On the Fourier coefficients of word maps on unitary groups

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arXiv:2210.04164v2 [math.PR] 25 Sep 2024

ON THE FOURIER COEFFICIENTS OF WORD MAPS ON UNITARY GROUPS

NIR AVNI AND ITAY GLAZER

In memory of Steve Zelditch

Abstract. Given a word w(x1,...,xr), i.e., an element in the free group on relements, and an

integer d≥1, we study the characteristic polynomial of the random matrix w(X1,...,Xr), where

Xiare Haar-random independent d×dunitary matrices. If cm(X)denotes the m-th coeﬃcient of

the characteristic polynomial of X, our main theorem implies that there is a positive constant ǫ(w),

depending only on w, such that

|E(cm(w(X1,...,Xr)))| ≤ d

m!1−ǫ(w)

for every dand every 1≤m≤d.

Our main computational tool is the Weingarten Calculus, which allows us to express integrals on

unitary groups such as the expectation above, as certain sums on symmetric groups. We exploit a

hidden symmetry to ﬁnd cancellations in the sum expressing E(cm(w)). These cancellations, coming

from averaging a Weingarten function over cosets, follow from Schur’s orthogonality relations.

1. Introduction

Let wbe a word on rletters, i.e., an element in the free group on the letters x1,...,xr. Let X1,...,Xr

be random d×dunitary matrices, chosen independently at random according to the Haar probability

measure, and consider the random matrix w(X1,...,Xr), obtained by substituting Xifor xiin w.

For example, if w=x1x2x−1

1x−1

2, then w(X1, X2) = X1X2X−1

1X−1

2. In this paper, we study the

distribution of the characteristic polynomial of w(X1,...,Xr).

To set notation, given a d×d-matrix Aand 1≤m≤d, let cm(A)be the coeﬃcient of td−m

in the characteristic polynomial det(t·Id −A)of A. Note that cm(A) = (−1)mtr (VmA), where

VmA:VmCd→VmCdis the m-th exterior power of A. If Ais unitary, all eigenvalues have

absolute value 1, so we get the trivial bound |cm(A)| ≤ d

m.

Our main theorem is the following:

Theorem 1.1. For every non-trivial word w∈Fr, there exists a constant ǫ(w)>0such that

E|cm(w(X1,...,Xr))|2≤d

m2(1−ǫ(w))

for every dand every 1≤m≤d. In particular, we have

E(|cm(w(X1,...,Xr))|)≤d

m1−ǫ(w)

Remark 1.2.

(1) In the proof of Theorem 1.1, we show that, if the length of wis ℓand d≥(25ℓ)7ℓ, then one

can take ǫ(w) = 1

72 (25ℓ)−2ℓ. We believe ǫ(w)−1can be taken to be a polynomial in ℓ, for

d≫ℓ1.

2020 Mathematics Subject Classiﬁcation. Primary 60B15, 60B20; Secondary 43A75, 20P05, 20B30.

Key words and phrases. Word maps, word measures, unitary groups, Weingarten Calculus, Fourier coeﬃcients, random

matrices, characteristic polynomial.

ON THE FOURIER COEFFICIENTS OF WORD MAPS ON UNITARY GROUPS 2

(2) On the other hand, it follows from [ET15, Theorem 5.2] that, for a ﬁxed d, one has to take

ǫ(w).e−√ℓ, for some arbitrarily long words, even for m= 1.

Theorem 1.1 relies on the following:

Theorem 1.3. For every m, ℓ ∈N, every d≥mℓ, and every word w∈Frof length ℓ, one has:

(1.1) E|cm(w(X1,...,Xr))|2≤(22ℓ)mℓ.

In particular, if d≥(22ℓ)ℓm, we have

E|cm(w(X1,...,Xr))|2≤d

m.

In addition, we show that similar bounds hold for symmetric powers:

Theorem 1.4. For every ℓ∈N, every d≥mℓ, and every word w∈Frof length ℓ, one has:

E|tr (Symmw(X1,...,Xr))|2≤(16ℓ)mℓ.

In particular, if d≥(16ℓ)ℓm, we have

E|tr (Symmw(X1,...,Xr))|2≤d+m−1

m= dim SymmCd,

and by the Cauchy–Schwarz inequality,

|E(tr (Symmw(X1,...,Xr)))| ≤ dim SymmCd1

Remark 1.5.Theorem 1.4 is an analogue of Theorem 1.3. It is also an analogue of Theorem 1.1 for

mat most linear in d. Contrary to exterior powers, the methods of this paper are insuﬃcient for

ﬁnding bounds similar to Theorem 1.1 for |E(tr (Symmw(X1,...,Xr)))|, in the regime where mis

superlinear in d.

1.1. Related work. Word maps on unitary groups and their eigenvalues have been extensively stud-

ied in the past few decades.

The case w=x, namely, the study of a Haar-random unitary matrix X, also known as the Circular

Unitary Ensemble (CUE), is an important object of study in random matrix theory (see e.g. [AGZ10,

Mec19] and the references within). The joint density of the eigenvalues of Xis given by the Weyl

Integration Formula [Wey39]. Schur’s orthogonality relations immediately imply that E|cm(X)|2=

1for all 1≤m≤d. Various other properties of the characteristic polynomial of a random unitary

matrix Xhave been extensively studied (see e.g. [KS00, HKO01, CFK+03, DG06, BG06, BHNY08,

ABB17, CMN18, PZ18]).

Diaconis and Shahshahani [DS94] have shown that, for a ﬁxed m∈N, the sequence of random

variables tr(X),tr(X2),...,tr(Xm)converges in distribution, as d→ ∞, to a sequence of independent

complex normal random variables. For the proof, which relies on the moment method, they computed

the joint moments of those random variables and showed that

(1.2) E



j=1

tr(Xj)ajtr(Xj)bj

=δa,b

j=1

jajaj!,

for d≥Pm

j=1(aj+bj)j. The rate of convergence was later shown to be super-exponential by Johansson

[Joh97].

ON THE FOURIER COEFFICIENTS OF WORD MAPS ON UNITARY GROUPS 3

When w=xℓ, (1.2) gives a formula for the moments of traces, and one can use Newton’s identities

relating elementary symmetric polynomials and power sums, to deduce that

Ecm(Xℓ)2=E|tr (Symmw)|2=ℓ+m−1

m,

for d≥2mℓ (see Appendix A). In [Rai97, Rai03], Rains partially extended (1.2) for small dand gave

an explicit formula for the joint density of the eigenvalues of Xℓ(see [Rai03, Theorem 1.3]).

We now move to general words w∈Fr. The case m= 1, namely, the asymptotics as d→ ∞ of the

distribution of the random variable tr (w(X1,...,Xr)), was studied in the context of Voiculescu’s free

probability (see e.g. [VDN92, MS17]). In particular, in [Voi91, R˘

06, MSS07] it was shown that, for

a ﬁxed w∈Fr, the sequence of random variables tr (w(X1,...,Xr)), for d= 1,2,..., converges in

distribution, as d→ ∞, to a complex normal random variable (with suitable normalization). As a

direct consequence, for a ﬁxed m∈N, the random variables cm(w(X1,...,Xr)) converge, as d→ ∞,

to a certain explicit polynomial of Gaussian random variables. This is done in Appendix A, Corollary

A.4, following [DG06].

In [MP19], Magee and Puder have shown that E(tr (w(X1,...,Xr))) coincides with a rational function

of d, if dis suﬃciently large, and bounded its degree in terms of the commutator length of w. They

also found a geometric interpretation for the coeﬃcients of the expansion of that rational function as

a power series in d−1, see [MP19, Corollaries 1.8 and 1.11]. See [Bro24] for additional work in this

direction.

1.2. Ideas of proofs. With a few exceptions, the results stated in §1.1 are asymptotic in d, but not

uniform in both mand d. We will try to explain some of the challenges in proving results that are

uniform in m, while explaining the idea of the proof of Theorem 1.1.

Our main tool (which is also used in the papers [R˘

06, MSS07, MP19] above) to study integrals on

unitary groups is the Weingarten Calculus ([Wei78, Col03, CS06]). Roughly speaking, the Weingarten

Calculus utilizes the Schur–Weyl Duality to express integrals on unitary groups as sums of so called

Weingarten functions over symmetric groups. In our case, in order to prove Theorem 1.1, we need to

estimate the integral

(1.3) E|cm(w)|2=ZUr

dtr ^mw(X1,...,Xr)2dX1. . . dXr.

Using Weingarten calculus (Theorem 2.12), we express (1.3) as a ﬁnite sum

(1.4) X

(π1,...,π2r)∈Q2r

i=1 Smℓi

F(π1,...,π2r)

i=1

Wg(i)

d(πiπ−1

i+r),

where ℓ1,...,ℓ2r∈Nand F:Q2r

i=1 Smℓi→Zare related to combinatorial properties of w, and each

Wg(i)

d:Smℓi→Ris a Weingarten function (see Deﬁnition 2.10). There are two main diﬃculties when

dealing with sums such as (1.4) in the region when mis unbounded:

(1) While the asymptotics of Weingarten functions Wgd:Sm→Rare well understood when

d≫m(see [Col03, Section 2.2] and [CM17, Therem 1.1]), much less is known in the regime

where mis comparable with d.

(2) Even if we have a good understanding of a single Weingarten function, the number of sum-

mands in (1.4) is large and it is not enough to bound each individual Weingarten function.

Luckily, there are plenty of cancellations in the sum (1.4). To understand these cancellations, we

identify a symmetry of (1.4). More precisely, we ﬁnd a group Hacting on Q2r

i=1 Smℓisuch that Fis

ON THE FOURIER COEFFICIENTS OF WORD MAPS ON UNITARY GROUPS 4

equivariant with respect to H, and such that the contribution of any H-orbit to the sum (1.4) is a

product of terms, each of which has the form

(1.5) 1

m!2ℓiX

h,h′∈Sℓi

sgn hh′Wg(i)

dh′πihπ−1

i+r,

where sgn(x)is the sign of xand the sum is over the Young subgroup Sℓi

m⊆Smℓi, see Corollary 5.3.

Weingarten functions are class functions, so they are linear combinations of irreducible characters of

Smℓi. Explicitly, we have (see [CS06, Eq. (13)]):

(1.6) Wg(i)

d(σ) = 1

(mℓi)!2X

λ⊢mℓi,ℓ(λ)≤d

χλ(1)2

ρλ(1) χλ(σ), σ ∈Smℓi,

where each λis a partition of mℓiwith at most dparts and χλand ρλare the corresponding irreducible

characters of Smℓiand Ud, respectively. The cancellations that we get in the sum (1.5) come from

averaging irreducible characters of Smℓiover Sℓi

m-cosets. Sℓi

mis a large subgroup of Smℓi, so these

cancellations will be signiﬁcant as well. For example, all terms in (1.6) for which λhas more than ℓi

columns vanish. See Lemmas 2.7 and 2.8 for the precise bounds.

After we bound the average contribution of each H-orbit in the sum (1.4) by a function C(m, d, w),

we bound (1.4) by |Z|·C(m, d, w)for some ﬁnite set Z. This becomes a counting problem, which we

solve in §6, see Proposition 6.1.

The proof of Theorem 1.1 occupies Sections 4, 5, 6 and 7. Since the combinatorics of general words

is a bit complicated, we prove a simpliﬁed version of Theorem 1.3 for the special case of the Engel

word [[x, y], y]in §3. The proof for this special case contains the main ideas of the paper, while being

easier to understand.

1.3. Further discussion and some open questions. The results of this paper ﬁt in the larger

framework of the study of word measures and their Fourier coeﬃcients.

Let Gbe a compact group, and let µGbe the Haar probability measure on G. To each word

w(x1,...,xr)∈Frwe associate the corresponding word map wG:Gr→G, deﬁned by (g1,...,gr)7→

w(g1,...,gr). The pushforward measure (wG)∗(µr

G)is called the word measure τw,G associated with

wand G. Let Irr(G)be the set of irreducible characters of G. The Fourier coeﬃcient of τw,G at

ρ∈Irr(G)is

(1.7) aw,G,ρ := ZGr

ρ(w(x1,...,xr))µr

G=ZG

ρ(y)τw,G.

If w6= 1 and Gis a compact semisimple Lie group, then by Borel’s theorem [Bor83], the map wG:

Gr→Gis a submersion outside a proper subvariety in Gr. It follows that τw,G is absolutely continuous

with respect to µGand, therefore, τw,G =fw,G·µG, where fw,G ∈L1(G)is the Radon–Nikodym

density. In this case, fw,G =Pρ∈Irr(G)aw,G,ρ ·ρ.

In [LST19, Theorem 4], Larsen, Shalev, and Tiep proved uniform L∞-mixing time for convolutions

of word measures on suﬃciently large ﬁnite simple groups. From this, the following can be deduced:

Theorem 1.6. For every w∈Fr, there exists N(w)∈Nsuch that if Gis a ﬁnite simple group with

at least N(w)elements, then

(1.8) |aw,G,ρ| ≤ (dim ρ)1−ǫ(w),

for ǫ(w) = C·ℓ(w)−4and some absolute constant C.

The proof of Theorem 1.6 is given at the end of §7.

ON THE FOURIER COEFFICIENTS OF WORD MAPS ON UNITARY GROUPS 5

We believe that a similar statement should be true for compact semisimple Lie groups.

Conjecture 1.7. For every 16=w∈Fr, there exists ǫ(w)>0such that, for every compact connected

semisimple Lie group Gand every ρ∈Irr(G),

|aw,G,ρ| ≤ (dim ρ)1−ǫ(w).

It is natural to estimate ǫ(w)in terms of the length ℓ(w)of the word w. For simple groups of bounded

rank, Item (2) of Remark 1.2 (i.e. [ET15, Theorem 5.2]) shows that there are arbitrarily long words

wfor which ǫ(w)cannot be larger than e−√ℓ(w). However, we believe that better Fourier decay can

be achieved for the high rank case.

Question 1.8. Can one take ǫ(w)to be a polynomial in ℓ(w), if rk(G)≫ℓ(w)1?

Theorem 1.1 gives evidence to Conjecture 1.7 for G= SUdand the collection of fundamental rep-

resentations VmCdd

m=1. Indeed, for every ρ∈Irr(Ud), since |ρ(λA)|=|ρ(A)|for A∈SUdand

λ∈U1, and since µUdis the pushforward of µU1×µSUdby the multiplication map (λ, A)7→ λA, we

have,

|aw,SUd,ρ|2≤EX1,...,Xr∈SUd|ρ(w(X1,...,Xr))|2

(1.9)

=E(λ1,X1),...,(λr,Xr)∈SUd×U1|ρ(w(λ1,...,λr)w(X1,...,Xr))|2

=E(λ1,X1),...,(λr,Xr)∈SUd×U1|ρ(w(λ1X1,...,λrXr))|2=EUd|ρ(w(X1,...,Xr))|2,

Theorem 1.4 deals with another family of irreducible representations SymmCd⌊d/(16ℓ)ℓ⌋

m=1 , giving

further evidence for Conjecture 1.7.

Verifying Conjecture 1.7 will imply that, for every word w, the random walks induced by the collection

of measures {τw,G}G, where Gruns over all compact connected simple Lie groups, admit a uniform

L∞-mixing time. Namely, using [GLM12, Theorem 1], it will show the existence of t(w)∈Nsuch

that

(1.10) 



τ∗t(w)

w,G

µG−1



∞

<1/2,

for every compact connected simple Lie group G. By the above discussion, t(w)grows at least

exponentially with pℓ(w)under no restriction on the rank. If the condition (1.10) is replaced by the

condition that τ∗t(w)

w,G has bounded density, one might hope for polynomial bounds.

Question 1.9. Let 16=w∈Fr. Can one ﬁnd t(w)∈Nsuch that for every compact connected

semisimple Lie group G,τ∗t(w)

w,G has bounded density with respect to µG? can t(w)be chosen to have

polynomial dependence on ℓ(w)?

Question 1.9 can be seen as an analytic specialization of a geometric phenomenon. Let ϕ:X→Y

be a polynomial map between smooth Q-varieties. We say that ϕis (FRS) if it is ﬂat and its ﬁbers

all have rational singularities. In [AA16, Theorem 3.4], Aizenbud and the ﬁrst author showed that

if ϕis (FRS), then for every non-Archimedean local ﬁeld Fand every smooth, compactly supported

measure µon X(F), the pushforward ϕ∗µhas bounded density. This result was extended in [Rei]

to the Archimedean case, F=Ror C, and, moreover, if one runs over a large enough family of

local ﬁelds, the condition of (FRS) is in fact necessary as well for the densities of pushforwards to be

bounded (see [AA16, Theorem 3.4] and [GHS, Corollary 6.2]).

To rephrase Question 1.9 in geometric term, we further need the following notion from [GH19, GH21].

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arXiv:2210.04164v2[math.PR]25Sep2024ONTHEFOURIERCOEFFICIENTSOFWORDMAPSONUNITARYGROUPSNIRAVNIANDITAYGLAZERInmemoryofSteveZelditchAbstract.Givenawordw(x1,...,xr),i.e.,anelementinthefreegrouponrelements,andanintegerd≥1,westudythecharacteristicpolynomialoftherandommatrixw(X1,...,Xr),whereXiareHaar-rando...

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