Introduction to Zonal Polynomials Lin Jiu Dalhousie University - - PowerPoint PPT Presentation

Introduction to Zonal Polynomials

Lin Jiu

Dalhousie University Number Theory Seminar

• Jan. 22, 2018

Hypergeometric function and Pochhammer symbol

sFt

a1, . . . , as b1, . . . , bt : z

• :=

∞

• n=0

(a1)n · · · (as)n (b1)n · · · (bt)n · zn n! ,

Hypergeometric function and Pochhammer symbol

sFt

a1, . . . , as b1, . . . , bt : z

• :=

∞

• n=0

(a1)n · · · (as)n (b1)n · · · (bt)n · zn n! , where (a)k = a (a + 1) · · · (a + k − 1).

Hypergeometric function and Pochhammer symbol

sFt

a1, . . . , as b1, . . . , bt : z

• :=

∞

• n=0

(a1)n · · · (as)n (b1)n · · · (bt)n · zn n! , where (a)k = a (a + 1) · · · (a + k − 1).

Examples

◮ 2F1

• a,b

c : z

• is the Gaussian hypergeometric function s. t.

z (1 − z) d2w dz2 + (c − (a + b + 1) z) dw dz − abw = 0.

Hypergeometric function and Pochhammer symbol

sFt

a1, . . . , as b1, . . . , bt : z

• :=

∞

• n=0

(a1)n · · · (as)n (b1)n · · · (bt)n · zn n! , where (a)k = a (a + 1) · · · (a + k − 1).

Examples

◮ 2F1

• a,b

c : z

• is the Gaussian hypergeometric function s. t.

z (1 − z) d2w dz2 + (c − (a + b + 1) z) dw dz − abw = 0.

◮ log (1 + z) = z 2F1

• 1,1

2 : −z

Hypergeometric function and Pochhammer symbol

sFt

a1, . . . , as b1, . . . , bt : z

• :=

∞

• n=0

(a1)n · · · (as)n (b1)n · · · (bt)n · zn n! , where (a)k = a (a + 1) · · · (a + k − 1).

Examples

◮ 2F1

• a,b

c : z

• is the Gaussian hypergeometric function s. t.

z (1 − z) d2w dz2 + (c − (a + b + 1) z) dw dz − abw = 0.

◮ log (1 + z) = z 2F1

• 1,1

2 : −z

• ◮ ez =

0F0( : z)

Hypergeometric function with matrix argument

Hypergeometric function with matrix argument

Given an m × m symmetric (postive definite) matrix Y ,

sFt

a1, . . . , as b1, . . . , bt : Y

• :=

∞

• n=0
• p∈Pn

(a1)p · · · (as)p (b1)p · · · (bt)p · Cp (Y ) n! ,

Hypergeometric function with matrix argument

Given an m × m symmetric (postive definite) matrix Y ,

sFt

a1, . . . , as b1, . . . , bt : Y

• :=

∞

• n=0
• p∈Pn

(a1)p · · · (as)p (b1)p · · · (bt)p · Cp (Y ) n! , where,

◮ Pn is the set of all partitions of n

Hypergeometric function with matrix argument

Given an m × m symmetric (postive definite) matrix Y ,

sFt

a1, . . . , as b1, . . . , bt : Y

• :=

∞

• n=0
• p∈Pn

(a1)p · · · (as)p (b1)p · · · (bt)p · Cp (Y ) n! , where,

◮ Pn is the set of all partitions of n

and a partition of n is a (p1, . . . , pl) ∈ Nl such that p1 ≥ · · · ≥ pl > 0 and p1 + · · · + pl = n(= |p|),

Hypergeometric function with matrix argument

Given an m × m symmetric (postive definite) matrix Y ,

sFt

a1, . . . , as b1, . . . , bt : Y

• :=

∞

• n=0
• p∈Pn

(a1)p · · · (as)p (b1)p · · · (bt)p · Cp (Y ) n! , where,

◮ Pn is the set of all partitions of n

and a partition of n is a (p1, . . . , pl) ∈ Nl such that p1 ≥ · · · ≥ pl > 0 and p1 + · · · + pl = n(= |p|), e.g., (5, 2, 2, 1) ∈ P10;

Hypergeometric function with matrix argument

Given an m × m symmetric (postive definite) matrix Y ,

sFt

a1, . . . , as b1, . . . , bt : Y

• :=

∞

• n=0
• p∈Pn

(a1)p · · · (as)p (b1)p · · · (bt)p · Cp (Y ) n! , where,

◮ Pn is the set of all partitions of n

and a partition of n is a (p1, . . . , pl) ∈ Nl such that p1 ≥ · · · ≥ pl > 0 and p1 + · · · + pl = n(= |p|), e.g., (5, 2, 2, 1) ∈ P10;

◮ for p = (p1, . . . , pl) ∈ Pn, (a)p = l

• i=1
• a − i−1

2

• pi;

Hypergeometric function with matrix argument

Given an m × m symmetric (postive definite) matrix Y ,

sFt

a1, . . . , as b1, . . . , bt : Y

• :=

∞

• n=0
• p∈Pn

(a1)p · · · (as)p (b1)p · · · (bt)p · Cp (Y ) n! , where,

◮ Pn is the set of all partitions of n

and a partition of n is a (p1, . . . , pl) ∈ Nl such that p1 ≥ · · · ≥ pl > 0 and p1 + · · · + pl = n(= |p|), e.g., (5, 2, 2, 1) ∈ P10;

◮ for p = (p1, . . . , pl) ∈ Pn, (a)p = l

• i=1
• a − i−1

2

• pi;

◮ Cp (Y ) is (C-normalization of) zonal polynomial, which is

homogeneous, symmetric, polynomial of degree n = |p|, in the eigenvalues of Y .

Zonal Polynomial Cp (Y )

Cp (Y )

Zonal Polynomial Cp (Y )

Cp (Y )

◮ It is defined on eigenvalues of Y

Zonal Polynomial Cp (Y )

Cp (Y )

◮ It is defined on eigenvalues of Y

Cp (y1, . . . , ym)

Zonal Polynomial Cp (Y )

Cp (Y )

◮ It is defined on eigenvalues of Y

Cp (y1, . . . , ym)

◮ For p = (p1, . . . , pl), if m < l, (will see why later)

Cp (Y ) = Cp (y1, . . . , ym, 0, . . . , 0)

Zonal Polynomial Cp (Y )

Cp (Y )

◮ It is defined on eigenvalues of Y

Cp (y1, . . . , ym)

◮ For p = (p1, . . . , pl), if m < l, (will see why later)

Cp (Y ) = Cp (y1, . . . , ym, 0, . . . , 0)

◮ An important fact

• p∈Pn

Cp (Y ) = (trY )n = (y1 + · · · + ym)n .

Zonal Polynomial Cp (Y )

0F0

• : Y
• =

∞

• n=0
• p∈Pn

·Cp (Y ) n! =

∞

• n=0

(trY )n n! = etrY

Zonal Polynomial Cp (Y )

0F0

• : Y
• =

∞

• n=0
• p∈Pn

·Cp (Y ) n! =

∞

• n=0

(trY )n n! = etrY ez =

0F0( : z)

Zonal Polynomial Cp (Y )

0F0

• : Y
• =

∞

• n=0
• p∈Pn

·Cp (Y ) n! =

∞

• n=0

(trY )n n! = etrY ez =

0F0( : z) 1F0

a : z

• = (1 − z)−a

1F0

a : Y

• = det(I − A)−a

Definition 1

Definition 1

Cp (Y ) = dpYp (Y ), where dp =

• i<j

(2pi − 2pj − i + j)

l

• i=1

(2pi + l − i)! · 2nn! (2n)!.

Definition 1

Cp (Y ) = dpYp (Y ), where dp =

• i<j

(2pi − 2pj − i + j)

l

• i=1

(2pi + l − i)! · 2nn! (2n)!.

◮ Define a linear space

Vn := {f : f is homogeneous, symmetric, of degree n, or f ≡ 0}

Definition 1

Cp (Y ) = dpYp (Y ), where dp =

• i<j

(2pi − 2pj − i + j)

l

• i=1

(2pi + l − i)! · 2nn! (2n)!.

◮ Define a linear space

Vn := {f : f is homogeneous, symmetric, of degree n, or f ≡ 0} where f is defined on eigenvalues of matrices.

◮ Basis for Vn:

Definition 1

Cp (Y ) = dpYp (Y ), where dp =

• i<j

(2pi − 2pj − i + j)

l

• i=1

(2pi + l − i)! · 2nn! (2n)!.

◮ Define a linear space

Vn := {f : f is homogeneous, symmetric, of degree n, or f ≡ 0} where f is defined on eigenvalues of matrices.

◮ Basis for Vn: define the elementary symmetric polynomial

ur (x1, . . . , xm) :=

• i1<···<ir

xi1 · · · xir . Then, for p = (p1, . . . , pl) ∈ Pn Up := up1−p2

1

up2−p3

2

· · · upl−1−pl

l−1

upl(−0)

l

,

Definition 1

Cp (Y ) = dpYp (Y ), where dp =

• i<j

(2pi − 2pj − i + j)

l

• i=1

(2pi + l − i)! · 2nn! (2n)!.

◮ Define a linear space

Vn := {f : f is homogeneous, symmetric, of degree n, or f ≡ 0} where f is defined on eigenvalues of matrices.

◮ Basis for Vn: define the elementary symmetric polynomial

ur (x1, . . . , xm) :=

• i1<···<ir

xi1 · · · xir . Then, for p = (p1, . . . , pl) ∈ Pn Up := up1−p2

1

up2−p3

2

· · · upl−1−pl

l−1

upl(−0)

l

,

◮ deg Up =

Definition 1

Cp (Y ) = dpYp (Y ), where dp =

• i<j

(2pi − 2pj − i + j)

l

• i=1

(2pi + l − i)! · 2nn! (2n)!.

◮ Define a linear space

Vn := {f : f is homogeneous, symmetric, of degree n, or f ≡ 0} where f is defined on eigenvalues of matrices.

◮ Basis for Vn: define the elementary symmetric polynomial

ur (x1, . . . , xm) :=

• i1<···<ir

xi1 · · · xir . Then, for p = (p1, . . . , pl) ∈ Pn Up := up1−p2

1

up2−p3

2

· · · upl−1−pl

l−1

upl(−0)

l

,

◮ deg Up = p1 − p2 + 2(p2 − p3) + · · · + lpl

Definition 1

Cp (Y ) = dpYp (Y ), where dp =

• i<j

(2pi − 2pj − i + j)

l

• i=1

(2pi + l − i)! · 2nn! (2n)!.

◮ Define a linear space

Vn := {f : f is homogeneous, symmetric, of degree n, or f ≡ 0} where f is defined on eigenvalues of matrices.

◮ Basis for Vn: define the elementary symmetric polynomial

ur (x1, . . . , xm) :=

• i1<···<ir

xi1 · · · xir . Then, for p = (p1, . . . , pl) ∈ Pn Up := up1−p2

1

up2−p3

2

· · · upl−1−pl

l−1

upl(−0)

l

,

◮ deg Up = p1 − p2 + 2(p2 − p3) + · · · + lpl = p1 + · · · + pl = n.

Definition 1

Cp (Y ) = dpYp (Y ), where dp =

• i<j

(2pi − 2pj − i + j)

l

• i=1

(2pi + l − i)! · 2nn! (2n)!.

◮ Define a linear space

Vn := {f : f is homogeneous, symmetric, of degree n, or f ≡ 0} where f is defined on eigenvalues of matrices.

◮ Basis for Vn: define the elementary symmetric polynomial

ur (x1, . . . , xm) :=

• i1<···<ir

xi1 · · · xir . Then, for p = (p1, . . . , pl) ∈ Pn Up := up1−p2

1

up2−p3

2

· · · upl−1−pl

l−1

upl(−0)

l

,

◮ deg Up = p1 − p2 + 2(p2 − p3) + · · · + lpl = p1 + · · · + pl = n. ◮ U :=

• U(n), U(n−1,1), . . . , U(1,1,...,1)

T forms a basis of Vn.

Definition 1

Definition 1

Y =      Y(n) Y(n−1,1) . . . Y(1n)      = ΞU = Ξ      U(n) U(n−1,1) . . . U(1n)     

Definition 1

Y =      Y(n) Y(n−1,1) . . . Y(1n)      = ΞU = Ξ      U(n) U(n−1,1) . . . U(1n)     

◮ Ξ is a matrix: nonsingular, upper triangular, such that

Tν = Ξ−1

(ν)ΛνΞ(ν), where Tν is related to Wishart distribution

and Λν is its diagonal matrix.

Definition 1

Y =      Y(n) Y(n−1,1) . . . Y(1n)      = ΞU = Ξ      U(n) U(n−1,1) . . . U(1n)     

◮ Ξ is a matrix: nonsingular, upper triangular, such that

Tν = Ξ−1

(ν)ΛνΞ(ν), where Tν is related to Wishart distribution

and Λν is its diagonal matrix.

◮ Define linear transform τν : Vn −

→ Vn by (τν (Up)) (A) :=

Definition 1

Y =      Y(n) Y(n−1,1) . . . Y(1n)      = ΞU = Ξ      U(n) U(n−1,1) . . . U(1n)     

◮ Ξ is a matrix: nonsingular, upper triangular, such that

Tν = Ξ−1

(ν)ΛνΞ(ν), where Tν is related to Wishart distribution

and Λν is its diagonal matrix.

◮ Define linear transform τν : Vn −

→ Vn by (τν (Up)) (A) := EW [Up (AW )] for W ∼ W (Ik, ν) .

Definition 1

Y =      Y(n) Y(n−1,1) . . . Y(1n)      = ΞU = Ξ      U(n) U(n−1,1) . . . U(1n)     

◮ Ξ is a matrix: nonsingular, upper triangular, such that

Tν = Ξ−1

(ν)ΛνΞ(ν), where Tν is related to Wishart distribution

and Λν is its diagonal matrix.

◮ Define linear transform τν : Vn −

→ Vn by (τν (Up)) (A) := EW [Up (AW )] for W ∼ W (Ik, ν) . Then, a lemma (due to basis) shows τνU = TνU.

Definition 1

Y =      Y(n) Y(n−1,1) . . . Y(1n)      = ΞU = Ξ      U(n) U(n−1,1) . . . U(1n)     

◮ Ξ is a matrix: nonsingular, upper triangular, such that

Tν = Ξ−1

(ν)ΛνΞ(ν), where Tν is related to Wishart distribution

and Λν is its diagonal matrix.

◮ Define linear transform τν : Vn −

→ Vn by (τν (Up)) (A) := EW [Up (AW )] for W ∼ W (Ik, ν) . Then, a lemma (due to basis) shows τνU = TνU.

◮ Z1, . . . , Zk ∼ N (0, 1) are independent (i. i. d. ), then

Q := Z1 + · · · + Zk ∼ χ2

k;

Definition 1

Y =      Y(n) Y(n−1,1) . . . Y(1n)      = ΞU = Ξ      U(n) U(n−1,1) . . . U(1n)     

◮ Ξ is a matrix: nonsingular, upper triangular, such that

Tν = Ξ−1

(ν)ΛνΞ(ν), where Tν is related to Wishart distribution

and Λν is its diagonal matrix.

◮ Define linear transform τν : Vn −

→ Vn by (τν (Up)) (A) := EW [Up (AW )] for W ∼ W (Ik, ν) . Then, a lemma (due to basis) shows τνU = TνU.

◮ Z1, . . . , Zk ∼ N (0, 1) are independent (i. i. d. ), then

Q := Z1 + · · · + Zk ∼ χ2

k; ◮ Let Xν×m be such that each row is independently drawn from

an m-variate normal distribution,

• x1

i , . . . , xm i

• ∼ Nm (0, V ) ⇒ S = X TX ∼ Wm (V , ν)

and ν is called the degree of freedom.

Computation

Computation

Mλ (y1, . . . , ym) =

• i1,...,il

distinct terms

yλ1

i1 · · · yλl il = yλ1 1 · · · yλl l +symmetric terms.

Computation

Mλ (y1, . . . , ym) =

• i1,...,il

distinct terms

yλ1

i1 · · · yλl il = yλ1 1 · · · yλl l +symmetric terms.

1. M(1) (Y ) = y1 + · · · + ym; 2. M(2) (Y ) = y2

1 + · · · + y2 m;

3. M(1,1) (Y ) =

• i<j

yiyj; 4. M(2,1) (Y ) =

• i,j

y2

i yj.

Computation

Computation

For p = (p1, . . . , pℓ) and λ = (λ1, . . . , λm), Cp (Y ) =

• λ≤p

cp,λMλ (Y ) .

Computation

For p = (p1, . . . , pℓ) and λ = (λ1, . . . , λm), Cp (Y ) =

• λ≤p

cp,λMλ (Y ) . for some constants cp,λ cp,λ =

• λ<µ≤p

(λi + t) − (λj − t) ρp − ρλ cp,µ. Here, ρp :=

ℓ

• j=1

pi (pi − j) and for λ = (λ1, . . . , λl), the sum is over all µ = (λ1, . . . , λi + t, . . . , λj − t, . . . , λl) for t = 1, . . . , λj such that by rearranging tuple µ in a descending order, it lies as λ < µ ≤ p.

Computation

Computation

◮ n = 5 p\λ (5) (4, 1) (3, 2) (3, 1, 1) (2, 2, 1) (2, 1, 1, 1) (1, 1, 1, 1, 1) (5) 1

5 9 10 21 20 63 2 7 4 21 8 63

(4, 1)

40 9 8 3 46 9

4

14 3 40 9

(3, 2)

48 7 32 7 176 21 64 7 80 7

(3, 1, 1) 10

20 3 130 7 200 7

(2, 2, 1)

32 3

16 32 (2, 1, 1, 1)

80 7 800 21

(1, 1, 1, 1, 1)

16 3

Computation

◮ n = 5 p\λ (5) (4, 1) (3, 2) (3, 1, 1) (2, 2, 1) (2, 1, 1, 1) (1, 1, 1, 1, 1) (5) 1

5 9 10 21 20 63 2 7 4 21 8 63

(4, 1)

40 9 8 3 46 9

4

14 3 40 9

(3, 2)

48 7 32 7 176 21 64 7 80 7

(3, 1, 1) 10

20 3 130 7 200 7

(2, 2, 1)

32 3

16 32 (2, 1, 1, 1)

80 7 800 21

(1, 1, 1, 1, 1)

16 3

◮

C(1,1) (a, b, c) = 4 3 (ab + bc + ac) C(2)(a, b, c) = a2 + b2 + c2 + 2 3 (ab + bc + ac)

Computation

Computation

Definition 2

Definition 2

Recall in R3, the following operators:

Definition 2

Recall in R3, the following operators:

◮ gradient

Definition 2

Recall in R3, the following operators:

◮ gradient

∇f := (fx, fy, fz) ;

Definition 2

Recall in R3, the following operators:

◮ gradient

∇f := (fx, fy, fz) ;

◮ divergence

divX = ∂X ∂x + ∂X ∂y + ∂X ∂z ;

Definition 2

Recall in R3, the following operators:

◮ gradient

∇f := (fx, fy, fz) ;

◮ divergence

divX = ∂X ∂x + ∂X ∂y + ∂X ∂z ;

◮ Laplace

∆f = ∂2f ∂x2 + ∂2f ∂y2 + ∂2f ∂z2 = (div • ∇) f ;

Definition 2

Recall in R3, the following operators:

◮ gradient

∇f := (fx, fy, fz) ;

◮ divergence

divX = ∂X ∂x + ∂X ∂y + ∂X ∂z ;

◮ Laplace

∆f = ∂2f ∂x2 + ∂2f ∂y2 + ∂2f ∂z2 = (div • ∇) f ; On a Riemannian manifold (M, g), the Laplace-Beltrami operator

• n f ∈ C ∞ (M) is given by

∆f := (div • grad) f = 1 √ G ∂k

• gik√

G∂if

• .

Definition 2

Definition 2

Let M = SPD (m)

Definition 2

Let M = SPD (m) and associate the canonical metric.

Definition 2

Let M = SPD (m) and associate the canonical metric. Then, for Y ∈ M with eigenvalues y1, . . . , ym: ∆ =

m

• i=1

 y2

i

∂2 ∂y2

i

− m − 3 2 yi ∂ ∂yi +

n

• j=1,j=i

y2

i

yi − yj ∂ ∂yi   .

Definition 2

Let M = SPD (m) and associate the canonical metric. Then, for Y ∈ M with eigenvalues y1, . . . , ym: ∆ =

m

• i=1

 y2

i

∂2 ∂y2

i

− m − 3 2 yi ∂ ∂yi +

n

• j=1,j=i

y2

i

yi − yj ∂ ∂yi   . Euler’s operator

m

• i=1

yi ∂

∂yi

Definition 2

Let M = SPD (m) and associate the canonical metric. Then, for Y ∈ M with eigenvalues y1, . . . , ym: ∆ =

m

• i=1

 y2

i

∂2 ∂y2

i

− m − 3 2 yi ∂ ∂yi +

n

• j=1,j=i

y2

i

yi − yj ∂ ∂yi   . Euler’s operator

m

• i=1

yi ∂

∂yi

Zonal polynomial Cp (y1, . . . , ym) are eigenfunctions of ∆Y , defined by ∆Y :=

m

• i=1

 y2

i

∂2 ∂y2

i

+

n

• j=1,j=i

y2

i

yi − yj ∂ ∂yi   . In particular ∆Y Cp (Y ) = (ρp + m (l − 1)) Cλ (Y ) , where ρp :=

l

• i=1

pi (pi − 1) .

Definition 3

Definition 3

Consider G = GL (m) and a representation of linear space Vn: g ∈ GL (m) : Vn → Vn ϕ (Y ) → ϕ

• g−1Y
• g−1T

Definition 3

Consider G = GL (m) and a representation of linear space Vn: g ∈ GL (m) : Vn → Vn ϕ (Y ) → ϕ

• g−1Y
• g−1T

As a representation, the linear space can be decomposed into invariant subspaces: Vn =

• p∈Pn

Vp.

Definition 3

Consider G = GL (m) and a representation of linear space Vn: g ∈ GL (m) : Vn → Vn ϕ (Y ) → ϕ

• g−1Y
• g−1T

As a representation, the linear space can be decomposed into invariant subspaces: Vn =

• p∈Pn

Vp. Now, note that (trY )n ∈ Vn. The projection (trY )n

• Vp

= Cp (Y ) .

Definition 4

Definition 4

◮ Macdonald polynomials Pλ(x; t, q) are a family of orthogonal

polynomials in several variables, introduced by Macdonald (1987).

Definition 4

◮ Macdonald polynomials Pλ(x; t, q) are a family of orthogonal

polynomials in several variables, introduced by Macdonald (1987).

◮

Definition 4

Definition 4

◮ If we put t = qα and let q tend to 1 the Macdonald

polynomials become Jack polynomials (with further conditions)

◮

Definition 4

Definition 4

◮ When α = 1, with some normalization, Jack polynomials

becomes Schur polynomials

Definition 4

◮ When α = 1, with some normalization, Jack polynomials

becomes Schur polynomials which are certain symmetric polynomials in multi-variables, indexed by partitions.

Definition 4

◮ When α = 1, with some normalization, Jack polynomials

becomes Schur polynomials which are certain symmetric polynomials in multi-variables, indexed by partitions.

◮ When α = 1/2, with some normalization, Jack polynomial

gives Cp.

Definition 4

◮ When α = 1, with some normalization, Jack polynomials

becomes Schur polynomials which are certain symmetric polynomials in multi-variables, indexed by partitions.

◮ When α = 1/2, with some normalization, Jack polynomial

gives Cp. Macdonald polynomial − → Jack polynomial − → zonal polynomial.

Definition 4

◮ When α = 1, with some normalization, Jack polynomials

becomes Schur polynomials which are certain symmetric polynomials in multi-variables, indexed by partitions.

◮ When α = 1/2, with some normalization, Jack polynomial

gives Cp. Macdonald polynomial − → Jack polynomial − → zonal polynomial.

sF (α) t

a1, . . . , as b1, . . . , bt : Y

• :=

∞

• n=0
• p∈Pn

(a1)p · · · (as)p (b1)p · · · (bt)p · C(α)

p

(Y ) n!