Random Unitary Representations of Surface Groups
Transcript of 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 Ideas
Plan
Free groups
Surface groups
Results
Ideas
Free groups Surface groups Results Ideas
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}
Free groups Surface groups Results Ideas
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}
Free groups Surface groups Results Ideas
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}
Free groups Surface groups Results Ideas
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}
Free groups Surface groups Results Ideas
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}
Free groups Surface groups Results Ideas
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}
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)’
Free groups Surface groups Results Ideas
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)’
Free groups Surface groups Results Ideas
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)’
Free groups Surface groups Results Ideas
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)’
Free groups Surface groups Results Ideas
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 .
Free groups Surface groups Results Ideas
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 .
Free groups Surface groups Results Ideas
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 .
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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
group
Rg = [a1, b1][a2, b2] · · · [ag ,bg ] ∈ F2g the relator of Γg
Free groups Surface groups Results Ideas
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
group
Rg = [a1, b1][a2, b2] · · · [ag ,bg ] ∈ F2g the relator of Γg
Free groups Surface groups Results Ideas
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
group
Rg = [a1, b1][a2, b2] · · · [ag ,bg ] ∈ F2g the relator of Γg
Free groups Surface groups Results Ideas
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
group
Rg = [a1, b1][a2, b2] · · · [ag ,bg ] ∈ F2g the relator of Γg
Free groups Surface groups Results Ideas
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
group
Rg = [a1, b1][a2, b2] · · · [ag ,bg ] ∈ F2g the relator of Γg
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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.
Free groups Surface groups Results Ideas
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 .
Free groups Surface groups Results Ideas
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 .
Free groups Surface groups Results Ideas
Algebraic properties
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
Free groups Surface groups Results Ideas
Algebraic properties
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
Free groups Surface groups Results Ideas
Algebraic properties
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
Free groups Surface groups Results Ideas
Estimates
TheoremLet g ≥ 2. If γ ∈ Γg is not the identity, then EΓg ,n[trγ ] = Oγ(1) as
n→∞.
Appears in arXiv:2101.03224
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
Free groups Surface groups Results Ideas
Estimates
TheoremLet g ≥ 2. If γ ∈ Γg is not the identity, then EΓg ,n[trγ ] = Oγ(1) as
n→∞.
Appears in arXiv:2101.03224
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
Free groups Surface groups Results Ideas
Estimates
TheoremLet g ≥ 2. If γ ∈ Γg is not the identity, then EΓg ,n[trγ ] = Oγ(1) as
n→∞.
Appears in arXiv:2101.03224
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
Free groups Surface groups Results Ideas
Convergence in probability?
Is it true that for fixed γ 6= id, for any ε > 0
µABG
{φ ∈ Hom(Γg ,SU(n)) :
|trγ(φ)|n
> ε
}→ 0
as n→∞?
For Γg replaced by Fr , true (Voiculescu 1991)
Levy family, Gromov-Milman
Free groups Surface groups Results Ideas
Convergence in probability?
Is it true that for fixed γ 6= id, for any ε > 0
µABG
{φ ∈ Hom(Γg ,SU(n)) :
|trγ(φ)|n
> ε
}→ 0
as n→∞?
For Γg replaced by Fr , true (Voiculescu 1991)
Levy family, Gromov-Milman
Free groups Surface groups Results Ideas
Convergence in probability?
Is it true that for fixed γ 6= id, for any ε > 0
µABG
{φ ∈ Hom(Γg ,SU(n)) :
|trγ(φ)|n
> ε
}→ 0
as n→∞?
For Γg replaced by Fr , true (Voiculescu 1991)
Levy family, Gromov-Milman
Free groups Surface groups Results Ideas
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
Free groups Surface groups Results Ideas
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
Free groups Surface groups Results Ideas
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
Free groups Surface groups Results Ideas
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
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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)
Free groups Surface groups Results Ideas
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|)
Free groups Surface groups Results Ideas
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|)
Free groups Surface groups Results Ideas
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|)
Free groups Surface groups Results Ideas
Tail
∫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)
Free groups Surface groups Results Ideas
Tail
∫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)
Free groups Surface groups Results Ideas
Tail
∫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)
Free groups Surface groups Results Ideas
Tail
∫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)
Free groups Surface groups Results Ideas
Tail
∫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)
Free groups Surface groups Results Ideas
Tail
∫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)
Free groups Surface groups Results Ideas
Tail
∫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)
Free groups Surface groups Results Ideas
Head
∫SU(n)2g tr(w(A1,B1, . . . ,Ag ,Bg ))tr(ρn
[λ,µ](Rg (A1,B1, . . . ,Ag ,Bg )))dA1 · · · dAgdB1 · · · dBg
λ, µ fixed.
Challenge: All known topological expansions do not give good enough
decay
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)
Free groups Surface groups Results Ideas
Head
∫SU(n)2g tr(w(A1,B1, . . . ,Ag ,Bg ))tr(ρn
[λ,µ](Rg (A1,B1, . . . ,Ag ,Bg )))dA1 · · · dAgdB1 · · · dBg
λ, µ fixed. Challenge: All known topological expansions do not give good enough
decay
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)
Free groups Surface groups Results Ideas
Head
∫SU(n)2g tr(w(A1,B1, . . . ,Ag ,Bg ))tr(ρn
[λ,µ](Rg (A1,B1, . . . ,Ag ,Bg )))dA1 · · · dAgdB1 · · · dBg
λ, µ fixed. Challenge: All known topological expansions do not give good enough
decay
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)
Free groups Surface groups Results Ideas
Head
∫SU(n)2g tr(w(A1,B1, . . . ,Ag ,Bg ))tr(ρn
[λ,µ](Rg (A1,B1, . . . ,Ag ,Bg )))dA1 · · · dAgdB1 · · · dBg
λ, µ fixed. Challenge: All known topological expansions do not give good enough
decay
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)
Free groups Surface groups Results Ideas
Head
∫SU(n)2g tr(w(A1,B1, . . . ,Ag ,Bg ))tr(ρn
[λ,µ](Rg (A1,B1, . . . ,Ag ,Bg )))dA1 · · · dAgdB1 · · · dBg
λ, µ fixed. Challenge: All known topological expansions do not give good enough
decay
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)
Free groups Surface groups Results Ideas
Head
∫SU(n)2g tr(w(A1,B1, . . . ,Ag ,Bg ))tr(ρn
[λ,µ](Rg (A1,B1, . . . ,Ag ,Bg )))dA1 · · · dAgdB1 · · · dBg
λ, µ fixed. Challenge: All known topological expansions do not give good enough
decay
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)
Free groups Surface groups Results Ideas
Thanks for your attention!