Random Unitary Representations of Surface Groups

Free groups Surface groups Results Ideas

Random Unitary Representations of

Surface Groups

Michael Magee

Durham University

29/03/2021

Free groups

Surface groups

Results

Set up

Fr a non-abelian free group on r generators

e.g. F2 = 〈a, b〉, elements a, b, a2b, aba−1b−1 etc

G a compact Lie group

Examples

G = U(n)=n × n complex unitary matrices

G = SU(n) = unit determinant elements of U(n)

G = Sn = permutations of {1, . . . , n}

Set up

Examples

Set up

Examples

Set up

Examples

Set up

Examples

Set up

Examples

Word maps and matrix integrals

Given w ∈ Fr , have word map w : G r → G

e.g. w = a2b−2 ∈ F2, w(g1, g2) = g21 g−22

Expected trace (main object of talk)

For fixed w ∈ Fr , interested in

Er ,n[tr ◦ w ]def=

∫U(n)r

tr(w(u1, u2, . . . , ur ))du1 · · · dur

du = prob. Haar measure on U(n)

E.g. r = 2, w = [a, b]def= aba−1b−1, E2,n[tr ◦ w ] = 1

n (Frobenius, 1896)

e.g. w = a2b−2 ∈ F2, w(g1, g2) = g21 g−22

∫U(n)r

tr(w(u1, u2, . . . , ur ))du1 · · · dur

n (Frobenius, 1896)

e.g. w = a2b−2 ∈ F2, w(g1, g2) = g21 g−22

∫U(n)r

tr(w(u1, u2, . . . , ur ))du1 · · · dur

n (Frobenius, 1896)

Algebraic properties

Rationality

Weingarten calculus =⇒ for n ≥ n0(w), Er ,n[tr ◦ w ] agrees with a

rational function of n

(that depends on w of course)

Trivial vanishing

If w /∈ [Fr ,Fr ], i.e. cannot be written as a product of elements of the

form [g1, g2], then

Er ,n[tr ◦ w ] = 0.

These facts appear explicitly in M-Puder ‘Matrix Group Integrals, Surfaces, and

Mapping Class Groups I: U(n)’

Rationality

Trivial vanishing

form [g1, g2], then

Er ,n[tr ◦ w ] = 0.

Rationality

Trivial vanishing

form [g1, g2], then

Er ,n[tr ◦ w ] = 0.

Rationality

Trivial vanishing

form [g1, g2], then

Er ,n[tr ◦ w ] = 0.

Analytic Estimates

Theorem (Voiculescu 1991)

If w ∈ Fr , w 6= id, then

Er ,n[tr ◦ w ] = o(n)

as n→∞.

Rationality of Er ,n[tr ◦ w ] =⇒ Er ,n[tr ◦ w ] = O(1) as n→∞.

M-Puder: Degree of rational function can be estimated in terms of algebraic

properties of w .

Analytic Estimates

Er ,n[tr ◦ w ] = o(n)

as n→∞.

properties of w .

Analytic Estimates

Er ,n[tr ◦ w ] = o(n)

as n→∞.

properties of w .

Perspectives and applications

asymptotic ∗-freeness of independent Haar unitary matrices

convergence of spectral distribution of e.g.

A1 + A2 + A3 + A−11 + A−1

2 + A−13 ∈ End(Cn),

where A1,A2,A3 i.i.d. by Haar measure in U(n)

Full Laurent expansion of Er ,n[tr ◦ w ] in variable 1n contains deep

algebraic invariants of w (M-Puder)

A1 + A2 + A3 + A−11 + A−1

2 + A−13 ∈ End(Cn),

A1 + A2 + A3 + A−11 + A−1

2 + A−13 ∈ End(Cn),

A1 + A2 + A3 + A−11 + A−1

2 + A−13 ∈ End(Cn),

Surface groups

g ≥ 2, Σg a closed surface of genus g

Γg is the fundamental group of Σg

Γg = 〈a1, b1, . . . , ag , bg | [a1, b1][a2, b2] · · · [ag ,bg ] 〉 called a surface

Rg = [a1, b1][a2, b2] · · · [ag ,bg ] ∈ F2g the relator of Γg

Surface groups

Generalizing previous

Recall Er ,n[tr ◦ w ]def=∫

U(n)rtr(w(u1, u2, . . . , ur ))du1 · · · dur

We have

U(n)r ∼= Hom(Fr ,U(n))

tr ◦ w ∼= trw : φ 7→ tr(φ(w)).

Now, instead, for given γ ∈ Γg we want to integrate

trγ : φ 7→ tr(φ(γ))

over Hom(Γg ,SU(n)) with respect to some nice measure.

We have

tr ◦ w ∼= trw : φ 7→ tr(φ(w)).

We have

tr ◦ w ∼= trw : φ 7→ tr(φ(w)).

We have

tr ◦ w ∼= trw : φ 7→ tr(φ(w)).

The Atiyah-Bott-Goldman measure

Hom(Γg ,SU(n)) is just the collection of

{ (A1,B1, . . . ,Ag ,Bg ) ∈ SU(n)2g : [A1,B1] · · · [Ag ,Bg ] = id }

Not a group!

On the manifold Hom(Γg ,SU(n))irr/SU(n) there is a naturalsymplectic form defined in works of Atiyah-Bott and Goldman. Thisalso gives a volume form.M. F. Atiyah and R. Bott. The Yang-Mills Equations over Riemann Surfaces.

Philosophical Transactions of the Royal Society of London. Series A,

Mathematical and Physical Sciences, 1983.

W. M. Goldman. The symplectic nature of fundamental groups of surfaces.

Advances in Mathematics, 1984.

Not a group!

On the manifold Hom(Γg ,SU(n))irr/SU(n) there is a naturalsymplectic form defined in works of Atiyah-Bott and Goldman. Thisalso gives a volume form.

M. F. Atiyah and R. Bott. The Yang-Mills Equations over Riemann Surfaces.

Not a group!

The measure: finite groups example

Suppose G a finite group, f is a G -conjugation invariant function on G 2g

(e.g. tr ◦ w for some w ∈ F2g )

How to evaluate∫Hom(Γg ,G)

f d(uniform measure)? (weighted sum)

This is the same as

|G |2g

|Hom(Γg ,G)|

∫G2g

f (A1,B1, . . . ,Ag ,Bg )1{Rg (A1,B1, . . . ,Ag ,Bg ) = idG}.

Schur orthogonality =⇒

1{x = idG} =1

|G |∑ρ∈G

(dim ρ)tr(ρ(x)),

where G is the set of equivalence classes of irreducible representations

of G ; here each ρ : G → Aut(Vρ) is a linear representation.

This is the same as

|G |2g

|Hom(Γg ,G)|

∫G2g

1{x = idG} =1

|G |∑ρ∈G

(dim ρ)tr(ρ(x)),

This is the same as

|G |2g

|Hom(Γg ,G)|

∫G2g

1{x = idG} =1

|G |∑ρ∈G

(dim ρ)tr(ρ(x)),

This is the same as

|G |2g

|Hom(Γg ,G)|

∫G2g

1{x = idG} =1

|G |∑ρ∈G

(dim ρ)tr(ρ(x)),

Finite groups example continued

Hence for G finite∫Hom(Γg ,G)

f d(uniform measure) =

|G |2g−1

|Hom(Γg ,G)|∑ρ∈G

(dim ρ)

∫G2g

f (A1,B1, . . . ,Ag ,Bg )tr(ρ(Rg (A1,B1, . . . ,Ag ,Bg ))

Theorem (Sengupta 2003)

When G = SU(n)

The average of f with respect to the Atiyah-Bott-Goldman measure is

proportional to the right hand side above if the right hand side is abso-

lutely convergent.

(Witten’s formula) The normalization constant is

ζSU(n)(2g − 2)def=

∑ρ∈ ˆSU(n)

1(dim ρ)2g−2 .

Finite groups example continued

Hence for G finite∫Hom(Γg ,G)

f d(uniform measure) =

|G |2g−1

|Hom(Γg ,G)|∑ρ∈G

(dim ρ)

∫G2g

f (A1,B1, . . . ,Ag ,Bg )tr(ρ(Rg (A1,B1, . . . ,Ag ,Bg ))

Theorem (Sengupta 2003)

When G = SU(n)

The average of f with respect to the Atiyah-Bott-Goldman measure is

proportional to the right hand side above if the right hand side is abso-

lutely convergent.

(Witten’s formula) The normalization constant is

ζSU(n)(2g − 2)def=

∑ρ∈ ˆSU(n)

1(dim ρ)2g−2 .

For γ ∈ Γg , let EΓg ,n[trγ ]def=∫Hom(Γg ,SU(n))

trγ dµABG .

TheoremFor any g ≥ 2 and γ ∈ Γg there is an infinite sequence of rational

numbers

a−1(γ), a0(γ), a1(γ), a2(γ), . . .

such that for any M ∈ N, as n→∞

EΓg ,n[trγ ] = a−1(γ)n + a0(γ) +a1(γ)

n+ · · ·+ aM−1(γ)

nM−1+ O(n−M).

Appears in arXiv:2101.00252

trγ dµABG .

numbers

a−1(γ), a0(γ), a1(γ), a2(γ), . . .

EΓg ,n[trγ ] = a−1(γ)n + a0(γ) +a1(γ)

n+ · · ·+ aM−1(γ)

nM−1+ O(n−M).

trγ dµABG .

numbers

a−1(γ), a0(γ), a1(γ), a2(γ), . . .

EΓg ,n[trγ ] = a−1(γ)n + a0(γ) +a1(γ)

n+ · · ·+ aM−1(γ)

nM−1+ O(n−M).

Estimates

TheoremLet g ≥ 2. If γ ∈ Γg is not the identity, then EΓg ,n[trγ ] = Oγ(1) as

n→∞.

Previous two theorems for SU(n) replaced by Sn and tr replaced by the standard

(n − 1)-dim character of Sn obtained in

M-Puder ‘The asymptotic statistics of random covering surfaces’ 2020

Estimates

n→∞.

Estimates

n→∞.

Convergence in probability?

Is it true that for fixed γ 6= id, for any ε > 0

{φ ∈ Hom(Γg ,SU(n)) :

|trγ(φ)|n

}→ 0

as n→∞?

For Γg replaced by Fr , true (Voiculescu 1991)

Levy family, Gromov-Milman

|trγ(φ)|n

}→ 0

as n→∞?

|trγ(φ)|n

}→ 0

as n→∞?

The shape of things, recap

To calculate EΓg ,n[trγ]:

Normalization factor = value of Witten zeta function

well understood, tends to 1 as n → ∞ (Guralnick, Larsen,

Manack)

pick w ∈ F2g that projects to γ ∈ Γg

EΓg ,n[trγ ] essentially of the form

∑ρ∈ ˆSU(n)

dim(ρ)×

∫SU(n)2g

tr(w(A1,B1, . . . ,Ag ,Bg ))tr(ρ([A1,B1] · · · [Ag ,Bg ])))dA1 · · · dAgdB1 · · · dBg

Manack)

∑ρ∈ ˆSU(n)

dim(ρ)×

∫SU(n)2g

Manack)

∑ρ∈ ˆSU(n)

dim(ρ)×

∫SU(n)2g

Manack)

∑ρ∈ ˆSU(n)

dim(ρ)×

∫SU(n)2g

Organizing representations I

We want to chop the sum into a ‘head’ and a ‘tail’.

Head = finitely many representations. Contribution from each of

these is rational function.

Tail = all other representations. Hope to control all

simultaneously. (Enough for first theorem)

Different ways to try this.

e.g. ˆSU(n) ∼=Young diagrams with ≤ n − 1 rows. Head of sum

corresponds to bounded number of boxes?

Not right perspective (Chiral)

simultaneously.

(Enough for first theorem)

Organizing representations II

Correct perspective: every pair of Young diagrams (λ, µ) gives rise, for

n ≥ `(λ) + `(µ) to a family of representations ρn[λ,µ]

of U(n) w/ highest weight

(λ1, λ2, . . . , λ`(λ), 0, . . . , 0,−µ`(µ), . . . ,−µ2,−µ1) ∈ Zn.

Figure: From Koike 1989

Head of sum ∼ (roughly speaking) to representations that come from λ, µ with

bounded total number of boxes. (dim(ρn[λ,µ]

) � n|λ|+|µ|)

(λ1, λ2, . . . , λ`(λ), 0, . . . , 0,−µ`(µ), . . . ,−µ2,−µ1) ∈ Zn.

bounded total number of boxes.

(dim(ρn[λ,µ]

) � n|λ|+|µ|)

(λ1, λ2, . . . , λ`(λ), 0, . . . , 0,−µ`(µ), . . . ,−µ2,−µ1) ∈ Zn.

bounded total number of boxes. (dim(ρn[λ,µ]

) � n|λ|+|µ|)

∫SU(n)2g tr(w(A1,B1, . . . ,Ag ,Bg ))tr(ρn

[λ,µ](Rg (A1,B1, . . . ,Ag ,Bg )))dA1 · · · dAgdB1 · · · dBg

|λ|+ |µ| � 1.

Challenge: Weingarten calculus not suited

Symmetry ∫SU(n)2g

F (A1,B1, . . . ,Ag ,Bg )tr(ρ(Rg (A1,B1, . . . ,Ag ,Bg ))dA1 · · · dAgdB1 · · · dBg

where F (A1,B1, . . . ,Ag ,Bg ) depends only on bottom left D × D of each

A1,B1, . . . ,Ag ,Bg , D = D(w).

This means F is SU(n − D) bi-invariant and now can first integrate

tr(ρ(Rg (A1,B1, . . . ,Ag ,Bg ))) over SU(n − D) double cosets

Rg is a very special word and so these integrals can be done using Weingarten

calculus + branching laws for SU(n) + Gelfand-Tsetlin bases.

(new methods)

|λ|+ |µ| � 1. Challenge: Weingarten calculus not suited

Symmetry ∫SU(n)2g

A1,B1, . . . ,Ag ,Bg , D = D(w).

(new methods)

Symmetry ∫SU(n)2g

A1,B1, . . . ,Ag ,Bg , D = D(w).

(new methods)

Symmetry ∫SU(n)2g

A1,B1, . . . ,Ag ,Bg , D = D(w).

(new methods)

Symmetry ∫SU(n)2g

A1,B1, . . . ,Ag ,Bg , D = D(w).

Rg is a very special word

and so these integrals can be done using Weingarten

(new methods)

Symmetry ∫SU(n)2g

A1,B1, . . . ,Ag ,Bg , D = D(w).

(new methods)

Symmetry ∫SU(n)2g

A1,B1, . . . ,Ag ,Bg , D = D(w).

(new methods)

λ, µ fixed.

Challenge: All known topological expansions do not give good enough

Solution part 1

ρn[λ,µ] is subrepresentation of (Cn)⊗|λ| ⊗ ˇ(Cn)⊗|µ|

, new ideas in

Weingarten calculus based on ‘mixed’ Schur -Weyl duality due to

Koike (1989)

refined topological expansions

Solution part 2

Choosing the initial w ∈ F2g projecting to γ ∈ Γg carefully to get best

results from topological expansions

(Dehn’s algorithm)

λ, µ fixed. Challenge: All known topological expansions do not give good enough

Solution part 1

, new ideas in

Koike (1989)

Solution part 2

Solution part 1

, new ideas in

Koike (1989)

Solution part 2

Solution part 1

, new ideas in

Koike (1989)

Solution part 2

Solution part 1

, new ideas in

Koike (1989)

Solution part 2

Solution part 1

, new ideas in

Koike (1989)

Solution part 2

Thanks for your attention!

Random Unitary Representations of Surface Groups

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