q-analogues of representation-theoreticmeasures

Cesar Cuenca

MIT

February 7, 2019

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 1 / 24

Partitions

A partition is an integer sequence λ = (λ1 ≥ λ2 ≥ λ3 ≥ . . . ) suchthat λk = 0, for large enough k . The size of λ is |λ| :=

∑i λi .

Denote Y the set of all partitions.

.

Figure: Partition (5, 2, 2, 0, 0, . . . )

In representation theory, partitions of size n parametrize theirreducible representations Vλ of the symmetric group S(n).

In algebraic combinatorics, partitions parametrize bases of the algebraof symmetric functions, e.g., the basis {sλ}λ of Schur functions.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 2 / 24

Partitions

A partition is an integer sequence λ = (λ1 ≥ λ2 ≥ λ3 ≥ . . . ) suchthat λk = 0, for large enough k . The size of λ is |λ| :=

∑i λi .

Denote Y the set of all partitions.

.

Figure: Partition (5, 2, 2, 0, 0, . . . )

In representation theory, partitions of size n parametrize theirreducible representations Vλ of the symmetric group S(n).

In algebraic combinatorics, partitions parametrize bases of the algebraof symmetric functions, e.g., the basis {sλ}λ of Schur functions.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 2 / 24

Random partitions: Poissonized Plancherel

1. Poissonized Plancherel measure: Given θ > 0, define

Mθ(λ) := e−θ(

dimλ

|λ|!

)2

θ|λ|

for λ ∈ Y, where dimλ is the dimension of Vλ.

Identify partitions with point configurations inZ′ := {· · · ,−3

2,−1

2, 12, 32, · · · } via

λ 7→ X (λ) :=

{. . . , λ3 −

5

2, λ2 −

3

2, λ1 −

1

2

}.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 3 / 24

Random partitions: Poissonized Plancherel

1. Poissonized Plancherel measure: Given θ > 0, define

Mθ(λ) := e−θ(

dimλ

|λ|!

)2

θ|λ|

for λ ∈ Y, where dimλ is the dimension of Vλ.

Identify partitions with point configurations inZ′ := {· · · ,−3

2,−1

2, 12, 32, · · · } via

λ 7→ X (λ) :=

{. . . , λ3 −

5

2, λ2 −

3

2, λ1 −

1

2

}.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 3 / 24

λ = (5, 2, 2, 0, 0, · · · ),

X (λ) =

{. . . , 0− 9

2, 0− 7

2, 2− 5

2, 2− 3

2, 5− 1

2

}.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 4 / 24

Random partitions: Poissonized Plancherel

Then Mθ becomes a random point process on Z′.This point process is determined by its correlation functions

ProbMθ({s1, . . . , sk} ⊂ random point configuration X (λ)) .

Theorem (Borodin-Olshanski-Okounkov ’00). For any k ≥ 1,and {s1, . . . , sk} ⊂ Z′,

ProbMθ({s1, . . . , sk} ⊂ X (λ)) = det[Kθ(si , sj)]ki ,j=1,

Kθ(x, y) =√θ · Jx(2

√θ)Jy+1(2

√θ)− Jx+1(2

√θ)Jy(2

√θ)

x− y

and Jx(η) :=∞∑

m=0

(−1)m

m! Γ(m + x + 1)

(η2

)2m+x

is the Bessel function.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 5 / 24

Random partitions: Poissonized Plancherel

Then Mθ becomes a random point process on Z′.This point process is determined by its correlation functions

ProbMθ({s1, . . . , sk} ⊂ random point configuration X (λ)) .

Theorem (Borodin-Olshanski-Okounkov ’00). For any k ≥ 1,and {s1, . . . , sk} ⊂ Z′,

ProbMθ({s1, . . . , sk} ⊂ X (λ)) = det[Kθ(si , sj)]ki ,j=1,

Kθ(x, y) =√θ · Jx(2

√θ)Jy+1(2

√θ)− Jx+1(2

√θ)Jy(2

√θ)

x− y

and Jx(η) :=∞∑

m=0

(−1)m

m! Γ(m + x + 1)

(η2

)2m+x

is the Bessel function.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 5 / 24

Random partitions: z–measures

2. z-measures: Given z , z ′ ∈ C and a positive integer N , define

Mz,z ′|N(λ) :=N!∏N−1

i=0 (zz ′ + i)

∏s∈λ

(z + c(s))(z ′ + c(s))

h(s)2

if |λ| = N (partitions of size N).

They appear in the representation theory of S(∞) (group of finitarypermutations of {1, 2, 3, . . . }), e.g. we have the coherence relation:∑

λ=µt�

Mz,z ′|N(λ)

dimλ=

Mz,z ′|N−1(µ)

dimµ.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 6 / 24

Random partitions: z–measures

2. z-measures: Given z , z ′ ∈ C and a positive integer N , define

Mz,z ′|N(λ) :=N!∏N−1

i=0 (zz ′ + i)

∏s∈λ

(z + c(s))(z ′ + c(s))

h(s)2

if |λ| = N (partitions of size N).

They appear in the representation theory of S(∞) (group of finitarypermutations of {1, 2, 3, . . . }), e.g. we have the coherence relation:∑

λ=µt�

Mz,z ′|N(λ)

dimλ=

Mz,z ′|N−1(µ)

dimµ.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 6 / 24

Random partitions: z–measures

The coherence relation can be written as: for all µ ∈ YN−1,∑|λ|=N

Mz,z ′|N(λ) ΛNN−1(λ, µ) = Mz,z ′|N−1(µ),

ΛNN−1(λ, µ) :=

{dimµdimλ

, if λ = µ t�,

0, otherwise.

For fixed λ, the values ΛNN−1(λ, µ) determine a probability measure

induced by the decomposition of the restriction representation

ResSNSN−1Vλ =

⊕µ=λ\�

Vµ

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 7 / 24

Random partitions: z–measures

The coherence relation can be written as: for all µ ∈ YN−1,∑|λ|=N

Mz,z ′|N(λ) ΛNN−1(λ, µ) = Mz,z ′|N−1(µ),

ΛNN−1(λ, µ) :=

{dimµdimλ

, if λ = µ t�,

0, otherwise.

For fixed λ, the values ΛNN−1(λ, µ) determine a probability measure

induced by the decomposition of the restriction representation

ResSNSN−1Vλ =

⊕µ=λ\�

Vµ

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 7 / 24

Random partitions: z–measures

The z–measures also lead to determinantal processes, e.g., embedpartitions of size N into the space of point configurations in Z′ via

λ 7→ X (λ) :=

{. . . , λ3 −

5

2, λ2 −

3

2, λ1 −

1

2

}and let M′z,z ′|N be the pushforward measure.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 8 / 24

Random partitions: z–measures

Theorem (Borodin-Olshanski ’05).There exists a limit (in distribution) limN→∞M′z,z ′|N = M′z,z ′ and

ProbM′z,z′

({s1, . . . , sk} ⊂ random X (λ)) = det[Kz,z ′(si , sj)]ki ,j=1

Kz,z ′(x, y) := C · P(x)Q(y)− Q(x)P(y)

x− y,

where C ,P(x),Q(x) are expressed in terms of the Gamma function,e.g.,

P(x) :=Γ(z + x + 1/2)√

Γ(z + x + 1/2) Γ(z ′ + x + 1/2).

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 9 / 24

q-analogues of random partitions

Our main result is the construction of q–analogues of the z–measures.

They depend on 4 parameters t0, t1, t2, t3, the quantum parameterq ∈ (0, 1), and a positive integer N ; then Mq,t0,t1,t2,t3|N is a measureon YN : the set of partitions λ with λN+1 = 0.

We call them the quantum BC type z–measures. Conjecturallythey are related to the representation theory of quantum groups oftype B, C (Uq(so∞),Uq(sp∞)). However, their construction is basedon the theory of hypergeometric orthogonal polynomials.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 10 / 24

q-analogues of random partitions

Our wish list:

(a) Probability measures Mq,t|N on the sets YN of partitionsλ = (λ1 ≥ · · · ≥ λN ≥ 0 ≥ 0 ≥ . . . ).

(b) Branching coefficients ΛNN−1(λ, µ) ≥ 0, such that:∑

µ∈YN−1

ΛNN−1(λ, µ) = 1;

∑λ∈YN

Mq,t|N(λ) ΛNN−1(λ, µ) = Mq,t|N−1(µ).

(c) Embeddings YN ↪→ L into a q-lattice, such that the pushforwardsM′q,t|N have a limit (in distribution) limN→∞M′q,t|N = M′q,t, and M′q,thas determinantal correlation functions.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 11 / 24

Interlude: brief history of random partitions

Baik-Deift-Johansson ’99 studied the distribution of the lengthL(N) of the longest increasing subsequence of a uniformly distributedrandom permutation of {1, 2, . . . ,N}. They showed

limN→∞

Prob

(L(N)− c1N

1/2

c2N1/6≤ x

)= TW (x)

(convergence in distribution to the Tracy–Widom distribution).

Borodin-Okounkov-Olshanski ’00 reproved (and generalized) theprevious result by considering the Plancherel measure on partitionsλ of size N (the distribution of λ1 and L(N) are the same).

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 12 / 24

Interlude: brief history of random partitions

Baik-Deift-Johansson ’99 studied the distribution of the lengthL(N) of the longest increasing subsequence of a uniformly distributedrandom permutation of {1, 2, . . . ,N}. They showed

limN→∞

Prob

(L(N)− c1N

1/2

c2N1/6≤ x

)= TW (x)

(convergence in distribution to the Tracy–Widom distribution).

Borodin-Okounkov-Olshanski ’00 reproved (and generalized) theprevious result by considering the Plancherel measure on partitionsλ of size N (the distribution of λ1 and L(N) are the same).

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 12 / 24

Interlude: history of random partitions

From 2000 on, Borodin and Olshanski studied the z–measures andtheir mixtures Mz,z ′,ξ :=

∑∞n=0 π(n)Mz,z ′|n, where π is the negative

binomial distribution (on Z≥0) with parameters (zz ′, ξ):

Mz,z ′,ξ(λ) = (1− ξ)zz′ξ|λ|∏s∈λ

(z + c(s))(z ′ + c(s))

h(s)2.

Okounkov ’04 defined the Schur measures, which are furthergeneralizations of the mixed z–measures:

Mx,y(λ) :=sλ(x1, x2, . . .)sλ(y1, y2, . . .)

Z.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 13 / 24

Interlude: history of random partitions

From 2000 on, Borodin and Olshanski studied the z–measures andtheir mixtures Mz,z ′,ξ :=

∑∞n=0 π(n)Mz,z ′|n, where π is the negative

binomial distribution (on Z≥0) with parameters (zz ′, ξ):

Mz,z ′,ξ(λ) = (1− ξ)zz′ξ|λ|∏s∈λ

(z + c(s))(z ′ + c(s))

h(s)2.

Okounkov ’04 defined the Schur measures, which are furthergeneralizations of the mixed z–measures:

Mx,y(λ) :=sλ(x1, x2, . . .)sλ(y1, y2, . . .)

Z.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 13 / 24

Interlude: history of random partitions

Since 2004, there have been two main directions of research:

1. Applications of Schur measures (RMT, statistical mechanics,Gromov-Witten theory), and its extensions (Macdonald measures,periodic Schur processes), etc.

2. Study of representation-theoretic measures, which are not specialcases of Schur measures, e.g., the zwzwzw-measures (related to U(∞)),BC type zzz-measures (related to O(∞), Sp(∞)), and applications.

The first point was developed fully (q-analogues of the Schurmeasures are the Macdonald measures), but what about the second?

Gorin-Olshanski ’17 found q-analogues of zw-measures.C-Olshanski ’18+ found q-analogues of BC type z-measures:they generalize all other measures coming from rep. theory.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 14 / 24

Interlude: history of random partitions

Since 2004, there have been two main directions of research:

1. Applications of Schur measures (RMT, statistical mechanics,Gromov-Witten theory), and its extensions (Macdonald measures,periodic Schur processes), etc.

2. Study of representation-theoretic measures, which are not specialcases of Schur measures, e.g., the zwzwzw-measures (related to U(∞)),BC type zzz-measures (related to O(∞), Sp(∞)), and applications.

The first point was developed fully (q-analogues of the Schurmeasures are the Macdonald measures), but what about the second?

Gorin-Olshanski ’17 found q-analogues of zw-measures.C-Olshanski ’18+ found q-analogues of BC type z-measures:they generalize all other measures coming from rep. theory.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 14 / 24

(a) The probability measures Mq,t|N

Recall our wish list:

(a) Probability measures Mq,t|N on the sets YN of partitionsλ = (λ1 ≥ · · · ≥ λN ≥ 0 ≥ 0 ≥ . . . ).

(b) Branching coefficients ΛNN−1(λ, µ) ≥ 0, such that:∑

µ∈YN−1

ΛNN−1(λ, µ) = 1;

∑λ∈YN

Mq,t|N(λ) ΛNN−1(λ, µ) = Mq,t|N−1(µ).

(c) Embeddings YN ↪→ L into a q-lattice, such that the pushforwardsM′q,t|N have a limit (in distribution) limN→∞M′q,t|N = M′q,t, and M′q,thas determinantal correlation functions.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 15 / 24

(a) The probability measures Mq,t|N

(Conditions: t0t1 < 1, t0t−11 < q−1, t2 = t3.)

Let L := {x0 < x1 < x2 < . . . }, xn := 12(t0qn + t−10 q−n).

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 16 / 24

(a) The probability measures Mq,t|N

(Conditions: t0t1 < 1, t0t−11 < q−1, t2 = t3.)

Let L := {x0 < x1 < x2 < . . . }, xn := 12(t0qn + t−10 q−n).

Identify partitions in YN with N-point subsets of L via

λ 7→ Xq(λ) := {xλN < . . . < xλ2+N−2 < xλ1+N−1}.

(In the figure: N = 3, λ = (4, 2, 2), and Xq(λ) = {x2, x3, x6}.)

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 17 / 24

(a) The probability measures Mq,t|N

A general NNN–point orthogonal polynomial ensemble on X is ameasure on N–subsets {x1, . . . , xN} ⊂ X, given by

ProbN{x1, . . . , xN} =1

ZN

∏1≤i<j≤N

(xi − xj)2

N∏k=1

W (xk),

for some weight function W : X→ R+.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 18 / 24

(a) The probability measures Mq,t|N

For us, X = L = {x0 < x1 < x2 < . . . }. The weight W is the weightfunction for the Askey–Wilson polynomials:

Wq,t(xn) :=1− t20q

2n

1− t20

(t20 , t0t1, t0t2, t0t3;q)n(q, t0q/t1, t0q/t2, t0q/t3;q)n

(q

t0t1t2t3

)n

where (x ;q)n :=∏n−1

i=0 (1− xqi) is the q-factorial.

The corresponding N-point orthogonal polynomial ensemble isdenoted Mq,t|N :

Mq,t|N{xn1 , . . . , xnN} =1

ZN

∏1≤i<j≤N

(xni − xnj )2

N∏k=1

Wq,t(xnk )

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 19 / 24

(a) The probability measures Mq,t|N

For us, X = L = {x0 < x1 < x2 < . . . }. The weight W is the weightfunction for the Askey–Wilson polynomials:

Wq,t(xn) :=1− t20q

2n

1− t20

(t20 , t0t1, t0t2, t0t3;q)n(q, t0q/t1, t0q/t2, t0q/t3;q)n

(q

t0t1t2t3

)n

where (x ;q)n :=∏n−1

i=0 (1− xqi) is the q-factorial.

The corresponding N-point orthogonal polynomial ensemble isdenoted Mq,t|N :

Mq,t|N{xn1 , . . . , xnN} =1

ZN

∏1≤i<j≤N

(xni − xnj )2

N∏k=1

Wq,t(xnk )

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 19 / 24

(b) Positive branching coefficients ΛNN−1(λ, µ)

The little q–Jacobi polynomials Pn(x ;q; a, b), n ≥ 0, are theorthogonal polynomials w.r.t. the weight

Wq,a,b(qk) := (aq)k(bq;q)k(q;q)k

, k ≥ 0,

on the q–lattice {. . . ,q2,q, 1}.

The multivariable little q–Jacobi polynomialsPλ|N(x1, . . . , xN ;q; a, b), λ ∈ YN , are

Pλ|N(x1, . . . , xN ;q; a, b) :=det[Pλi+N−i(xj ;q; a, b)]Ni ,j=1∏

1≤i<j≤N (xi − xj).

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 20 / 24

(b) Positive branching coefficients ΛNN−1(λ, µ)

The little q–Jacobi polynomials Pn(x ;q; a, b), n ≥ 0, are theorthogonal polynomials w.r.t. the weight

Wq,a,b(qk) := (aq)k(bq;q)k(q;q)k

, k ≥ 0,

on the q–lattice {. . . ,q2,q, 1}.

The multivariable little q–Jacobi polynomialsPλ|N(x1, . . . , xN ;q; a, b), λ ∈ YN , are

Pλ|N(x1, . . . , xN ;q; a, b) :=det[Pλi+N−i(xj ;q; a, b)]Ni ,j=1∏

1≤i<j≤N (xi − xj).

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 20 / 24

(b) Positive branching coefficients ΛNN−1(λ, µ)

Set a := t0t1q−1, b := t0t

−11 ; the branching coefficients are given by

Pλ|N(x1, . . . , xN−1, 0)

Pλ|N(0, . . . , 0)=

∑µ∈YN−1

ΛNN−1(λ, µ)

Pµ|N−1(x1, . . . , xN−1)

Pµ|N−1(0, . . . , 0).

Theorem (C-Olshanski, 2019+).The coefficients ΛN

N−1(λ, µ) are nonnegative and∑µ∈YN−1

ΛNN−1(λ, µ) = 1, for all λ ∈ YN ,∑

λ∈YN

Mq,t|N(λ) ΛNN−1(λ, µ) = Mq,t|N−1(µ), for all µ ∈ YN−1.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 21 / 24

(b) Positive branching coefficients ΛNN−1(λ, µ)

Set a := t0t1q−1, b := t0t

−11 ; the branching coefficients are given by

Pλ|N(x1, . . . , xN−1, 0)

Pλ|N(0, . . . , 0)=

∑µ∈YN−1

ΛNN−1(λ, µ)

Pµ|N−1(x1, . . . , xN−1)

Pµ|N−1(0, . . . , 0).

Theorem (C-Olshanski, 2019+).The coefficients ΛN

N−1(λ, µ) are nonnegative and∑µ∈YN−1

ΛNN−1(λ, µ) = 1, for all λ ∈ YN ,∑

λ∈YN

Mq,t|N(λ) ΛNN−1(λ, µ) = Mq,t|N−1(µ), for all µ ∈ YN−1.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 21 / 24

(c) Determinantal point processes

For each N , consider the map

L→ qNL := {qNx : x ∈ L} ⊂ Rx 7→ qNx .

and denote the pushforward measure (on the real line) by M′q,t|N .

Theorem (C-Olshanski, 2019+).The limit (in distribution) limN→∞M′q,t|N = M′q,t exists, and it is aprobability measure supported on

L′ := {. . . , t−10 q, t−10 , t−10 q−1, . . .}.

· · · · · · · · · · · ·

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 22 / 24

(c) Determinantal point processes

For each N , consider the map

L→ qNL := {qNx : x ∈ L} ⊂ Rx 7→ qNx .

and denote the pushforward measure (on the real line) by M′q,t|N .

Theorem (C-Olshanski, 2019+).The limit (in distribution) limN→∞M′q,t|N = M′q,t exists, and it is aprobability measure supported on

L′ := {. . . , t−10 q, t−10 , t−10 q−1, . . .}.

· · · · · · · · · · · ·

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 22 / 24

(c) Determinantal point processes

For any {s1, . . . , sk} ⊂ L′, the correlation function is

ProbM′q,t ({s1, . . . , sk} ⊂ random subset X ) = det[Kq,t(si , sj)]ki ,j=1,

Kq,t(x, y) = C · F(x)G(y)−G(x)F(y)

x− y,

and C ,F(x),G(x) are expressed in terms of q-factorials and basichypergeometric series, e.g.,

F(x) ∼ 2φ1

[1/(t0t3), t2/x

1/(t0x); q; 1/(t1t2)

].

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 23 / 24

Summary

The study of probability measures on partitions began fromstriking analogies with Random Matrix Theory, and developedinto a subject of its own right.

A rich class of random partitions comes from the representationtheory of infinite-dimensional groups S(∞), U(∞), etc.

We found two new examples (we showed one here) of randompartitions that are q-analogues of measures of rep-theoreticorigin. They are conjectured to be related to quantum groups.

Our measures are the most general ones of rep-theoretic origin.The proofs are tricky and don’t explain their nature; many openquestions remain, including a possible (affine) generalization ofSchur measures.

Cesar Cuenca (MIT) q-analogues of rep-theoretic measures February 7, 2019 24 / 24