Zonal Polynomials and Hypergeometric Functions of Matrix Argument

Zonal Polynomials and Hypergeometric Functions of
Matrix Argument
Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 1/2

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 2/2
Some matrix spaces
G = GL(n, R): The general linear group, containing all n × n
nonsingular real matrices
S = S(n, R): The space of real symmetric matrices
G “acts” on S: x ∈ G acts on s ∈ S by x ◦ s ≡ xsx ′
x 1 x 2 ◦ s = (x 1 x 2 )s(x 1 x 2 ) ′ = x 1 x 2 sx ′ 2x ′ 1
= x 1 (x 2 sx ′ 2)x ′ 1 = x 1 (x 2 ◦ s)x ′ 1 = x 1 ◦ (x 2 ◦ s)
x 1 x 2 ◦ s = x 1 ◦ (x 2 ◦ s)
This is a group action
right-action vs. left-action

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 3/2
Notation: For x = (x ij ) ∈ G,
∆(x) := det(x)
⎛
⎞
x 11 · · · x 1j
⎜
⎟
∆ j (x) = ∆ ⎝ . . ⎠ ,
x j1 · · · x jj
j = 1, . . . , n
The standard bitriangular structure of G: Each x ∈ G can be
expressed as x = vcu where
c is diagonal
u is upper triangular with 1’s on the main diagonal
v is lower triangular with 1’s on the main diagonal

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 4/2
The map from (u, c, v) → vcu is smooth when restricted to the
subset of matrices x ∈ G such that
n∏
∆ j (x) ≠ 0
j=1
O(n): the group of n × n orthogonal matrices
O(n) is a maximal compact subgroup of G
P (n, R): The cone of positive definite n × n matrices
Each r ∈ P (n, R) is of the form r = xx ′ , x ∈ G

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 5/2
Polynomials on G
φ: A polynomial function on G
φ(x) is a polynomial in the entries x ij of x
P(G): The space of polynomials on G
P d (G): The space of polynomials homogeneous of degree d
P d (G) is a vector space of dimension ( )
N+d−1
N−1 , where N ≡ n
2
G is an open subset of R n×n
φ extends uniquely to a polynomial on R n×n
A polynomial on G restricts uniquely to a polynomial on P (n, R)

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 6/2
For a ∈ G, define R a : P(G) → P(G) by:
R a φ(x) = φ(xa),
x ∈ G
R a is called the right regular representation of G on P(G)
What is R a if a = I n
R ab φ = R b R a φ
P d (G) is invariant under R a : R a P d (G) = P d (G)

Regard R n×n as R N
Each polynomial φ on R n×n is a polynomial on R N
φ(x) =
∑
a α1 ,...,α N
x α 1
1 xα 2
2 · · · xα N
N
α 1 ,...,α N
Shorthand notation:
φ(x) = ∑ a α X α
α
α! = α 1 ! · · · α N !
Conjugate of φ:
˜φ(x) =
∑
α āαX α Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 7/2

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 8/2
The differentiation inner product on P(G)
For φ = ∑ α a αX α and ψ = ∑ α b αX α , define the inner product
〈φ|ψ〉 = ∑ α
α!a α¯bα
This inner product arises from differentiation of ψ by φ
HW: Prove that for k ∈ O(n), R k is a unitary operator on P(G):
〈R k φ|R k ψ〉 = 〈φ|ψ〉
Hint: Show that 〈R a ′φ|R a −1ψ〉 = 〈φ|ψ〉, a ∈ G
Under the inner product,
∞∑
P(G) = ⊕P d (G)
d=0

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 9/2
Representations of G
G: GL(n, R)
V : A space of linear transformations
ρ : G → V is a representation of G on V if
ρ(xy) = ρ(x)ρ(y), x, y ∈ G
The dimension of ρ is the dimension of V
Example: The right regular representation
Example: ρ(x) = ∆(x) k ∆(x) l is a one-dimensional
representation of G
All one-dimensional representations of G are powers of ∆(x)

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 10/2
Given representations ρ 1 , ρ 2 , form the direct sum
(
)
ρ 1 (x) 0
(ρ 1 ⊕ ρ 2 (x) =
0 ρ 2 (x)
This gives us a new representation
Irreducible representations: Those which cannot be written as a
direct sum of lower-dimensional representations
The basic problem in representation theory: Describe all
irreducible representations of a group
G = GL(n, R): We use the standard bitriangular decomposition
to describe some irreducible polynomial representations of G

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 11/2
Each irreducible, finite-dimensional, polynomial representation
of GL(n, R) is parametrized by an n-tuple (m 1 , . . . , m n ) of
integers where m 1 ≥ · · · ≥ m n ≥ 0
π m : The representation corresponding to the signature
m = (m 1 , . . . , m n )
Return to the (U, C, V ) decomposition: x = vcu
For each signature m and c = diag(c 1 , . . . , c n ), let
µ m (c) = |c 1 | 2m 1
|c 2 | 2m2 · · · |c n | 2m n
µ m is a character of C
µ m is a one-dimensional representation of C

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 12/2
P 2m (G): The collection of all φ ∈ P(G) such that
φ(vcx) = µ 2m (c)φ(x), v ∈ V, c ∈ C, x ∈ R n×n
Each φ in P 2m (G) is homogeneous:
P 2m (G) ⊂ P d (G),
d = 2|m|, |m| = m 1 + · · · + m n
A crucial example of a polynomial in P 2m (G):
φ 2m (x) = ∆(x) 2m n
n−1
∏
∆ j (x) 2(m j−m j+1 )
j=1
Calculate ∆ j (vcx) to see why φ 2m ∈ P 2m (G)
More comments on the representation theory of G

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 13/2
Orthogonally invariant polynomials
I(G): The space of left-invariant polynomials φ on G,
φ(kx) = φ(x), k ∈ O(n), x ∈ G
If φ is homogeneous then it is of even degree, because
−I n ∈ O(n)
I 2d (G): The class of φ ∈ I(G) which are homogeneous of
degree 2d
I(G) =
∞∑
⊕I 2d (G)
d=0
The spherical transform: Given φ ∈ P(G), construct φ # ∈ I(G):
∫
φ # (x) = φ(kx) dk
O(n)

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 14/2
Apply the spherical transform to each φ ∈ P 2m (G) to get the
space I 2m (G)
The spherical transform decomposes I 2d (G) into irreducible,
orthogonal subspaces
I 2d (G) = ∑
⊕I 2m (G)
|m|=d
S = S(n, R): The space of real symmetric matrices
Each polynomial φ on G gives rise to a polynomial q on S:
q(xx ′ ) = φ(x)
The spherical transform converts each left-invariant polynomial p
on G into a biinvariant polynomial q on S

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 15/2
The zonal polynomials
Apply the spherical transform to I 2d (G) and I 2m (G)
P d (S) = ∑
|m|=d
⊕P m (S)
This decomposition is multiplicity free: Up to constant multiples,
there exists only one nontrivial, biinvariant polynomial in P m (S)
This unique polynomial is called the zonal polynomial
Let φ ∈ P d (S), φ(ksk ′ ) = φ(s), k ∈ O(n), s ∈ S. Then there exist
unique φ m ∈ P m (S) such that
φ = ∑
|m|=d
φ m
The {φ m } are orthogonal w.r.t. the differentiation inner product

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 16/2
Is each subspace P m (S) nontrivial Is there an explicit formula
for each zonal polynomial
Apply the spherical transform to the “crucial example” in P 2m (G)
q m (s) =
∫
O(n)
∆(ksk ′ ) m n
n−1
∏
j=1
∆ j (ksk ′ ) m j−m j+1
dk
q m ∈ P m (S) and q m > 0 on the cone of positive definite matrices
Conclude: Each P m (S) is nontrivial
q m is the zonal polynomial of weight m

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 17/2
In multivariate statistical analysis, we choose the zonal
polynomials Z m to be such that
(tr s) d = ∑
Z m (s)
|m|=d
∫
Z m (s) = Z m (I n )
O(n)
∆(ksk ′ ) m n
n−1
∏
j=1
∆ j (ksk ′ ) m j−m j+1
dk
This requires that we be able to calculate Z m (I n )
James (1964), Muirhead (1982), Gross and D.R. (1987), etc.

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 18/2
Theorem: Z m is an eigenfunction of the Laplace-Beltrami
operator and, more generally, all G-invariant differential
operators on S
D s : A G-invariant differential operator
D s = D x′ sx, s ∈ S, x ∈ G
If s = (s ij ) ∈ S, set
(
∂ s = 1
2 (1 + δ ij) ∂ )
∂s ij
Examples of invariant differential operators
tr (s∂ s ) k , k = 1, 2, . . . ; ∆(s∂ s )

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 19/2
tr (s∂ s ) 2 is the Laplace-Beltrami operator
It can be shown that
q m (s) = ∆(s) m n
n−1
∏
j=1
∆ j (s) m j−m j+1
is an eigenfunction of every invariant differential operator D s
Since D s is G-invariant then D s = D ksk
′
D.R. (SIAM J. Math. Analysis, 1985): Applications of invariant
differential operators ...

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 20/2
D s Z m (s) ∝ D s
∫
=
=
∝
∫
∫
∫
O(n)
O(n)
O(n)
= Z m (s)
O(n)
q m (ksk ′ ) dk
D s q m (ksk ′ ) dk
D ksk ′q m (ksk ′ ) dk
q m (ksk ′ ) dk

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 21/2
Theorem: For s, t ∈ S,
∫
Z m (ksk ′ t) dk = Z m(s) Z m (t)
Z m (I n )
O(n)
Denote the LHS by f(s)
Check that f is an eigenfunction of every invariant D s
Therefore f(s) ∝ Z m (s)
Evaluate f at the identity matrix
We can also evaluate Laplace transforms of Z m and q m

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 22/2
The invariant measure on P (n, R): d ∗ s = ∆(s) −(p+1)/2 ds
For Re(α) > (n − 1)/2,
∫
e −tr r ∆(r) α q m (r)d ∗ r = [α] m Γ n (α)
P
where (α) k = α(α + 1) · · · (α + k − 1) = Γ(α + k)/Γ(α)
n∏ (
[α] m = α −
1
2 (j − 1)) m j
j=1
Γ n (α) = π n(n−1)/4
n ∏
j=1
Γ ( α − 1 2
(j − 1))
Proof: Apply the standard bitriangular structure (a.k.a. Bartlett
decomposition), etc.

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 23/2
For z ∈ P (n, R), s ∈ S,
∫
e −tr rz ∆(r) α Z m (rs)d ∗ r = [α] m Γ n (α)∆(z) −α Z m (sz −1 )
P
Denote this integral by F (s, z)
We already know F (I n , I n )
Check that F (s, I n ) is O(n) invariant and is in P m (S)
Therefore F (s, I n ) ∝ Z m (s)
Use a change of variables to show that
F (s, z) = ∆(z) −α F (z −1/2 sz −1/2 , I n )

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 24/2
For z ∈ S,
∫
0

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 25/2
Hypergeometric functions of matrix argument
Recall:
[α] m =
n∏
j=1
(
α −
1
2 (j − 1)) m j
The generalized hypergeometric function with argument s ∈ S:
pF q (α 1 , . . . , α p ; β 1 , . . . , β q ; s) =
When do these series converge
∞∑ ∑
d=0 |m|=d
[α 1 ] m · · · [α p ] m Z m (s)
[β 1 ] m · · · [β q ] m d!
Reinhardt domain of convergence: Gross and D.R. (1987),
Faraut and Korányi (1994)

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 26/2
Example 1: The series for 0 F 0 converges everywhere on S
0F 0 (s) =
∞∑
= ∑ d=0
∑
d=0 |m|=d
1
d!
∑
|m|=d
= ∑ 1
(tr s)d
d!
d=0
= exp(tr s)
Z m (s)
d!
Z m (s)

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 27/2
Example 2: The series for 1 F 0 converges for ||s|| < 1
∆(I n − s) −α = 1
Γ n (α)
= 1
Γ n (α)
= 1
Γ n (α)
= ∑ d=0
1
d!
∫
∫
P
P
∑
d=0
= 1 F 0 (α; s)
∑
e −tr (I n−s)r ∆(r) α d ∗ r
e −tr r 0F 0 (rs)∆(r) α d ∗ r
1
d!
|m|=d
∑
|m|=d
∫
P
[α] m Z m (s)
d!
e −tr r ∆(r) α Z m (rs) d ∗ r

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 28/2
Laplace and beta integrals
For p ≤ q and Re(α p+1 ) > (n − 1)/2,
1
Γ n (α p+1 )
∫
P
e −tr rz pF q (α 1 , . . . , α p ; β 1 , . . . , β q ; r)∆(r) α p+1
d ∗ r
= ∆(z) −α p+1
p+1 F q (α 1 , . . . , α p , a p+1 ; β 1 , . . . , β q ; z −1 )
Note: If p = q then the RHS converges for ||z −1 || < 1
∫
0

Zonal Polynomials and Hypergeometric Functions of Matrix Argument – p. 29/2
A few references
Herz (1955)
Constantine (1962)
James (1964)
Maass (1971)
Muirhead (1982)
Gross and D.R. (1987)
D.R. (Editor). Contemp. Math., Vol. 138, 1992
Faraut and Korányi (1994)