Zonal Polynomials and Hypergeometric Functions of Matrix Argument. Zonal Polynomials and Hypergeometric Functions of Matrix Argument p.

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

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

3 Zonal Polynomials and Hypergeometric Functions of Matrix Argument p. 3/2 Notation: For x = (x ij ) G, (x) := det(x) j (x) = x 11 x 1j.., j = 1,..., n x j1 x jj 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

4 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

5 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)

6 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)

7 Zonal Polynomials and Hypergeometric Functions of Matrix Argument p. 7/2 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 α1 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

8 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

9 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)

10 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

11 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

12 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 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

13 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)

14 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

15 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

16 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

17 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.

18 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 )

19 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...

20 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

21 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

22 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.

23 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 )

24 Zonal Polynomials and Hypergeometric Functions of Matrix Argument p. 24/2 For z S, 0<r<I n (r) α (I n r) β α 1 2 (n+1) Z m (rz) d r = Γ n(α)γ n (β α) Γ n (β) [α] m [β] m Z m (z) As in the classical setting, apply the convolution formula for the Laplace transform

25 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)

26 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)

27 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

28 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<r<I n (r) α p+1 (I n r) β q+1 α p (n+1) p F q (α 1,..., α p ; β 1,..., β q ; rs) d r = Γ n(α p+1 )Γ n (β q+1 α p+1 ) Γ n (β q+1 ) p+1f q+1 (α 1,..., α p+1 ; β 1,..., β q+1 ; s)

29 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)