v}\in[d]^{m}$ . Similarly, given $\overrightarrow{\!v},\overrightarrow{w}\in[d]^{m}$ and $\tau\in S_{2m}$ , we denote by $(\overrightarrow{\!v},\overrightarrow{w})$ the element in $[d]^{2m}$ given by $(\overrightarrow{\!v},\overrightarrow{w})_{i}=\small{\begin{cases}\overrightarrow{\!v}_{i} & \text{if }i\leq m\\\overrightarrow{w}_{i-m} & \text{if }m<i\leq2m\end{cases}}$ , and denote by $\tau(\overrightarrow{\!v},\overrightarrow{w})_{i}=(\overrightarrow{\!v},\overrightarrow{w})_{\tau^{-1}(i)}$ . In particular, writing $X_{\overrightarrow{\!v},\overrightarrow{\!u}}:=\prod_{i=1}^{m}X_{v_{i},u_{i}}$ for $\overrightarrow{\!v},\overrightarrow{\!u}\in[d]^{m}$ , we have

(3.3) \begin{align}\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X,Y)\Big)&=\sum_{\overrightarrow{\!a}\in{\mathcal I}_{m,d}}\sum_{\overrightarrow{\!b},\ldots,\overrightarrow{\!{\kern-.5pt}D}\in[d]^{m}}\sum_{\pi\in S_{m}}(-1)^{\pi}\big(X_{\overrightarrow{\!a},\overrightarrow{\!{\kern-.5pt}D}}X_{\overrightarrow{\!b},\overrightarrow{\!{\kern.5pt}C}}\overline{X_{\overrightarrow{\!A},\overrightarrow{\!c}}}\overline{X_{\overrightarrow{\!{\kern-1pt}B},\overrightarrow{\!{\kern-.5pt}d}}}\big)\nonumber\\&\quad \times \big(Y_{\overrightarrow{\!A},\overrightarrow{\!b}}Y_{\overrightarrow{\!{\kern-.5pt}D},\overrightarrow{\!c}}\overline{Y_{\pi^{-1}(\overrightarrow{\!a}),\overrightarrow{\!{\kern-1pt}B}}Y_{\overrightarrow{\!{\kern-.5pt}d},\overrightarrow{\!{\kern.5pt}C}}}\big).\end{align}

We now rewrite the expected value of (3.3) using Weingarten calculus. For this, define

\[S(\overrightarrow{\!a},\ldots,\overrightarrow{\!{\kern-.5pt}D}):=\bigg\{(\sigma_{1},\sigma_{2},\tau_{1},\tau_{2})\in S_{2m}^{4}:\begin{array}{c}(\overrightarrow{\!A},\overrightarrow{\!{\kern-1pt}B})=\sigma_{1}(\overrightarrow{\!a},\overrightarrow{\!b}),\,\,\,(\overrightarrow{\!c},\overrightarrow{\!{\kern-.5pt}d})=\tau_{1}(\overrightarrow{\!{\kern-.5pt}D},\overrightarrow{\!{\kern.5pt}C})\\(\overrightarrow{\!a},\overrightarrow{\!{\kern-.5pt}d})=\sigma_{2}(\overrightarrow{\!A},\overrightarrow{\!{\kern-.5pt}D}),\,\,\,(\overrightarrow{\!{\kern-1pt}B},\overrightarrow{\!{\kern.5pt}C})=\tau_{2}(\overrightarrow{\!b},\overrightarrow{\!c})\end{array}\bigg\}\]

and

(3.4) \begin{equation}Z:=\{(\overrightarrow{\!a},\ldots,\overrightarrow{\!{\kern-.5pt}D},\sigma_{1},\sigma_{2},\tau_{1},\tau_{2})\in{\mathcal I}_{m,d}\times[d]^{7m}\times S_{2m}^{4}:(\sigma_{1},\sigma_{2},\tau_{1},\tau_{2})\in S(\overrightarrow{\!a},\ldots,\overrightarrow{\!{\kern-.5pt}D})\} .\end{equation}

Lemma 3.2 We have

(3.5) \begin{equation}\mathbb{E}\Big(\mathrm{tr}\bigwedge\nolimits^{\!\!m}w(X,Y)\Big)=\sum_{(\overrightarrow{\!a},\ldots,\overrightarrow{\!{\kern-.5pt}D},\sigma_{1},\sigma_{2},\tau_{1},\tau_{2})\in Z}\sum_{\pi\in S_{m}}(-1)^{\pi}\mathrm{Wg}(\sigma_{1}^{-1}\tau_{1})\mathrm{Wg}(\sigma_{2}^{-1}(\mathrm{\pi}\times\mathrm{Id})\tau_{2}).\end{equation}

Proof. Using Weingarten calculus, i.e. Theorem 2.12, and (3.3),

(3.6) \begin{align}&\mathbb{E}\Big(\mathrm{tr}\bigwedge\nolimits^{\!\!m}w(X,Y)\Big)\nonumber\\&\quad =\sum_{\overrightarrow{\!a}\in{\mathcal I}_{m,d}}\sum_{\overrightarrow{\!b},\ldots,\overrightarrow{\!{\kern-.5pt}D}\in[d]^{m}}\sum_{\pi\in S_{m}}(-1)^{\pi}\sum_{\sigma_{1},\widetilde{\sigma}_{2},\tau_{1},\tau_{2}\in S_{2m}}\delta_{(\overrightarrow{\!a},\overrightarrow{\!b}),\sigma_{1}^{-1}(\overrightarrow{\!A},\overrightarrow{\!{\kern-1pt}B})}\cdot\delta_{(\overrightarrow{\!{\kern-.5pt}D},\overrightarrow{\!{\kern.5pt}C}),\tau_{1}^{-1}(\overrightarrow{\!c},\overrightarrow{\!{\kern-.5pt}d})}\mathrm{Wg}(\sigma_{1}^{-1}\tau_{1})\nonumber\\&\qquad \cdot\delta_{(\overrightarrow{\!A},\overrightarrow{\!{\kern-.5pt}D}),\widetilde{\sigma}_{2}^{-1}(\pi^{-1}(\overrightarrow{\!a}),\overrightarrow{\!{\kern-.5pt}d})}\cdot\delta_{(\overrightarrow{\!b},\overrightarrow{\!c}),\tau_{2}^{-1}(\overrightarrow{\!{\kern-1pt}B},\overrightarrow{\!{\kern.5pt}C})}\mathrm{Wg}(\widetilde{\sigma}_{2}^{-1}\tau_{2}).\end{align}

Applying the change of coordinate $\sigma_{2}:=(\mathrm{\pi}\times\mathrm{Id})\circ\widetilde{\sigma}_{2}$ , and observing that $\widetilde{\sigma}_{2}^{-1}(\pi^{-1}(\overrightarrow{\!a}),\overrightarrow{\!{\kern-.5pt}d})=\sigma_{2}^{-1}(\overrightarrow{\!a},\overrightarrow{\!{\kern-.5pt}d})$ , (3.6) becomes

\[\mathbb{E}\Big(\mathrm{tr}\bigwedge\nolimits^{\!\!m}w(X,Y)\Big)=\sum_{(\overrightarrow{\!a},\ldots,\overrightarrow{\!{\kern-.5pt}D},\sigma_{1},\sigma_{2},\tau_{1},\tau_{2})\in Z}\sum_{\pi\in S_{m}}(-1)^{\pi}\mathrm{Wg}(\sigma_{1}^{-1}\tau_{1})\cdot\mathrm{Wg}(\sigma_{2}^{-1}(\mathrm{\pi}\times\mathrm{Id})\tau_{2}).\]

In order to bound (3.5), we consider a natural action of $S_{m}^{7}$ on Z, and find a suitable change of coordinates such that the average of the product of the Weingarten functions in (3.5) over any $S_{m}^{7}$ -orbit is equal to a product of averages of individual Weingarten functions over cosets (see (3.8)). We then use Lemma 2.7 to estimate the contribution in (3.5) of each $S_{m}^{7}$ -orbit. To conclude the estimates of (3.5), we will further provide estimates for $|Z|$ .

We first describe the action of $S_{m}^{7}$ . The element $(\pi_{b},\pi_{c},\ldots,\pi_{D})\in S_{m}^{7}$ acts on $(\overrightarrow{\!a},\ldots,\overrightarrow{\!{\kern-.5pt}D})$ by $(\overrightarrow{\!a},\pi_{b}(\overrightarrow{\!b}),\pi_{c}(\overrightarrow{\!c}),\ldots,\pi_{D}(\overrightarrow{\!{\kern-.5pt}D}))$ and it acts on $(\sigma_{1},\sigma_{2},\tau_{1},\tau_{2})$ by

\[\sigma_{1}\mapsto(\pi_{A}\times\pi_{B})\circ\sigma_{1}\circ(\mathrm{Id}\times\pi_{b}^{-1}),\]

\[\tau_{1}\mapsto(\pi_{c}\times\pi_{d})\circ\tau_{1}\circ(\pi_{D}^{-1}\times\pi_{C}^{-1}),\]

\[\sigma_{2}\mapsto(\mathrm{Id}\times\pi_{d})\circ\sigma_{2}\circ(\pi_{A}^{-1}\times\pi_{D}^{-1}),\]

\[\tau_{2}\mapsto(\pi_{B}\times\pi_{C})\circ\tau_{2}\circ(\pi_{b}^{-1}\times\pi_{c}^{-1}).\]

This gives rise to an action of $S_{m}^{7}$ on Z. The action on the input of the Weingarten functions becomes

(3.7) \begin{equation}\mathrm{Wg}((\pi_{D}^{-1}\times\pi_{C}^{-1}\pi_{b})\sigma_{1}^{-1}(\pi_{A}^{-1}\pi_{c}\times\pi_{B}^{-1}\pi_{d})\tau_{1})\text{and }\mathrm{Wg}(\pi_{b}^{-1}\pi_{A}\times\pi_{c}^{-1}\pi_{D})\sigma_{2}^{-1}(\pi\pi_{B}\times\pi_{d}^{-1}\pi_{C})\tau_{2}),\end{equation}

where we used the conjugacy invariance of Wg to move permutations from right to left. Consider the bijection $\psi:S_{m}^{8}\rightarrow S_{m}^{8}$ , defined by $(x_{1},\ldots,x_{8})\mapsto(x_{1},x_{1}x_{2},\ldots,x_{1}x_{2},\ldots, x_{8})$ . Under the change of coordinates $(\theta_{D},\theta_{c},\theta_{A},\theta_{b},\theta_{C},\theta_{d},\theta_{B},\theta):=\psi^{-1}(\pi_{D},\pi_{c},\pi_{A},\pi_{b},\pi_{C},\pi_{d},\pi_{B},\pi^{-1})$ , the summation of (3.5) over an $S_{m}^{7}$ -orbit splits into a product of two separate sums:

(3.8) \begin{align}& \sum_{(\pi_{D},\ldots,\pi)\in S_{m}^{8}}(-1)^{\pi}\mathrm{Wg}((\pi_{D}^{-1}\times\pi_{C}^{-1}\pi_{b})\sigma_{1}^{-1}(\pi_{A}^{-1}\pi_{c}\times\pi_{B}^{-1}\pi_{d})\tau_{1})\nonumber\\&\qquad \times \mathrm{Wg}(\pi_{b}^{-1}\pi_{A}\times\pi_{c}^{-1}\pi_{D})\sigma_{2}^{-1}(\mathrm{\pi}\pi_{B}\times\pi_{d}^{-1}\pi_{C})\tau_{2})\nonumber\\&\quad = \sum_{(\theta_{D},\ldots,\theta)\in S_{m}^{8}}(-1)^{\theta_{D}\cdots\theta}\mathrm{Wg}((\theta_{D}^{-1}\times\theta_{C}^{-1})\sigma_{1}^{-1}(\theta_{A}^{-1}\times\theta_{B}^{-1})\tau_{1})\mathrm{Wg}(\theta_{b}^{-1}\times\theta_{c}^{-1})\sigma_{2}^{-1}(\mathrm{\theta}^{-1}\times\theta_{d}^{-1})\tau_{2})\nonumber\\&\quad = \sum_{\eta_{1},\eta'_{1}\in S_{m}^{2}}(-1)^{\eta_{1}\eta'_{1}}\mathrm{Wg}(\eta_{1}\sigma_{1}^{-1}\eta'_{1}\tau_{1})\sum_{\eta_{2},\eta'_{2}\in S_{m}^{2}}(-1)^{\eta_{2}\eta'_{2}}\mathrm{Wg}(\eta_{2}\sigma_{2}^{-1}\eta'_{2}\tau_{2}).\end{align}

We can now use the Fourier expansion of Wg (2.11) and the estimates in § 2.1.3 to bound the contribution of an $S_{m}^{7}$ -orbit in Z to (3.5).

Proposition 3.3. Let $\widetilde{v}:=(\widetilde{\overrightarrow{\!a}},\ldots,\,\widetilde{\overrightarrow{\!{\kern-.5pt}D}},\widetilde{\sigma}_{1},\widetilde{\sigma}_{2},\widetilde{\tau}_{1},\widetilde{\tau}_{2})\in Z$ and let ${\mathcal O}_{\widetilde{v}}:=S_{m}^{7}\widetilde{v}$ be its $S_{m}^{7}$ -orbit. Then,

(3.9) \begin{equation}\bigg|\frac{1}{|{\mathcal{O}}_{\widetilde{v}}|}\sum_{(\overrightarrow{\!a},\ldots,\tau_{2})\in{\mathcal{O}}_{\widetilde{v}}}\sum_{\pi\in S_{m}}(-1)^{\pi}\,\mathrm{Wg}(\sigma_{1}^{-1}\tau_{1})\,\mathrm{Wg}(\sigma_{2}^{-1}(\mathrm{\pi}\times\mathrm{Id})\tau_{2})\bigg|\leq\frac{1}{(2m)!^{2}m!^{3}}\binom{d}{2m}^{\!\!-2}.\end{equation}

Proof. By the orbit-stabilizer theorem, the left-hand side of (3.9) is the same as summing over all $(\pi_{D},\ldots,\pi_{B})\in S_{m}^{7}$ and dividing by $m!^{7}$ . By (3.8), the left-hand side of (3.9) is equal to

\[\frac{1}{m!^{7}}\bigg|\sum_{\eta_{1},\eta'_{1}\in S_{m}^{2}}(-1)^{\eta_{1}\eta'_{1}}\,\mathrm{Wg}(\eta_{1}\widetilde{\sigma}_{1}^{-1}\eta'_{1}\widetilde{\tau}_{1})\sum_{\eta_{2},\eta'_{2}\in S_{m}^{2}}(-1)^{\eta_{2}\eta'_{2}}\,\mathrm{Wg}(\eta_{2}\widetilde{\sigma}_{2}^{-1}\eta'_{2}\widetilde{\tau}_{2})\bigg|.\]

Note that $(S_{2m},S_{m}\times S_{m})$ is a sgn-twisted Gelfand pair, that is, the representation $\mathrm{Ind}_{S_{m}^{2}}^{S_{2m}}\,\mathrm{sgn}$ is multiplicity-free. By Frobenius reciprocity, each irreducible subrepresentation $(V_{\lambda},\pi_{\lambda})$ of $\mathrm{Ind}_{S_{m}^{2}}^{S_{2m}}\mathrm{sgn}$ has a unique $(S_{m}^{2},\mathrm{sgn})$ -invariant unit vector, so $\langle \chi_{\lambda}|_{S_{m}^{2}},\mathrm{sgn}\rangle_{S_{m}^{2}}=1$ . By Lemma 2.7, for each $\sigma\in S_{2m}$ , we have

(3.10) \begin{equation}\bigg|\!\sum_{h\in S_{m}^{2}}(-1)^{h}\chi_{\lambda}(h\sigma)\!\bigg|\leq m!^{2}\langle \chi_{\lambda}|_{S_{m}^{2}},\mathrm{sgn}\rangle_{S_{m}^{2}}=\begin{cases}m!^{2} & \text{if }\pi_{\lambda}\hookrightarrow\mathrm{Ind}_{S_{m}^{2}}^{S_{2m}}\,\mathrm{sgn},\\0 & \text{otherwise}.\end{cases}\end{equation}

By Lemma 2.3 it follows that $\pi_{\lambda}\hookrightarrow\mathrm{Ind}_{S_{m}^{2}}^{S_{2m}}\mathrm{sgn}$ if and only if the Young diagram of $\lambda\vdash2m$ has at most two columns. Combining with (3.10), we have

(3.11) \begin{align}& \bigg|\!\sum_{\eta_{1},\eta'_{1}\in S_{m}^{2}}(-1)^{\eta_{1}\eta'_{1}}\chi_{\lambda}(\eta_{1}\widetilde{\sigma}_{1}^{-1}\eta'_{1}\widetilde{\tau}_{1})\bigg|=\bigg|\!\sum_{\eta_{1}'\in S_{m}^{2}}(-1)^{\eta'_{1}}\sum_{\eta_{1}\in S_{m}^{2}}(-1)^{\eta_{1}}\chi_{\lambda}(\eta_{1}\widetilde{\sigma}_{1}^{-1}\eta'_{1}\widetilde{\tau}_{1})\bigg|\nonumber\\&\quad \leq \sum_{\eta_{1}'\in S_{m}^{2}}\bigg|\!\sum_{\eta_{1}\in S_{m}^{2}}(-1)^{\eta_{1}}\chi_{\lambda}(\eta_{1}\widetilde{\sigma}_{1}^{-1}\eta'_{1}\widetilde{\tau}_{1})\bigg|\leq\begin{cases}m!^{4} & \text{if }\lambda\vdash2m,\ \lambda\text{ has } \leq2\text{ columns},\\0 & \text{otherwise}.\end{cases}\end{align}

By (2.11), (3.11), (2.3), and by our assumption that $d\geq2m$ , we have

\begin{align*}&\bigg|\!\sum_{\eta_{1},\eta'_{1}\in S_{m}^{2}}(-1)^{\eta_{1}\eta'_{1}}\mathrm{Wg}(\eta_{1}\widetilde{\sigma}_{1}^{-1}\eta'_{1}\widetilde{\tau}_{1})\bigg|=\frac{1}{(2m)!^{2}}\,\bigg|\!\sum_{\lambda\vdash2m}\frac{\chi_{\lambda}(1)^{2}}{\rho_{\lambda}(1)}\sum_{\eta_{1},\eta'_{1}\in S_{m}^{2}}(-1)^{\eta_{1}\eta'_{1}}\chi_{\lambda}(\eta_{1}\widetilde{\sigma}_{1}^{-1}\eta'_{1}\widetilde{\tau}_{1})\bigg|\\&\quad \leq \frac{m!^{4}}{(2m)!^{2}}\sum_{\lambda\vdash2m,\,\lambda\text{ has}\leq2\text{ columns}}\frac{\chi_{\lambda}(1)^{2}}{\rho_{\lambda}(1)}=\frac{m!^{4}}{(2m)!}\sum_{\lambda\vdash2m,\,\lambda\text{has }\leq2\text{ columns}}\frac{\chi_{\lambda}(1)}{\prod_{(i,j)\in\lambda}(d+j-i)}\\&\quad \leq \frac{m!^{4}}{(2m)!}\cdot\frac{1}{d\cdot\cdots\cdot(d-2m+1)}\sum_{\lambda\vdash2m,\,\lambda\text{ has }\leq2\text{columns}}\chi_{\lambda}(1)=\frac{m!^{4}}{(2m)!}\frac{\dim\mathrm{Ind}_{S_{m}^{2}}^{S_{2m}}\,\mathrm{sgn}}{d\cdot\cdots\cdot(d-2m+1)}\\&\quad =\frac{m!^{2}}{(2m)!}\binom{d}{2m}^{\!\!-1}.\end{align*}

This concludes the proposition.

We now turn to the last ingredient in the proof of Theorem 3.1.

Definition 3.4. Let $f:S\rightarrow[d]$ be a function on a set S. We define the shape $\nu_{f}:[d]\rightarrow\mathbb{N}$ of f as

\[\nu_{f}=(\nu_{f,1},\ldots,\nu_{f,d}):=(|f^{-1}(1)|,\ldots,|f^{-1}(d)|),\]

and denote $\nu_{f}!:=\prod_{u=1}^{d}\nu_{f,u}$ .

Proposition 3.5. Let Z be as in (3.4). Then,

\[|Z|\leq m!^{7}\binom{2m}{m}^{\!\!4}\binom{d}{m}\binom{d+m-1}{m}^{\!\!3}.\]

1. (1) There are $\frac{m!^{3}}{\nu_{\overrightarrow{\!b}}!\nu_{\overrightarrow{\!c}}!\nu_{\overrightarrow{\!{\kern-.5pt}d}}!}$ options for $\overrightarrow{\!b},\overrightarrow{\!c},\overrightarrow{\!{\kern-.5pt}d}$ with the above shapes.

2. (2) There are $(2m)!^{2}$ options for $\sigma_{2}$ and $\tau_{2}$ .

3. (3) There are at most $\binom{2m}{m}^{2}$ options for choosing $\tau_{1}^{-1}([m])$ and $\sigma_{1}([m])$ , as subsets of [2m]. Note that we count both valid and invalid options.

4. (4) After fixing the subsets $\tau_{1}^{-1}([m])$ and $\sigma_{1}([m])$ , there are at most $\nu_{\overrightarrow{\!c}}!\nu_{\overrightarrow{\!{\kern-.5pt}d}}!$ options for $\tau_{1}$ and $\nu_{\overrightarrow{\!b}}!$ options for $\sigma_{1}$ .

Summarizing the above items, we get there are at most $(\frac{m!^{3}(2m)!^{2}\nu_{\overrightarrow{\!b}}!\nu_{\overrightarrow{\!c}}!\nu_{\overrightarrow{\!{\kern-.5pt}d}}!}{\nu_{\overrightarrow{\!b}}!\nu_{\overrightarrow{\!c}}!\nu_{\overrightarrow{\!{\kern-.5pt}d}}!})\binom{2m}{m}^{\!2}=m!^{7}\binom{2m}{m}^{\!4}$ options for $(\overrightarrow{\!a},\ldots,\overrightarrow{\!{\kern-.5pt}D},\sigma_{1},\sigma_{2},\tau_{1},\tau_{2})\in Z$ with the initial data . Note that there are $\binom{d}{m}$ possible options for $\overrightarrow{\!a}$ , and $\binom{d+m-1}{m}^{\!3}$ options for . This gives the desired upper bound.

We can now finish the proof of Theorem 3.1.

Proof of Theorem 3.1. Note that for every $k\geq1$ and $n\geq k$ we have

(3.12) \begin{equation}\bigg(\frac{n}{k}\bigg)^{\!\!k}\leq\prod_{j=0}^{k-1}\bigg(\frac{n-j}{k-j}\bigg)=\binom{n}{k}\leq\frac{n^{k}}{k!}\leq\bigg(\frac{n}{k}\bigg)^{\!\!k}e^{k},\end{equation}

where the rightmost inequality follows from Stirling’s approximation. By Lemma 3.2 and by Propositions 3.3 and 3.5,

(3.13) \begin{align}\Big|\mathbb{E}\Big(\mathrm{tr}\bigwedge\nolimits^{\!\!m}w(X,Y)\Big)\!\Big|& =|Z|\cdot\bigg|\frac{1}{|Z|}\sum_{(\overrightarrow{\!a},\ldots,\overrightarrow{\!{\kern-.5pt}D},\sigma_{1},\sigma_{2},\tau_{1},\tau_{2})\in Z}\sum_{\pi\in S_{m}}(-1)^{\pi}\,\mathrm{Wg}(\sigma_{1}^{-1}\tau_{1})\cdot\mathrm{Wg}(\sigma_{2}^{-1}(\mathrm{\pi}\times\mathrm{Id})\tau_{2})\bigg|\nonumber \\&\leq|Z|\cdot\frac{1}{(2m)!^{2}m!^{3}}\binom{d}{2m}^{\!\!-2}\leq\binom{2m}{m}^{\!\!2}\binom{d}{m}\binom{d+m}{m}^{\!\!3}\binom{d}{2m}^{\!\!-2}.\end{align}

By (3.13) and (3.12), by the inequality $\binom{2m}{m}\leq2^{2m}$ , and by our assumption that $d\geq2m$ ,

\[\Big|\mathbb{E}\Big(\mathrm{tr}\bigwedge\nolimits^{\!\!m}w(X,Y)\Big)\!\Big|\leq\frac{2^{4m}e^{4m}\big(\frac{d}{m}\big)^{m}\big(\frac{d+m}{m}\big)^{3m}}{\big(\frac{d}{2m}\big)^{4m}}\leq\frac{2^{7m}e^{4m}\big(\frac{d}{m}\big)^{4m}}{\big(\frac{d}{2m}\big)^{4m}}\leq2^{11m}e^{4m}\leq2^{17m}.\]

We encode the expression

(3.14) \begin{equation}X_{a,D}Y_{D,c}X_{c,A}^{-1}Y_{A,b}X_{b,C}Y_{C,d}^{-1}X_{d,B}^{-1}Y_{B,a}^{-1}\end{equation}

from (3.2), graphically, by the $4\times4$ matrix

(3.15) \begin{equation}\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}\cdot & C & D & \cdot\\c & \cdot & \cdot & b\\d & \cdot & \cdot & a\\\cdot & B & A & \cdot\end{array}\right),\end{equation}

which is constructed as follows. The rows and columns are indexed by $x,y,x^{-1},y^{-1}$ . We order the rows by $x<y<y^{-1}<x^{-1}$ and order the columns by $x^{-1}<y^{-1}<y<x$ . To find the $(x,y^{-1})$ -entry of this matrix (i.e. the (1,2)-entry), we look for the subword $XY^{-1}$ in (3.14) and record the letter of the common index, which is C. All other entries are determined in similar fashion. Note that we do not have elements in the main diagonal since w is cyclically reduced.

We denote $\eta_{1}=\tau_{1}$ , $\eta_{2}=\tau_{2}$ , $\eta_{3}=\sigma_{2}^{-1}$ and $\eta_{4}=\sigma_{1}^{-1}$ . Note that $\eta_{i}$ sends the ith row of (3.15) into a permuted copy of its ith column. The alternative counting argument in Proposition 3.5 goes as follows. We fix the upper triangular part, i.e. $\overrightarrow{\!{\kern.5pt}C},\overrightarrow{\!{\kern-.5pt}D},\overrightarrow{\!a},\overrightarrow{\!b}$ (instead of $\overrightarrow{\!a},\overrightarrow{\!b},\overrightarrow{\!c},\overrightarrow{\!{\kern-.5pt}d}$ in the proof above). We then choose $\eta_{1}$ (with $2m!$ options), which gives us $\overrightarrow{\!c},\overrightarrow{\!{\kern-.5pt}d}$ and, in particular, reveals the second row. Next, we choose all possible $\eta_{2}:(\overrightarrow{\!b},\overrightarrow{\!c})\rightarrow(\overrightarrow{\!{\kern-1pt}B},\overrightarrow{\!{\kern.5pt}C})$ , taking into consideration the fact that $\overrightarrow{\!{\kern.5pt}C}$ is already known. We then proceed to the next row and guess $\eta_{3}$ , taking into consideration that we already know $\overrightarrow{\!{\kern-.5pt}D}$ . At this point, the vectors $\overrightarrow{\!a},\overrightarrow{\!b},\ldots,\overrightarrow{\!{\kern.5pt}C},\overrightarrow{\!{\kern-.5pt}D}$ and the permutations $\eta_{1},\eta_{2},\eta_{3}$ are known, and the number of options for $\eta_{4}$ is determined by the shapes of $\overrightarrow{\!a},\overrightarrow{\!b}$ . This argument will be generalized in § 6 for arbitrary words, where, instead of a $4\times4$ matrix, we will have a $2r\times2r$ matrix and, each time we choose $\eta_{1},\ldots,\eta_{k}$ , the $k+1$ th row is revealed, allowing us to proceed by induction.

4 Rewriting Theorem 1.3 using Weingarten calculus

In this section, we rewrite the expression $\mathbb{E}\big(|\mathrm{tr}\bigwedge\nolimits^{\!m}w(X_{1},\ldots,X_{r})|^{2}\big)$ of Theorem 1.3 as a finite sum of Weingarten functions.

Let $\ell,m,d,w$ be as in Theorem 1.3. We may assume that w is cyclically reduced, i.e. it does not contain a subword of the form $x_{j}x_{j}^{-1}$ and the first and last letters of w are not inverse of each other. For $u\in[\ell]$ , let

\[w(u)=\begin{cases}a & \text{if the } u\text{th letter of } w \text{ is } x_{a},\\-a & \text{if the } u\text{th letter of } w \text{ is } x_{a}^{-1}.\end{cases}\]

If we denote $x_{-a}=x_{a}^{-1}$ , then $w=\prod_{u}x_{w(u)}$ . We write $w^{-1}$ for the inverse word,

(4.1) \begin{equation}w^{-1}:=x_{-w(\ell)}x_{-w(\ell-1)}\cdots x_{-1}.\end{equation}

We start by noting that

(4.2) \begin{align}&\mathbb{E}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)\nonumber\\&\quad =\mathbb{E}\Big(\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big)\cdot\overline{\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big)}\Big)\nonumber\\&\quad =\mathbb{E}\Big(\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big)\cdot\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w^{-1}(X_{1},\ldots,X_{r})\Big)\Big).\end{align}

Define $\widetilde{T}\in\mathrm{Sym}([\ell]\times[m])$ by

(4.3) \begin{equation}\widetilde{T}(u,k)=\begin{cases}(u+1,k) & u\neq\ell,\\(1,k) & u=\ell.\end{cases}\end{equation}

Recall that ${\mathcal I}_{m,d}=\{a_{1}<\cdots<a_{m}:a_{i}\in[d]\}$ . We have

(4.4) \begin{align}&\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big)\nonumber\\&\quad =\sum_{\overrightarrow{\!a}\in{\mathcal I}_{m,d}}\sum_{\pi\in S_{m}}(-1)^{\pi}\prod_{k=1}^{m}w(X_{1},\ldots,X_{r})_{a_{k},a_{\pi(k)}}\nonumber\\&\quad =\sum_{\overrightarrow{\!a}\in{\mathcal I}_{m,d}}\sum_{\pi\in S_{m}}(-1)^{\pi}\prod_{k=1}^{m}\sum_{\substack{f_{k}:[\ell+1]\rightarrow[d]\\f_{k}(1)=a_{k},f_{k}(\ell+1)=a_{\pi(k)} }}\prod_{u=1}^{\ell}(X_{w(u)})_{f_{k}(u),f_{k}(u+1)}\nonumber\\[4pt]&\quad =\sum_{\overrightarrow{\!a}\in{\mathcal I}_{m,d}}\sum_{\pi\in S_{m}}(-1)^{\pi}\sum_{\substack{f:[\ell+1]\times[m]\rightarrow[d]\\f(1,k')=a_{k'},f(\ell+1,k')=a_{\pi(k')},\forall k'}}\prod_{(u,k)\in[\ell]\times[m]}(X_{w(u)})_{f(u,k),f(u+1,k)}\nonumber\\[4pt]&\quad =\sum_{\pi\in S_{m}}(-1)^{\pi}\sum_{\substack{f:[\ell+1]\times[m]\rightarrow[d]\\f(\ell+1,k')=f(1,\pi(k')),\forall k'\\ f(1,-)\text{ increasing}}}\prod_{(u,k)\in[\ell]\times[m]}(X_{w(u)})_{f(u,k),f(u+1,k)}\nonumber\\[4pt]&\quad =\sum_{\pi\in\mathrm{Sym}(\{ \ell\}\times[m])}(-1)^{\pi}\sum_{\substack{F:[\ell]\times[m]\rightarrow[d]\\ F(1,-)\text{increasing}}}\prod_{(u,k)\in[\ell]\times[m]}(X_{w(u)})_{F(u,k),F(\widetilde{T}\pi(u,k))},\end{align}

where in the last equality we use the natural embedding $\mathrm{Sym}(\{ \ell\}\times[m])\hookrightarrow\mathrm{Sym}([\ell]\times[m])$ obtained by acting trivially on $[\ell-1]\times[m]$ . Applying this to $w^{-1}$ , we get

(4.5) \begin{align}&\overline{\mathrm{tr}\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})}\nonumber\\[4pt]&\quad =\sum_{\pi'\in\mathrm{Sym}(\{\ell\} \times[m])}(-1)^{\pi'}\sum_{\substack{F':[\ell]\times[m]\rightarrow[d]\\F'(1,-)\text{ increasing}}}\prod_{(u,k)\in[\ell]\times[m]}(X_{w^{-1}(u)})_{F'(u,k),F'(\widetilde{T}\pi'(u,k))}.\end{align}

Set $\Omega=[2]\times[\ell]\times[m]$ , $\Omega_{s,u}=\{ s\} \times\{ u\} \times[m]$ , and for $\gamma\in\Omega$ , define

\[\widetilde{w}(\gamma)=\begin{cases}w(u) & \gamma=(1,u,k),\\w^{-1}(u) & \gamma=(2,u,k).\end{cases}\]

Define $T\in\mathrm{Sym}(\Omega)$ by

(4.6) \begin{equation}T(s,u,k):=(s,\widetilde{T}(u,k)).\end{equation}

By combining (4.4) and (4.5), we get

(4.7) \begin{equation}\Big|\mathrm{tr}\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big|^{2}=\sum_{(\pi,\pi')\in\prod_{s=1}^{2}\mathrm{Sym}(\Omega_{s,\ell})}(-1)^{\pi\pi'}\sum_{\substack{F:\Omega\rightarrow[d]\\F(1,1,-)\ \text{increasing}\\ F(2,1,-)\ \text{increasing}}}\prod_{\gamma\in\Omega}(X_{\widetilde{w}(\gamma)})_{F(\gamma),F(T\pi\pi'(\gamma))}.\end{equation}

The map $\pi\mapsto T\pi T^{-1}$ is an isomorphism $\mathrm{Sym}(\Omega_{s,\ell})\overset{\simeq}{\rightarrow}\mathrm{Sym}(\Omega_{s,1})$ , for $s\in[2]$ . Hence,

(4.8) \begin{align}&\Big|\mathrm{tr}\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big|^{2}\nonumber\\&\quad =\sum_{(\pi,\pi')\in\mathrm{Sym}(\Omega_{1,1})\times\mathrm{Sym}(\Omega_{2,1})}(-1)^{\pi\pi'}\sum_{\substack{F:\Omega\rightarrow[d]\nonumber\\F(1,1,-)\ \text{increasing}\\ F(2,1,-)\ \text{increasing}}}\prod_{\gamma\in\Omega}(X_{\widetilde{w}(\gamma)})_{F(\gamma),F(\pi\pi'T(\gamma))}\nonumber\\&\quad =\sum_{(\pi,\pi')\in\mathrm{Sym}(\Omega_{1,1})\times\mathrm{Sym}(\Omega_{2,1})}(-1)^{\pi\pi'}\sum_{\substack{F:\Omega\rightarrow[d]\\F(1,1,-)\ \text{increasing}\\ F(2,1,-)\ \text{increasing}}}\prod_{\gamma\in\Omega}(X_{\widetilde{w}(\gamma)})_{F(\gamma),F((\pi\pi')^{-1}T(\gamma))}\nonumber\\&\quad =\sum_{(\pi,\pi')\in\mathrm{Sym}(\Omega_{1,1})\times\mathrm{Sym}(\Omega_{2,1})}(-1)^{\pi\pi'}\sum_{\substack{F:\Omega\rightarrow[d]\\F\circ\pi(1,1,-)\ \text{increasing}\\ F\circ\pi'(2,1,-)\ \text{increasing}}}\prod_{\gamma\in\Omega}(X_{\widetilde{w}(\gamma)})_{F(\pi\pi'\gamma),F(T(\gamma))},\end{align}

where, in the last equality, we replaced F by $F\circ\big(\pi'\pi\big)^{-1}$ .

Let $\Phi=(A_{1},\ldots,A_{r},B_{1},\ldots,B_{r})$ be the partition given by

(4.9) \begin{equation}A_{i}=\{ (s,u,k)\mid\widetilde{w}(s,u,k)=i\} \quad B_{i}=\{ (s,u,k)\mid\widetilde{w}(s,u,k)=-i\} .\end{equation}

For each $(\pi,\pi')\in\mathrm{Sym}(\Omega_{1,1})\times\mathrm{Sym}(\Omega_{2,1})$ , set

(4.10) \begin{equation}Z_{\pi,\pi'}:=\left\{ (F,\Sigma):\substack{F:\Omega\rightarrow[d],\Sigma\in S_{\Phi}\\F\circ\pi(1,1,-),F\circ\pi'(2,1,-)\text{ increasing}\\F\circ T=F\circ\pi\pi'\circ\Sigma} \right\} .\end{equation}

The sets $Z_{\pi,\pi'}$ are disjoint. We use the notation

(4.11) \begin{equation}Z:=\bigcup_{\pi,\pi'}Z_{\pi,\pi'}.\end{equation}

Rewriting (4.8) using Weingarten calculus (Corollary 2.15), we have the following result.

Proposition 4.2. Let $w\in F_{r}$ be a cyclically reduced word. Then,

(4.12) \begin{equation}\mathbb{E}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)=\sum_{(F,\Sigma)\in Z}(-1)^{\pi_{F}\pi'_{F}}\widetilde{\mathrm{Wg}}(\Sigma^{2}).\end{equation}

5. Estimating the contribution of a single orbit in Z

In this section we introduce an action of $H:=\prod_{(s,u)\in[2]\times[\ell]}\mathrm{Sym}(\Omega_{s,u})$ on Z, and estimate (4.12) restricted to each H-orbit. The action can be described as follows.

For every $(s,u)\in[2]\times[\ell]$ , the group $\mathrm{Sym}(\Omega_{s,u})$ acts on Z in the following way: if $u\neq1$ , the action of $\pi_{s,u}\in\mathrm{Sym}(\Omega_{s,u})$ is

(5.1) \begin{equation}\pi_{s,u}\cdot(F,\Sigma)=(F\circ\pi_{s,u}^{-1},\pi_{s,u}\circ\Sigma\circ T^{-1}\pi_{s,u}^{-1}T).\end{equation}

If $s\in[2]$ and $\pi_{s,1}\in\mathrm{Sym}(\Omega_{s,1})$ , then

(5.2) \begin{equation}\pi_{s,1}\cdot (F,\Sigma)=(F\circ\pi_{s,1}^{-1},\Sigma\circ T^{-1}\pi_{s,1}^{-1}T).\end{equation}

The above group actions commute, which gives rise to an action of H. Note that $(\pi_{1,1},\pi_{2,1})\cdot Z_{\pi,\pi'}=Z_{\pi_{1,1}\pi,\pi_{2,1}\pi'}$ . If $u\neq1$ , then $\pi_{s,u}\cdot(Z_{\pi,\pi'})=Z_{\pi,\pi'}$ .

Definition 5.1. For each $u,v\in[\ell]$ , we define $*:\mathrm{Sym}(\Omega_{s,u})\times\mathrm{Sym}(\Omega_{s,v})\rightarrow\mathrm{Sym}(\Omega_{s,v})$ by

(5.3) \begin{equation}\pi_{s,u}*\pi_{s,v}:=T^{v-u}\pi_{s,u}T^{u-v}\pi_{s,v}.\end{equation}

Note that $*$ is associative.

Let $h:=\prod_{(s,u)}\pi_{s,u}\in H$ and denote $\overline{h}:=\prod_{(s,u)\neq(1,1),(2,1)}\pi_{s,u}$ . Then $h\cdot\Sigma=\overline{h}\circ\Sigma\circ T^{-1}h^{-1}T$ . Since $\widetilde{\mathrm{Wg}}$ is invariant under conjugation in H,

\[\widetilde{\mathrm{Wg}}((h\cdot(\Sigma))^{2})=\widetilde{\mathrm{Wg}}(\Psi_{h}\circ\Sigma\circ\Psi_{h}\circ\Sigma),\]

where $\Psi_{h}=T^{-1}h^{-1}T\overline{h}\in H$ . On each $\Omega_{s,u}$ , $\Psi_{h}$ has the following form.

Lemma 5.2. We have

\[\Psi_{h}|_{\Omega_{s,u}}=\begin{cases}T^{-1}\pi_{s,2}^{-1}T & \text{if }u=1,\\\pi_{s,u+1}^{-1}*\pi_{s,u} & \text{if }u\neq1,\ell,\\\pi_{s,1}^{-1}*\pi_{s,\ell} & \text{if }u=\ell.\end{cases}\]

Corollary 5.3. Let $(\widehat{F},\widehat{\Sigma})$ be a representative of an H-orbit ${\mathcal O}_{(\widehat{F},\widehat{\Sigma})}$ , with $(\pi_{\widehat{F}},\pi'_{\widehat{F}})=(\mathrm{Id},\mathrm{Id})$ . Then,

(5.4) \begin{align}&\frac{1}{\big|{\mathcal{O}}_{(\widehat{F},\widehat{\Sigma})}\big|}\sum_{(F,\Sigma)\in{\mathcal{O}}_{(\widehat{F},\widehat{\Sigma})}}(-1)^{\pi_{F}\pi'_{F}}\,\widetilde{\mathrm{Wg}}(\Sigma^{2})\nonumber\\&\quad =\frac{1}{m!^{2\ell}}\prod_{i=1}^{r}\sum_{\substack{h_{i}\in\prod_{\widetilde{w}=i}\mathrm{Sym}(\Omega_{s,u})\\h'_{i}\in\prod_{\widetilde{w}=-i}\mathrm{Sym}(\Omega_{s,u}) }}(-1)^{h_{i}h'_{i}}\,\mathrm{Wg}(h_{i}\widehat{\Sigma}|_{B_{i}}h'_{i}\widehat{\Sigma}|_{A_{i}}).\end{align}

Proof. For each $h=\prod_{(s,u)}\pi_{s,u}\in H$ , write $\nu(h):=(-1)^{\pi_{1,1}\pi_{2,1}}$ . Consider the bijection $\psi:H\rightarrow H$ , $\psi\big(\prod_{(s,u)}\pi_{s,u}\big)=\prod_{(s,u)}\theta_{s,u}$ , where, for $s=1,2$ ,

\[(\theta_{s,2},\ldots,\theta_{s,\ell},\theta_{s,1})=(\pi_{s,2},\pi_{s,2}*\pi_{s,3},\ldots,\pi_{s,2}*\cdots*\pi_{s,\ell},\pi_{s,2}*\cdots*\pi_{s,\ell}*\pi_{s,1}),\]

and observe that $\nu(\psi(h))=(-1)^{h}=(-1)^{T^{-1}h^{-1}T}$ . Further note that

\[(\pi_{s,2}*\cdots*\pi_{s,u+1})^{-1}*\pi_{s,2}*\cdots*\pi_{s,u}=T^{-1}\pi_{s,u+1}^{-1}T,\]

and hence $\Psi_{\psi(h)}=\prod_{(s,u)}T^{-1}\pi_{s,u}^{-1}T$ . Changing variables using $\psi$ , the left-hand side of (5.4) is

\begin{align*}\frac{1}{|H|}\!\sum_{h\in H}\!\nu(h)\widetilde{\mathrm{Wg}}(\Psi_{h}\circ\widehat{\Sigma}\circ\Psi_{h}\circ\widehat{\Sigma})& =\frac{1}{m!^{2\ell}}\!\sum_{h\in H}\!\nu(\psi(h))\widetilde{\mathrm{Wg}} \bigg(\prod_{(s,u)}\! T^{-1}\pi_{s,u}^{-1}T\circ\widehat{\Sigma}\circ\!\prod_{(s,u)}\! T^{-1}\pi_{s,u}^{-1}T\circ\widehat{\Sigma}\bigg)\\& =\frac{1}{m!^{2\ell}}\sum_{h\in H}(-1)^{h}\widetilde{\mathrm{Wg}}\bigg(\prod_{(s,u)}\pi_{s,u}\circ\widehat{\Sigma}\circ\prod_{(s,u)}\pi_{s,u}\circ\widehat{\Sigma}\bigg)\\& =\frac{1}{m!^{2\ell}}\sum_{h\in H}(-1)^{h}\prod_{i=1}^{r}\mathrm{Wg}\bigg(\prod_{(s,u):\widetilde{w}=i}\pi_{s,u}\widehat{\Sigma}|_{B_{i}}\!\prod_{(s,u):\widetilde{w}=-i}\!\pi_{s,u}\widehat{\Sigma}|_{A_{i}}\bigg),\end{align*}

where, in each line above, $h=\prod_{(s,u)}\pi_{s,u}$ .

Corollary 5.4. Set $\ell_{i}:=\frac{|A_{i}|}{m}$ for each $i\in[r]$ . Then the following holds:

(5.5) \begin{equation}\mathbb{E}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)\leq\frac{|Z|}{m!^{\ell}}\prod_{i=1}^{r}\frac{1}{d\cdots(d-m\ell_{i}+1)}.\end{equation}

Proof. By Proposition 4.2, Corollary 5.3, (2.11), Lemma 2.7, and by (2.3),

(5.6) \begin{align}\!\!\!\mathbb{E}\Big(\Big|\mathrm{tr}\!\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\!\Big)\!\Big|^{2}\Big)&= \sum_{(\widehat{F},\widehat{\Sigma})\in Z}(-1)^{\pi_{\widehat{F}}\pi'_{\widehat{F}}}\widetilde{\mathrm{Wg}}(\widehat{\Sigma}^{2})\nonumber\\&= \sum_{(\widehat{F},\widehat{\Sigma})\in Z}\frac{1}{\big|{\mathcal{O}}_{(\widehat{F},\widehat{\Sigma})}{\kern-1pt}\big|}\sum_{(F,\Sigma)\in{\mathcal{O}}_{(\widehat{F},\widehat{\Sigma})}}(-1)^{\pi_{F}\pi'_{F}}\,\widetilde{\mathrm{Wg}}(\Sigma^{2})\nonumber\\&\leq \sum_{(\widehat{F},\widehat{\Sigma})\in Z}\frac{1}{m!^{2\ell}}\prod_{i=1}^{r}\left|\sum_{\substack{h_{i}\in\prod_{\widetilde{w}=i}\mathrm{Sym}(\Omega_{s,u})\\h'_{i}\in\prod_{\widetilde{w}=-i}\mathrm{Sym}(\Omega_{s,u}) }}(-1)^{h_{i}h'_{i}}\mathrm{Wg}(h_{i}\widehat{\Sigma}|_{B_{i}}h'_{i}\widehat{\Sigma}|_{A_{i}})\right|\nonumber\\&\leq \sum_{(\widehat{F},\widehat{\Sigma})\in Z}\frac{1}{m!^{2\ell}}\prod_{i=1}^{r}\frac{m!^{2\ell_{i}}}{(m\ell_{i})!^{2}}\sum_{\lambda\vdash m\ell_{i}:\chi_{\lambda}\subseteq\mathrm{Ind}_{\mathrm{S}_{m}^{\ell_{i}}}^{S_{m\ell_{i}}}(\mathrm{sgn})}\!\langle\chi_{\lambda},\mathrm{sgn}\rangle_{\mathrm{S}_{m}^{\ell_{i}}}\frac{\chi_{\lambda}(1)^{2}}{\rho_{\lambda}(1)}\nonumber\\&=\frac{|Z|}{m!^{\ell}}\prod_{i=1}^{r}\frac{m!^{\ell_{i}}}{(m\ell_{i})!}\sum_{\lambda\vdash m\ell_{i}:\chi_{\lambda}\subseteq\mathrm{Ind}_{\mathrm{S}_{m}^{\ell_{i}}}^{S_{m\ell_{i}}}(\mathrm{sgn})}\frac{\langle\chi_{\lambda},\mathrm{sgn}\rangle_{\mathrm{S}_{m}^{\ell_{i}}}\chi_{\lambda}(1)}{\prod_{(a,b)\in\lambda}(d+b-a)}.\end{align}

Note that the irreducible characters $\chi_{\lambda}$ in $\mathrm{Ind}_{\mathrm{S}_{m}^{\ell_{i}}}^{S_{m\ell_{i}}}(\mathrm{sgn})$ correspond to Young diagrams $\lambda\vdash m\ell_{i}$ with at most $\ell_{i}$ columns. If the columns of $\lambda$ are of lengths $j_{1}\geq\cdots\geq j_{\ell_{i}}$ , then

(5.7) \begin{align}\prod_{(a,b)\in\lambda}(d+b-a)&\geq d\cdots(d-j_{1}+1)\cdot d\cdots(d-j_{2}+1)\cdot \ldots\cdot d\cdots(d-j_{\ell_{i}}+1)\nonumber\\&\geq d\cdots(d-m\ell_{i}+1).\end{align}

Combining (5.6) with (5.7) implies the corollary.

6. Estimates on $|Z|$

In this section we give upper bounds on $|Z|$ , defined in (4.11). We first set some notation. For each $0\neq i,j\in[-r,r]$ , set

\[R_{i}:=\{ \gamma\in\Omega:\widetilde{w}\circ T^{-1}(\gamma)=i\} =\begin{cases}T(A_{i}) & i>0,\\T(B_{-i}) & i<0,\end{cases}\]

\[C_{j}:=\{ \gamma\in\Omega:\widetilde{w}(\gamma)=-j\} =\begin{cases}B_{j} & j>0,\\A_{-j} & j<0,\end{cases}\]

\[V_{ij}:=\{ \gamma\in\Omega:\widetilde{w}\circ T^{-1}(\gamma)=i,\widetilde{w}(\gamma)=-j\}=R_{i}\cap C_{j}.\]

Following Remark 3.6, it is helpful to picture a $2r\times2r$ matrix, whose (i,j)th entry is the set $V_{ij}$ , with $R_{-r},\ldots,R_{r}$ correspond to rows, and $C_{-r},\ldots,C_{r}$ correspond to columns. Denote

(6.1) \begin{equation}\ell_{i,j}:=\frac{|V_{ij}|}{m}\quad \text{and}\quad\ell_{i}:=\frac{|R_{i}|}{m}=\frac{|C_{i}|}{m}.\end{equation}

Observe that $\ell_{i}=\ell_{-i}$ , $\ell_{i,j}=\ell_{j,i}$ and note that $\ell_{i}=\frac{|A_{i}|}{m}$ if $i>0$ , so that (6.1) extends the definition of $\ell_{i}$ in Corollary 5.4. For each $0\neq j\in[-r,r]$ set

\[C_{j}^{+}:=\bigcup_{i<j}V_{ij},\quad C_{+}:=\bigcup_{j}C_{j}^{+}.\]

For each $i\in[r]$ and each $\Sigma\in S_{\Phi}$ , denote $\eta_{i}:=T\circ(\Sigma^{-1})|_{B_{i}}$ and $\eta_{-i}:=T\circ(\Sigma^{-1})|_{A_{i}}$ . Note that $\eta_{i}(C_{i})=R_{i}$ for all i. Define the following sets:

(6.2) \begin{equation}W':=\{ (F:\Omega\rightarrow[d],\Sigma\in S_{\Phi}):F\circ T=F\circ\Sigma\} ,\end{equation}

and

\[W:=\{ (F,\Sigma)\in W':F(s,1,-)\text{ is one-to-one } \forall s\in[2]\} .\]

Proposition 6.1. We have

\[|Z|=|W|\leq|W'|\leq\binom{d+m\ell}{m\ell}(m\ell)!\prod_{0\neq k=-r}^{r}\frac{(m\ell_{k})!}{\big(\sum_{i<k}m\ell_{i,k}\big)!}.\]

In order to prove the last inequality, we use the map $\Phi_{+}:W'\rightarrow\{f:C_{+}\rightarrow[d]\}$ , sending $(F,\Sigma)\in W'$ to $F|_{C_{+}}$ . We estimate $|W'|$ by analyzing the fibers of $\Phi_{+}$ . Let $f\in\Phi_{+}(W')$ and suppose it has a shape $\nu_{+}$ (see Definition 3.4). We write $\nu_{j,+}$ for the shapes of $f|_{C_{j}^{+}}$ . We reveal $(F,\Sigma)\in\Phi_{+}^{-1}(f)$ row by row, starting with the $-r$ th row $R_{-r}$ and making sure that, in each step, $F\circ T|_{T^{-1}(R_{k})}=F\circ\Sigma|_{T^{-1}(R_{k})}$ , or, equivalently, $F|_{R_{k}}=F\circ\eta_{k}^{-1}|_{R_{k}}$ .

1. 1. There are at most $(m\ell_{-r})!$ options for $\eta_{-r}$ . Note that $R_{-r\subseteq C_+}$ and, hence, by (6.2), the choice of $\eta_{-r}$ determines $F|_{C_{-r}}$ . At this point, $F|_{R_{-r+1}}$ is determined as well.

2. 2. Note that $C_{-r+1}^{+}=V_{-r,-r+1}$ . There are at most:

   1. (a) $\binom{m\ell_{-r+1}}{m\ell_{-r,-r+1}}$ options for the sets $\eta_{-r+1}(C_{-r+1}^{+})$ and $\eta_{-r+1}(C_{-r+1}\backslash C_{-r+1}^{+})$ ;

   2. (b) $(m(\ell_{-r+1}-\ell_{-r,-r+1}))!$ options for $\eta_{-r+1}|_{C_{-r+1}\backslash C_{-r+1}^{+}}:C_{-r+1}\backslash C_{-r+1}^{+}\rightarrow\eta_{-r+1}(C_{-r+1}\backslash C_{-r+1}^{+})$ ;

   3. (c) $(\nu_{-r+1,+})!$ options for $\eta_{-r+1}:C_{-r+1}^{+}\rightarrow\eta_{-r+1}(C_{-r+1}^{+})$ .

3. 3. More generally, assume, by induction, that we have fixed $(\eta_{i})_{i<k}$ , and, thus, we have already determined $F|_{R_{i}}$ for $i\leq k$ , $F|_{C_{i}}$ for $i<k$ , and $F|_{C_{+}}$ . Then there are at most:

4. (a) $\binom{m\ell_{k}}{\sum_{i<k}m\ell_{i,k}}$ options for the sets $\eta_{k}(C_{k}^{+})$ and $\eta_{k}(C_{k}\backslash C_{k}^{+})$ ;

5. (b) $\big(m(\ell_{k}-\sum_{i<k}\ell_{i,k})\big)!$ options for $\eta_{k}|_{C_{k}\backslash C_{k}^{+}}:C_{k}\backslash C_{k}^{+}\rightarrow\eta_{k}(C_{k}\backslash C_{k}^{+})$ ;

6. (c) $(\nu_{k,+})!$ options for $\eta_{k}|_{C_{k}^{+}}:C_{k}^{+}\rightarrow\eta_{k}(C_{k}^{+})$ .

After choosing $\eta_{-r},\ldots,\eta_{r}$ , we have determined F. Furthermore, since $\sum_{0\neq k=-r}^{r}\nu_{k,+}=\nu_{+}$ , we have $\prod_{0\neq k=-r}^{r}(\nu_{k,+})!\leq\nu_{+}!$ . Hence,

(6.3) \begin{align}|\Phi_{+}^{-1}(f)| & \leq\prod_{0\neq k=-r}^{r}\bigg(\!\binom{m\ell_{k}}{\sum_{i<k}m\ell_{i,k}}\bigg(m\bigg(\ell_{k}-\sum_{i<k}\ell_{i,k}\bigg)\!\bigg)!(\nu_{k,+})!\bigg)\nonumber\\& =\prod_{0\neq k=-r}^{r}(\nu_{k,+})!\frac{(m\ell_{k})!}{\big(\sum_{i<k}m\ell_{i,k}\big)!}\leq\nu_{+}!\prod_{0\neq k=-r}^{r}\frac{(m\ell_{k})!}{\big(\sum_{i<k}m\ell_{i,k}\big)!}.\end{align}

Since $|C_{+}|=m\ell$ , we have

(6.4) \begin{equation}|\{ f\in\Phi_{+}(W'):f\text{ is of shape }\nu_{+}\} |\leq\frac{(m\ell)!}{\nu_{+}!},\end{equation}

and there are at most $\binom{d+m\ell}{m\ell}$ possible shapes $\nu_{+}$ . Combining (6.3) and (6.4) we conclude

\begin{align*}|W'| & \leq\sum_{\nu_{+}}|\{ f\in\Phi_{+}(W'):f\text{ is of shape}\nu_{+}\} |\cdot|\Phi_{+}^{-1}(f)|\\& \leq\sum_{\nu_{+}}\frac{(m\ell)!}{\nu_{+}!}\cdot\nu_{+}!\prod_{0\neq k=-r}^{r}\frac{(m\ell_{k})!}{\big(\sum_{i<k}m\ell_{i,k}\big)!}\leq\binom{d+m\ell}{m\ell}(m\ell)!\prod_{0\neq k=-r}^{r}\frac{(m\ell_{k})!}{\big(\sum_{i<k}m\ell_{i,k}\big)!}.\quad\quad\end{align*}

7. Proof of Theorems 1.1 and 1.3

In this section we use the results of §§ 4, 5 and 6 to prove Theorems 1.1 and 1.3. We end the section with the proof of Theorem 1.6.

Proof of Theorem 1.3. Assume that $d=am$ for $a\geq\ell\geq2$ . By (3.12), we have

(7.1) \begin{equation}\binom{d}{m\ell}=\binom{am}{m\ell}\geq\frac{a^{m\ell}}{\ell^{m\ell}},\end{equation}

(7.2) \begin{equation}\binom{d+m\ell}{m\ell}\leq\bigg(\frac{a+\ell}{\ell}\bigg)^{\!\!m\ell}e^{m\ell}\leq\frac{a^{m\ell}(2e)^{m\ell}}{\ell^{m\ell}}.\end{equation}

We remind the reader the definition of $\ell_{i}$ and $\ell_{i,j}$ in (6.1). Concretely, for each $0\neq i,j\in[-r,r]$ , $\ell_{i}$ is the combined number of appearances of the letter $x_{i}$ (with the convention that $x_{-i}=x_{i}^{-1}$ ) in w and $w^{-1}$ , and $\ell_{i,j}$ is the combined number of appearances of the string ‘ $x_{i}x_{j}^{-1}$ ’ in w and in $w^{-1}$ . In particular, we have $\sum_{i=1}^{r}\ell_{i}=\ell$ , $\ell_{i,i}=0$ and $\sum_{0\neq i\in[-r,r]}\ell_{i,k}=\ell_{k}$ and, therefore,

(7.3) \begin{equation}\begin{aligned}&\prod_{i=1}^{r}d\cdots(d-m\ell_{i}+1)\geq d\cdots(d-m\ell+1)\\ \text{and}\quad &\frac{(m\ell_{k})!}{\big(\sum_{i\lt k}m\ell_{i,k}\big)!\big(\sum_{i>k}m\ell_{i,k}\big)!}=\binom{m\ell_{k}}{\sum_{i>k} m\ell_{i,k}} \leq2^{m\ell_{k}}.\end{aligned}\end{equation}

By Corollary 5.4, Proposition 6.1 and by (7.3), (7.1) and (7.2), we obtain

\begin{align*}&\mathbb{E}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)\leq\frac{|Z|}{m!^{\ell}}\prod_{i=1}^{r}\frac{1}{d\cdots(d-m\ell_{i}+1)}\\&\quad \leq \binom{d+m\ell}{m\ell}\cdot\frac{(m\ell)!}{\prod_{i=1}^{r}d\cdots(d-m\ell_{i}+1)}\cdot\frac{1}{m!^{\ell}}\cdot\prod_{0\neq k=-r}^{r}\frac{(m\ell_{k})!}{\big(\sum_{i<k}m\ell_{i,k}\big)!}\\&\ \leq \binom{d+m\ell}{m\ell}\cdot\frac{(m\ell)!}{d\cdots(d-m\ell+1)}\cdot\frac{\prod_{0\neq k=-r}^{r}\big(\sum_{i>k}m\ell_{i,k}\big)!}{m!^{\ell}}\cdot\prod_{0\neq k=-r}^{r}\frac{(m\ell_{k})!}{\big(\sum_{i<k}m\ell_{i,k}\big)!\big(\sum_{i>k}m\ell_{i,k}\big)!}\\[5pt]&\ \leq \binom{d+m\ell}{m\ell}\cdot\binom{d}{m\ell}^{-1}\cdot\frac{(m\ell)!}{m!^{\ell}}\cdot\prod_{0\neq k=-r}^{r}2^{m\ell_{k}}\leq(2e)^{m\ell}\ell^{m\ell}\cdot2^{2m\ell}\leq(8e\ell)^{m\ell}\leq(22\ell)^{m\ell}.\end{align*}

Finally, note that if $d\geq(22\ell)^{\ell}m$ , then $(22\ell)^{m\ell}\leq(\frac{d}{m})^{m}\leq\binom{d}{m}$ .

We now turn to the proof of Theorem 1.1. We first deal with the case when the rank is bounded (and prove Conjecture 1.7 in this case) and then prove Theorem 1.1 in the unbounded case.

We remind the reader that for a compact group G, and a word $w\in F_{r}$ , we denote by $\tau_{w,G}:=(w_{G})_{*}(\mu_{G}^{r})$ the word measure associated to w and G, and the Fourier coefficient of $\tau_{w,G}$ at $\rho\in\mathrm{Irr}(G)$ is $a_{w,G,\rho}:=\int_{G^{r}}\rho(w(x_{1},\ldots,x_{r}))\mu_{G}^{r}=\int_{G}\rho(y)\tau_{w,G}$ . If G is a compact connected semisimple Lie group, by [Reference BorelBor83], the map $w_{G}:G^{r}\rightarrow G$ is a submersion outside a proper subvariety in $G^{r}$ . It follows that in this case, or e.g. when G is a finite group, $\tau_{w,G}$ is absolutely continuous with respect to $\mu_{G}$ , and we can write $\tau_{w,G}=f_{w,G}\cdot \mu_{G}$ , with $f_{w,G}\in L^{1}(G)$ . Since $\tau_{w,G}$ is conjugate invariant, $f_{w,G}$ is a class function, and it can be written as a linear combination of characters $f_{w,G}=\sum_{\rho\in\mathrm{Irr}(G)}\overline{a_{w,G,\rho}}\cdot\rho$ .

By Definition 7.1, we see that $\tau_{w_{1}*w_{2},G}=\tau_{w_{1},G}*\tau_{w_{2},G}$ for every $w_{1}\in F_{r_{1}}$ and $w_{2}\in F_{r_{2}}$ . Since $\rho_{1}*\rho_{2}=\frac{\delta_{\rho_{1},\rho_{2}}}{\rho_{1}(1)}\cdot\rho_{1}$ for every $\rho_{1},\rho_{2}\in\mathrm{Irr}(G)$ , we have

(7.4) \begin{equation}a_{w_{1}*w_{2},G,\rho}=\int_{G}\rho(g)\tau_{w_{1}*w_{2},G}(g)=\int_{G}\rho(g)\tau_{w_{1},G}*\tau_{w_{2},G}(g)=\frac{a_{w_{1},G,\rho}\cdot a_{w_{2},G,\rho}}{\rho(1)}.\end{equation}

1. (1) for every compact connected semisimple Lie group G of rank d and every $\rho\in\mathrm{Irr}(G)$ , we have $|a_{w,G,\rho}|\leq\rho(1)^{1-\epsilon(d,w)}$ ;

2. (2) in particular, for every $1\leq m\leq d$ ,

   \[\mathbb{E}_{\mathrm{U}_{d}}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)=\mathbb{E}_{\mathrm{SU}_{d}}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)\leq\binom{d}{m}^{\!\!2(1-\epsilon(d,w))}.\]

Proof. We first prove item (1). Fix $w\in F_{r}$ and a compact connected semisimple Lie group G. Let $\tau_{w,G}=f_{w,G}\mu_{G}$ be the word measure. By (7.4), and since $a_{w^{-1},G,\rho}=\overline{a_{w,G,\rho}}$ for each $\rho\in\mathrm{Irr}(G)$ , we have

(7.5) \begin{equation}a_{w*w^{-1},G,\rho}=\frac{|a_{w,G,\rho}|^{2}}{\rho(1)}.\end{equation}

Replacing w by $w*w^{-1}$ , we may assume that all Fourier coefficients $a_{w,G,\rho}$ are in $\mathbb{R}_{\geq0}$ .

It follows from [Reference Glazer, Hendel and SodinGHS24, Theorem 1.1] that $f_{w,G}\in L^{1+\epsilon'}(G)$ for some $\epsilon'=\epsilon'(G,w)>0$ . By Young’s convolution inequality, it follows that $f_{w,G}^{*t}\in L^{\infty}(G)$ for all $t\geq t_{0}(G,w):=\lceil\frac{1+\epsilon'(G,w)}{\epsilon'(G,w)}\rceil $ (see, e.g., [Reference Glazer, Hendel and SodinGHS24, Section 1.1, end of p.3]). In particular, by (7.4), we deduce that

\[f_{w,G}^{*t_{0}}(1)=\sum_{\rho\in\mathrm{Irr}(G)}\rho(1)^{2-t_{0}}a_{w,G,\rho}^{t_{0}}<\infty.\]

Since $a_{w,G,\rho}\geq0$ , we deduce that $a_{w,G,\rho}<\rho(1)^{1-\frac{2}{t_{0}(G,w)}}$ for all but finitely many $\rho\in\mathrm{Irr}(G)$ . To deal with the remaining finitely many (non-trivial) representations of G, we simply use the bound $a_{w,G,\rho}<\rho(1)$ , which follows e.g. by the Itô–Kawada equidistribution theorem [Reference Kawada and ItôKI40] (see also [Reference ApplebaumApp14, Theorem 4.6.3]), since $\mathrm{Supp}(\tau_{w,G})$ generates G. Since there are only finitely many compact semisimple connected Lie groups of rank d, this implies item (1).

Note that the character $\rho_{(1^{m})}\otimes\rho_{(1^{m})}^{\vee}$ of the representation $\bigwedge\nolimits^{\!m}\mathbb{C}^{d}\otimes\big(\bigwedge\nolimits^{\!m}\mathbb{C}^{d}\big)^{\vee}$ of $\mathrm{SU}_{d}$ is given by $\big|\mathrm{tr}\big(\bigwedge\nolimits^{\!m}(A)\big)\big|^{2}$ . Since $\rho_{(1^{m})}\otimes\rho_{(1^{m})}^{\vee}$ is a sum of irreducible characters, by applying the Itô–Kawada equidistribution theorem to each irreducible character, for each $1\leq m\leq d$ , we have

\begin{align*}\mathbb{E}_{\mathrm{SU}_{d}}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!m}(w(X_{1},\ldots,X_{r}))\Big)\!\Big|^{2}\Big)&=\mathbb{E}_{\mathrm{SU}_{d}}(\rho_{(1^{m})}\otimes\rho_{(1^{m})}^{\vee}(w(X_{1},\ldots,X_{r})))\\&<\rho_{(1^{m})}\otimes\rho_{(1^{m})}^{\vee}(1)=\binom{d}{m}^{\!\!2}.\end{align*}

Since there are only finitely many such m’s, this implies item (2).

Theorem 1.1 now follows from Proposition 7.2 and the following theorem.

Theorem 7.3. For every $\ell\in\mathbb{N}$ , there exist $\epsilon(\ell),C(\ell)>0$ such that, for every $d\geq C(\ell)$ , every $1\leq m\leq d$ , and every word $w\in F_{r}$ of length $\ell$ , one has

\[\mathbb{E}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!m}w(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)\leq\binom{d}{m}^{\!\!2(1-\epsilon(\ell))}.\]

In order to prove Theorem 7.3, we need the following technical lemma.

1. (1) for every $d\in\mathbb{N}$ and every $0<x<1$ such that $dx\in\mathbb{N}$ , we have $({2^{dH(x)}}/{\sqrt{8dx(1-x)}})\leq\binom{d}{xd}\leq({2^{dH(x)}}/{\sqrt{\pi dx(1-x)}})\leq2^{dH(x)}$ ;

2. (2) let $0<\delta\leq\frac{1}{2}$ , then for every $b\in[\delta,\frac{1}{2}]$ , $a\in[\delta,b]$ , and $d>({1}/{\delta^{4}})$ such that bd,ad,d are integers, one has

   \[\binom{d}{(b-a)d}\leq\binom{d}{bd}^{\!\!1-\delta^{2}}.\]

Proof. Item (1) follows, e.g., from [Reference Cover and ThomasCT06, Lemma 17.5.1]. The Taylor series of H(x) around $1/2$ is

(7.6) \begin{equation}H(x)=1-\frac{1}{2\ln2}\sum_{n=1}^{\infty}\frac{(1-2x)^{2n}}{n(2n-1)}.\end{equation}

Since $H'(x)=\log(\frac{1-x}{x})$ , H(x) is monotone increasing in $(0,1/2)$ and, therefore,

\begin{align*}H(b)-H(b-a) & \geq H(b)-H(b-\delta)=\frac{1}{2\ln2}\bigg(\sum_{n=1}^{\infty}\frac{(1-2b+2\delta)^{2n}-(1-2b)^{2n}}{n(2n-1)}\bigg)\\&\geq\frac{1}{2\ln2}((1-2b+2\delta)^{2}-(1-2b)^{2})=\frac{1}{2\ln2}(4\delta^{2}+4\delta(1-2b))\geq2\delta^{2}.\end{align*}

Since $d>\frac{1}{\delta^{4}}\geq16$ , we have $\frac{\log(d)}{d}\leq\frac{1}{\sqrt{d}}\leq\delta^{2}$ . Combining with item (1), we have

\begin{align*}\hphantom{000000000}\binom{d}{(b-a)d} &\leq2^{dH(b-a)}\leq\sqrt{8db(1-b)}2^{d(H(b-a)-H(b))}\binom{d}{bd}\leq2^{-2d\delta^{2}+\log(d)}\binom{d}{bd}\\&\leq2^{-d\delta^{2}}\binom{d}{bd}=(2^{-dH(b)})^{\frac{\delta^{2}}{H(b)}}\binom{d}{bd}\leq\binom{d}{bd}^{\!\!1-\frac{\delta^{2}}{H(b)}}\leq\binom{d}{bd}^{\!\!1-\delta^{2}}.\hphantom{000000000}\end{align*}

Proof of Theorem 7.3. Since $\bigwedge\nolimits^{\!m}V\simeq\big(\bigwedge\nolimits^{\!d-m}V\big)^{\vee}\otimes\chi_{\mathrm{det}}$ , we may assume that $2m\leq d$ . Let $\delta(\ell):=(25\ell)^{-\ell}$ , let $C(\ell)=\delta(\ell)^{-7}$ , and suppose that $d\geq C(\ell)$ . By Theorem 1.3, we may assume that $d\leq\delta(\ell)^{-1}m$ , and, in particular, $m\geq\delta(\ell)^{-6}$ . As in the proof of Proposition 7.2, by replacing w by $w*w^{-1}$ , we may assume that $a_{w,\mathrm{U}_{d},\rho}\in\mathbb{R}_{\geq0}$ for all $\rho\in\mathrm{Irr}(\mathrm{U}_{d})$ . By Theorem 2.5, we have for all $c\leq \frac{d}{2}$ :

\[\bigwedge\nolimits^{\!c}V\otimes\bigwedge\nolimits^{\!c}V^{\vee}\simeq\Big(\!\bigwedge\nolimits^{\!c}V\otimes\bigwedge\nolimits^{\!d-c}V\Big)\otimes\chi_{\mathrm{det}}^{-1}\simeq\bigoplus_{j=0}^{c}V_{\lambda_{(j)}},\]

where $\lambda_{(j)}=(1,\ldots,1,0,\ldots,0,-1,\ldots,-1)$ , with $-1$ and 1 appearing j times. Moreover, $V_{\lambda_{(c)}}$ is the largest irreducible subrepresentation of $\bigwedge\nolimits^{\!c}V\otimes\bigwedge\nolimits^{\!c}V^{\vee}$ , and we have $\rho_{\lambda_{(c)}}(1)\geq (\frac{1}{c+1})\binom{d}{c}^{\!2}\geq\binom{d}{c}^{\!3/2}$ . By Theorem 1.3, and since all $a_{w,\mathrm{U}_{d},\rho}$ are non-negative, if $c\leq\lceil\delta(\ell)d\rceil\leq(22\ell)^{-\ell}d$ , then

\[\mathbb{E}\big(\rho_{\lambda_{(c)}}\circ w\big)\leq\sum_{j=0}^{c}\mathbb{E}\big(\rho_{\lambda_{(j)}}\circ w\big)=\mathbb{E}\big(\rho_{\bigwedge\nolimits^{\!c}V\otimes\bigwedge\nolimits^{\!c}V^{\vee}}\circ w\big)=\mathbb{E}\big(\big|\big(\rho_{\bigwedge\nolimits^{\!c}V}\circ w\big)\big|^{2}\big)\leq\binom{d}{c}\leq\rho_{\lambda_{(c)}}(1)^{2/3}.\]

Applying the last inequality for $w^{*9}$ , recalling that $a_{w^{*t},\mathrm{U}_{d},\rho}=\frac{a_{w,\mathrm{U}_{d},\rho}^{t}}{\rho(1)^{t-1}}$ for all $\rho\in\mathrm{Irr}(\mathrm{U}_{d})$ , we get

(7.7) \begin{align}\mathbb{E}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\lceil\delta(\ell)d\rceil}w^{*9}(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)&=\sum_{j=0}^{\lceil\delta(\ell)d\rceil}\mathbb{E}\big(\rho_{\lambda_{(j)}}\circ w^{*9}\big)\leq\sum_{j=0}^{\lceil\delta(\ell)d\rceil}\rho_{\lambda_{(j)}}(1)^{-2}\nonumber\\&\leq\sum_{j=1}^{\infty}\frac{1}{j^{2}}<2.\end{align}

Note that, for each $\delta(\ell)d\leq m\leq\frac{d}{2}$ , $\bigwedge\nolimits^{\!m}V$ is a subrepresentation of $\bigwedge\nolimits^{\lceil\delta(\ell)d\rceil}V\otimes\bigwedge\nolimits^{\!m-\lceil\delta(\ell)d\rceil}V$ , so

\[\bigwedge\nolimits^{\!\!m}V\otimes\Big(\!\bigwedge\nolimits^{\!\!m}V\!\Big)^{\vee}\!\hookrightarrow\!\Big(\!\bigwedge\nolimits^{\lceil\delta(\ell)d\rceil}V\otimes\Big(\!\bigwedge\nolimits^{\lceil\delta(\ell)d\rceil}V\!\Big)^{\vee}\Big)\otimes\Big(\!\bigwedge\nolimits^{\!\!m-\lceil\delta(\ell)d\rceil}V\otimes\Big(\!\bigwedge\nolimits^{\!\!m-\lceil\delta(\ell)d\rceil}V\!\Big)^{\vee}\Big).\]

Finally, by the positivity of the Fourier coefficients of w, by (7.7), by Lemma 7.4 (note that $m\geq\lceil\delta(\ell)d\rceil$ ) and by (3.12) (note that $\delta(\ell)^{2}m\geq1$ ),

\begin{align*}&\mathbb{E}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w^{*9}(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)\\&\quad \leq\mathbb{E}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\lceil\delta(\ell)d\rceil}w^{*9}(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m-\lceil\delta(\ell)d\rceil}w^{*9}(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)\\&\quad \leq\mathbb{E}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\lceil\delta(\ell)d\rceil}w^{*9}(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)\cdot\binom{d}{m-\lceil\delta(\ell)d\rceil}^{\!\!2}\\&\quad \leq2\binom{d}{m-\lceil\delta(\ell)d\rceil}^{\!\!2}\leq\frac{d}{m}\binom{d}{m}^{\!\!2-2\delta(\ell)^{2}}\leq\binom{d}{m}^{\!\!2-\delta(\ell)^{2}}.\end{align*}

By (3.12), $m+1\leq2^{2\sqrt{m}}\leq\binom{d}{m}^{\!2/\sqrt{m}}$ for each $m\leq\frac{d}{2}$ . Hence,

(7.8) \begin{equation}\rho_{\lambda_{(m)}}(1)\geq\frac{1}{m+1}\binom{d}{m}^{\!\!2}\geq\binom{d}{m}^{\!\!2(1-\frac{1}{\sqrt{m}})}\geq\binom{d}{m}^{\!\!2-2\delta(\ell)^{3}}.\end{equation}

Consequently, we get

\begin{align*}\big(\mathbb{E}\big(\rho_{\lambda_{(m)}}\circ w\big)\big)^{9} &=\mathbb{E}\big(\rho_{\lambda_{(m)}}\circ w^{*9}\big)\rho_{\lambda_{(m)}}(1)^{8}\leq\mathbb{E}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w^{*9}(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)\rho_{\lambda_{(m)}}(1)^{8}\\&\leq\binom{d}{m}^{\!\!2-\delta(\ell)^{2}}\rho_{\lambda_{(m)}}(1)^{8}\leq\rho_{\lambda_{(m)}}(1)^{9-\frac{\delta(\ell)^{2}}{4}},\end{align*}

and, thus, $\mathbb{E}\big(\rho_{\lambda_{(m)}}\circ w\big)\leq\rho_{\lambda_{(m)}}(1)^{1-\frac{\delta(\ell)^{2}}{36}}$ . Taking $\epsilon(\ell):=\frac{\delta(\ell)^{2}}{72}$ , and using $m+1\leq\binom{d}{m}^{\!2\delta(\ell)^{3}}$ , we get

\[\mathbb{E}\Big(\Big|\mathrm{tr}\Big(\!\bigwedge\nolimits^{\!\!m}w(X_{1},\ldots,X_{r})\Big)\!\Big|^{2}\Big)=\sum_{j=0}^{m}\mathbb{E}\big(\rho_{\lambda_{(j)}}\circ w\big)\leq(m+1)\binom{d}{m}^{\!\!2-4\epsilon(\ell)}\leq\binom{d}{m}^{\!\!2(1-\epsilon(\ell))}.\]

We end the section with a proof of Theorem 1.6.

Proof of Theorem 1.6. Let $w\in F_{r}$ . Denote $\widetilde{w}:=w*w^{-1}$ . Recall that $\tau_{\widetilde{w},G}=f_{\widetilde{w},G}\mu_{G}$ and note that for every $t\in\mathbb{N}$ ,

\[ f_{\widetilde{w}^{*t},G}=f_{\widetilde{w},G}^{*t}.\]

Applying [Reference Larsen, Shalev and TiepLST19, Theorem 4], there are $C',M(w)\in\mathbb{N}$ such that, for $N(w):=C'\ell(w)^{4}$ and for every finite simple group G of size $>M(w)$ , one has

\[\bigg|\!\sum_{1\neq\rho\in\mathrm{Irr}(G)}\frac{a_{\widetilde{w},G,\rho}^{N(w)}}{\rho(1)^{N(w)-1}}\rho(1)\bigg|=\bigg|\!\sum_{1\neq\rho\in\mathrm{Irr}(G)}a_{\widetilde{w}^{*N(w)},G,\rho}\rho(1)\bigg|=|f_{\widetilde{w}^{*N(w)},G}(1)-1|=|f_{\widetilde{w},G}^{*N(w)}(1)-1|<1,\]

where the first equality follows from (7.4). Since $a_{\widetilde{w},G,\rho}=\frac{|a_{w,G,\rho}|^{2}}{\rho(1)}\geq0$ , we deduce that for each $1\neq\rho\in\mathrm{Irr}(G)$

\[\frac{|a_{w,G,\rho}|^{2N(w)}}{\rho(1)^{2N(w)-2}}=\frac{|a_{w,G,\rho}|^{2N(w)}}{\rho(1)^{2N(w)-1}}\rho(1)=\frac{a_{\widetilde{w},G,\rho}^{N(w)}}{\rho(1)^{N(w)-1}}\rho(1)<1,\]

from which the theorem follows for $\epsilon=\frac{1}{N(w)}=\frac{1}{C'\ell(w)^{4}}$ .

8. Fourier coefficients of symmetric powers

In this section, we prove Theorem 1.4. Denote ${\mathcal J}_{m,d}=\{c_{1}\leq\cdots\leq c_{m}:c_{i}\in[d]\}$ . We first claim that, for each $A\in\mathrm{End}(\mathbb{C}^{d})$ and $m\geq1$ ,

\[\mathrm{tr}(\mathrm{Sym}^{m}A)=\frac{1}{m!}\sum_{\overrightarrow{\!a}\in[d]^{m}}\sum_{\pi\in S_{m}}A_{a_{1}a_{\pi(1)}}\cdots A_{a_{m}a_{\pi(m)}}.\]

Indeed, for each $\overrightarrow{\!c}\in{\mathcal J}_{m,d}$ , let $\nu_{\overrightarrow{\!c}}$ be the shape of $\overrightarrow{\!c}$ (see Definition 3.4) and set

\[v_{\overrightarrow{\!c}}:=\sqrt{\frac{1}{m!\cdot\nu_{\overrightarrow{\!c}}!}}\sum_{\pi\in S_{m}}e_{c_{\pi(1)}}\otimes\cdots\otimes e_{c_{\pi(m)}}.\]

Then $\{ v_{\overrightarrow{\!c}}\}_{\overrightarrow{\!c}\in{\mathcal J}_{m,d}}$ is an orthonormal basis for $\mathrm{Sym}^{m}(\mathbb{C}^{d})$ . Given $A\in\mathrm{End}(\mathbb{C}^{d})$ , we have

\begin{align*}\mathrm{tr}(\mathrm{Sym}^{m}A) & =\sum_{\overrightarrow{\!c}\in{\mathcal J}_{m,d}}\langle A\cdot v_{\overrightarrow{\!c}},v_{\overrightarrow{\!c}}\rangle\\&=\sum_{\overrightarrow{\!c}\in{\mathcal J}_{m,d}}\frac{1}{m!\cdot\nu_{\overrightarrow{\!c}}!}\sum_{\pi,\pi'\in S_{m}}\big\langle Ae_{c_{\pi(1)}}\otimes\cdots\otimes Ae_{c_{\pi(m)}},e_{c_{\pi'(1)}}\otimes\cdots\otimes e_{c_{\pi'(m)}}\big\rangle\\& =\sum_{\overrightarrow{\!c}\in{\mathcal J}_{m,d}}\frac{1}{\nu_{\overrightarrow{\!c}}!}\sum_{\pi\in S_{m}}\big\langle Ae_{c_{1}}\otimes\cdots\otimes Ae_{c_{m}},e_{c_{\pi(1)}}\otimes\cdots\otimes e_{c_{\pi(m)}}\big\rangle\\& =\sum_{\overrightarrow{\!c}\in{\mathcal J}_{m,d}}\frac{1}{\nu_{\overrightarrow{\!c}}!}\sum_{\pi\in S_{m}}A_{c_{1}c_{\pi(1)}}\cdots A_{c_{m}c_{\pi(m)}}=\frac{1}{m!}\sum_{\overrightarrow{\!a}\in[d]^{m}}\sum_{\pi\in S_{m}}A_{a_{1}a_{\pi(1)}}\cdots A_{a_{m}a_{\pi(m)}},\end{align*}

where the last equality follows since $\sum_{\pi\in S_{m}}A_{c_{1}c_{\pi(1)}}\cdots A_{c_{m}c_{\pi(m)}}$ is invariant under permuting $c_{1},\ldots,c_{m}$ , and since there are $\frac{m!}{\nu_{\overrightarrow{\!c}}!}$ vectors $\overrightarrow{\!a}\in[d]^{m}$ of a shape $\nu_{\overrightarrow{\!c}}$ . In particular, for any word w,

(8.1) \begin{equation}\mathrm{tr}(\mathrm{Sym}^{m}w(X_{1},\ldots,X_{r}))=\frac{1}{m!}\sum_{\overrightarrow{\!a}\in[d]^{m}}\sum_{\pi\in S_{m}}w(X_{1},\ldots,X_{r})_{a_{1}a_{\pi(1)}}\cdots w(X_{1},\ldots,X_{r})_{a_{m}a_{\pi(m)}}.\end{equation}

Proposition 8.1. Let $w\in F_{r}$ be a cyclically reduced word. With $\Phi,T,\Omega,\Omega_{s,u}$ as in § 4, we have

(8.2) \begin{equation}\mathbb{E}(|\mathrm{tr}(\mathrm{Sym}^{m}w(X_{1},\ldots,X_{r}))|^{2})=\frac{1}{m!^{2}}\sum_{(\pi,\pi',F,\Sigma)\in\widetilde{Z}}\widetilde{\mathrm{Wg}}(\Sigma^{2}),\end{equation}

where

\[\widetilde{Z}:=\bigg\{ (\pi,\pi',F,\Sigma):\substack{F:\Omega\rightarrow[d],\Sigma\in S_{\Phi}\\ \pi,\pi'\in\mathrm{Sym}(\Omega_{1,1})\times\mathrm{Sym}(\Omega_{2,1})\\ F\circ T=F\circ\pi\pi'\circ\Sigma } \bigg\} .\]

Proof. Similarly to (4.4), we have

\begin{align*}\mathrm{tr}(\mathrm{Sym}^{m}w(X_{1},\ldots,X_{r})) &=\frac{1}{m!}\sum_{\overrightarrow{\!a}\in[d]^{m}}\sum_{\pi\in S_{m}}\sum_{\substack{f:[\ell+1]\times[m]\rightarrow[d]\\f(1,k)=a_{k},f(\ell+1,k)=a_{\pi(k)} }}\prod_{(u,k)\in[\ell]\times[m]}(X_{w(u)})_{f(u,k),f(u+1,k)}\\& =\sum_{\pi\in\mathrm{Sym}(\{ \ell\}\times[m])}\sum_{F:[\ell]\times[m]\rightarrow[d]}\prod_{(u,k)\in[\ell]\times[m]}(X_{w(u)})_{F(u,k),F(\widetilde{T}\pi(u,k))}.\end{align*}

Consequently, as in (4.8), we have

\[\mathbb{E}(|\mathrm{tr}(\mathrm{Sym}^{m}w(X_{1},\ldots,X_{r}))|^{2})=\frac{1}{m!^{2}}\sum_{(\pi,\pi')\in\mathrm{Sym}(\Omega_{1,1})\times\mathrm{Sym}(\Omega_{2,1})}\sum_{F:\Omega\rightarrow[d]}\prod_{\gamma\in\Omega}(X_{\widetilde{w}(\gamma)})_{F(\pi\pi'\gamma),F(T(\gamma))}.\]

The proposition now follows from Corollary 2.15.

We next define an action of $H:=\prod_{(s,u)\in[2]\times[\ell]}\mathrm{Sym}(\Omega_{s,u})$ on $\widetilde{Z}$ in the same way as in § 5. For $(s,u)\in[2]\times([\ell]\backslash\{1\})$ and $\pi_{s,u}\in\mathrm{Sym}(\Omega_{s,u})$ ,

\[\pi_{s,u}\cdot(\pi,\pi',F,\Sigma):=(\pi,\pi',F\circ\pi_{s,u}^{-1},\pi_{s,u}\circ\Sigma\circ T^{-1}\pi_{s,u}^{-1}T),\]

and if $(\pi_{1,1},\pi_{2,1})\in\mathrm{Sym}(\Omega_{1,1})\times\mathrm{Sym}(\Omega_{2,1})$ ,

\[(\pi_{1,1},\pi_{2,1})\cdot(\pi,\pi',F,\Sigma):=(\pi_{1,1}\pi,\pi_{2,1}\pi',F\circ\pi_{1,1}^{-1}\pi_{2,1}^{-1},\Sigma\circ T^{-1}\pi_{1,1}^{-1}\pi_{2,1}^{-1}T).\]

Proof of Theorem 1.4. The proof is similar to the proof of Theorem 1.3. The only difference is that now, summing over the H-orbit kills all representations that do not appear in $\mathrm{Ind}_{S_{m}^{\ell_{i}}}^{S_{m\ell_{i}}}(1)$ , rather than the representations not in $\mathrm{Ind}_{S_{m}^{\ell_{i}}}^{S_{m\ell_{i}}}(\mathrm{sgn})$ . By Lemma 2.3, the irreducible subrepresentations $\chi_{\lambda}$ of $\mathrm{Ind}_{S_{m}^{\ell_{i}}}^{S_{m\ell_{i}}}(1)$ correspond to partitions $\lambda=(\lambda_{1},\ldots,\lambda_{\ell_{i}})$ with at most $\ell_{i}$ rows, and, therefore, $\prod_{(a,b)\in\lambda}(d+b-a)\geq(d-\ell)^{m\ell_{i}}$ . As in Corollary 5.3 and (5.6), the average of $\widetilde{\mathrm{Wg}}(\Sigma^{2})$ over an H-orbit $H\cdot(\widehat{\pi},\widehat{\pi'},\widehat{F},\widehat{\Sigma})$ is bounded by

(8.3) \begin{align}&\frac{1}{m!^{2\ell}}\prod_{i=1}^{r}\left|\sum_{\substack{h_{i}\in\prod_{\widetilde{w}=i}\mathrm{Sym}(\Omega_{s,u})\\h'_{i}\in\prod_{\widetilde{w}=-i}\mathrm{Sym}(\Omega_{s,u})}}\mathrm{Wg}\big(h_{i}\widehat{\Sigma}|_{B_{i}}h'_{i}\widehat{\Sigma}|_{A_{i}}\big)\right|\nonumber\\&\quad \leq \frac{1}{m!^{\ell}}\prod_{i=1}^{r}\frac{m!^{\ell_{i}}}{(m\ell_{i})!}\sum_{\lambda\vdash m\ell_{i}:\chi_{\lambda}\subseteq\mathrm{Ind}_{\mathrm{S}_{m}^{\ell_{i}}}^{S_{m\ell_{i}}}(1)}\frac{\chi_{\lambda}(1)\langle\chi_{\lambda},1\rangle_{\mathrm{S}_{m}^{\ell_{i}}}}{\prod_{(a,b)\in\lambda}(d+b-a)}\leq\frac{1}{m!^{\ell}}\frac{1}{(d-\ell)^{m\ell}}.\end{align}

Denote $\widetilde{Z}_{\pi,\pi'}:=\{ (F,\Sigma):(\pi,\pi',F,\Sigma)\in\widetilde{Z}\}$ . Since $\widetilde{Z}_{\mathrm{Id},\mathrm{Id}}=W'$ , Proposition 6.1 implies that

(8.4) \begin{equation}|\widetilde{Z}|=m!^{2}|\widetilde{Z}_{\mathrm{Id},\mathrm{Id}}|=m!^{2}|W'|\leq m!^{2}\binom{d+m\ell}{m\ell}(m\ell)!\prod_{0\neq k=-r}^{r}\frac{(m\ell_{k})!}{\big(\sum_{i<k}m\ell_{i,k}\big)!}.\end{equation}

As in the proof of Theorem 1.3, if $d\geq m\ell$ , then

\begin{align*}\mathbb{E}(|\mathrm{tr}(\mathrm{Sym}^{m}w(X_{1},\ldots,X_{r}))|^{2}) &=\frac{1}{m!^{2}}\sum_{(\pi,\pi',F,\Sigma)\in\widetilde{Z}}\widetilde{\mathrm{Wg}}(\Sigma^{2})\leq|\widetilde{Z}|\frac{1}{m!^{\ell+2}}\frac{1}{(d-\ell)^{m\ell}}\\& \leq\frac{(d+m\ell)\cdots(d+1)}{(d-\ell)^{m\ell}m!^{\ell}}\prod_{0\neq k=-r}^{r}\frac{(m\ell_{k})!}{\big(\sum_{i<k}m\ell_{i,k}\big)!}\\& \leq4^{m\ell}\ell^{m\ell}\prod_{0\neq k=-r}^{r}\binom{m\ell_{k}}{m\ell_{k}/2}4^{m\ell}\ell^{m\ell}2^{2m\ell}=(16\ell)^{m\ell}.\qquad \qquad \end{align*}