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 ∞ · z n � a 1 , . . . , a s � ( a 1 ) n · · · ( a s ) n � s F t : z := n ! , b 1 , . . . , b t ( b 1 ) n · · · ( b t ) n n = 0

Hypergeometric function and Pochhammer symbol ∞ · z n � a 1 , . . . , a s � ( a 1 ) n · · · ( a s ) n � s F t : z := n ! , b 1 , . . . , b t ( b 1 ) n · · · ( b t ) n n = 0 where ( a ) k = a ( a + 1 ) · · · ( a + k − 1 ) .

Hypergeometric function and Pochhammer symbol ∞ · z n � a 1 , . . . , a s � ( a 1 ) n · · · ( a s ) n � s F t : z := n ! , b 1 , . . . , b t ( b 1 ) n · · · ( b t ) n n = 0 where ( a ) k = a ( a + 1 ) · · · ( a + k − 1 ) . Examples � � a , b ◮ 2 F 1 c : z is the Gaussian hypergeometric function s. t. z ( 1 − z ) d 2 w d z 2 + ( c − ( a + b + 1 ) z ) d w d z − abw = 0 .

Hypergeometric function and Pochhammer symbol ∞ · z n � a 1 , . . . , a s � ( a 1 ) n · · · ( a s ) n � s F t : z := n ! , b 1 , . . . , b t ( b 1 ) n · · · ( b t ) n n = 0 where ( a ) k = a ( a + 1 ) · · · ( a + k − 1 ) . Examples � � a , b ◮ 2 F 1 c : z is the Gaussian hypergeometric function s. t. z ( 1 − z ) d 2 w d z 2 + ( c − ( a + b + 1 ) z ) d w d z − abw = 0 . � � 1 , 1 ◮ log ( 1 + z ) = z 2 F 1 2 : − z

Hypergeometric function and Pochhammer symbol ∞ · z n � a 1 , . . . , a s � ( a 1 ) n · · · ( a s ) n � s F t : z := n ! , b 1 , . . . , b t ( b 1 ) n · · · ( b t ) n n = 0 where ( a ) k = a ( a + 1 ) · · · ( a + k − 1 ) . Examples � � a , b ◮ 2 F 1 c : z is the Gaussian hypergeometric function s. t. z ( 1 − z ) d 2 w d z 2 + ( c − ( a + b + 1 ) z ) d w d z − abw = 0 . � � 1 , 1 ◮ log ( 1 + z ) = z 2 F 1 2 : − z ◮ e z = 0 F 0 ( : z )

Hypergeometric function with matrix argument

Hypergeometric function with matrix argument Given an m × m symmetric (postive definite) matrix Y , ∞ ( a 1 ) p · · · ( a s ) p � a 1 , . . . , a s � · C p ( Y ) � � : Y := , s F t b 1 , . . . , b t ( b 1 ) p · · · ( b t ) p n ! n = 0 p ∈P n

Hypergeometric function with matrix argument Given an m × m symmetric (postive definite) matrix Y , ∞ ( a 1 ) p · · · ( a s ) p � a 1 , . . . , a s � · C p ( Y ) � � : Y := , s F t b 1 , . . . , b t ( b 1 ) p · · · ( b t ) p n ! n = 0 p ∈P n where, ◮ P n is the set of all partitions of n

Hypergeometric function with matrix argument Given an m × m symmetric (postive definite) matrix Y , ∞ ( a 1 ) p · · · ( a s ) p � a 1 , . . . , a s � · C p ( Y ) � � : Y := , s F t b 1 , . . . , b t ( b 1 ) p · · · ( b t ) p n ! n = 0 p ∈P n where, ◮ P n is the set of all partitions of n and a partition of n is a ( p 1 , . . . , p l ) ∈ N l such that p 1 ≥ · · · ≥ p l > 0 and p 1 + · · · + p l = n (= | p | ) ,

Hypergeometric function with matrix argument Given an m × m symmetric (postive definite) matrix Y , ∞ ( a 1 ) p · · · ( a s ) p � a 1 , . . . , a s � · C p ( Y ) � � : Y := , s F t b 1 , . . . , b t ( b 1 ) p · · · ( b t ) p n ! n = 0 p ∈P n where, ◮ P n is the set of all partitions of n and a partition of n is a ( p 1 , . . . , p l ) ∈ N l such that p 1 ≥ · · · ≥ p l > 0 and p 1 + · · · + p l = n (= | p | ) , e.g., ( 5 , 2 , 2 , 1 ) ∈ P 10 ;

Hypergeometric function with matrix argument Given an m × m symmetric (postive definite) matrix Y , ∞ ( a 1 ) p · · · ( a s ) p � a 1 , . . . , a s � · C p ( Y ) � � : Y := , s F t b 1 , . . . , b t ( b 1 ) p · · · ( b t ) p n ! n = 0 p ∈P n where, ◮ P n is the set of all partitions of n and a partition of n is a ( p 1 , . . . , p l ) ∈ N l such that p 1 ≥ · · · ≥ p l > 0 and p 1 + · · · + p l = n (= | p | ) , e.g., ( 5 , 2 , 2 , 1 ) ∈ P 10 ; l a − i − 1 ◮ for p = ( p 1 , . . . , p l ) ∈ P n , ( a ) p = � � � p i ; 2 i = 1

Hypergeometric function with matrix argument Given an m × m symmetric (postive definite) matrix Y , ∞ ( a 1 ) p · · · ( a s ) p � a 1 , . . . , a s � · C p ( Y ) � � : Y := , s F t b 1 , . . . , b t ( b 1 ) p · · · ( b t ) p n ! n = 0 p ∈P n where, ◮ P n is the set of all partitions of n and a partition of n is a ( p 1 , . . . , p l ) ∈ N l such that p 1 ≥ · · · ≥ p l > 0 and p 1 + · · · + p l = n (= | p | ) , e.g., ( 5 , 2 , 2 , 1 ) ∈ P 10 ; l a − i − 1 ◮ for p = ( p 1 , . . . , p l ) ∈ P n , ( a ) p = � � � p i ; 2 i = 1 ◮ C p ( Y ) is ( C -normalization of ) zonal polynomial, which is homogeneous, symmetric, polynomial of degree n = | p | , in the eigenvalues of Y .

Zonal Polynomial C p ( Y ) C p ( Y )

Zonal Polynomial C p ( Y ) C p ( Y ) ◮ It is defined on eigenvalues of Y

Zonal Polynomial C p ( Y ) C p ( Y ) ◮ It is defined on eigenvalues of Y C p ( y 1 , . . . , y m )

Zonal Polynomial C p ( Y ) C p ( Y ) ◮ It is defined on eigenvalues of Y C p ( y 1 , . . . , y m ) ◮ For p = ( p 1 , . . . , p l ) , if m < l , (will see why later) C p ( Y ) = C p ( y 1 , . . . , y m , 0 , . . . , 0 )

Zonal Polynomial C p ( Y ) C p ( Y ) ◮ It is defined on eigenvalues of Y C p ( y 1 , . . . , y m ) ◮ For p = ( p 1 , . . . , p l ) , if m < l , (will see why later) C p ( Y ) = C p ( y 1 , . . . , y m , 0 , . . . , 0 ) ◮ An important fact C p ( Y ) = ( tr Y ) n = ( y 1 + · · · + y m ) n . � p ∈P n

Zonal Polynomial C p ( Y ) ∞ ∞ ( tr Y ) n ·C p ( Y ) � � � � � = e tr Y 0 F 0 : Y = = n ! n ! n = 0 p ∈P n n = 0

Zonal Polynomial C p ( Y ) ∞ ∞ ( tr Y ) n ·C p ( Y ) � � � � � = e tr Y 0 F 0 : Y = = n ! n ! n = 0 p ∈P n n = 0 e z = 0 F 0 ( : z )

Zonal Polynomial C p ( Y ) ∞ ∞ ( tr Y ) n ·C p ( Y ) � � � � � = e tr Y 0 F 0 : Y = = n ! n ! n = 0 p ∈P n n = 0 e z = 0 F 0 ( : z ) � a : z � = ( 1 − z ) − a 1 F 0 � a : Y � = det( I − A ) − a 1 F 0

Definition 1

Definition 1 � ( 2 p i − 2 p j − i + j ) · 2 n n ! i < j C p ( Y ) = d p Y p ( Y ) , where d p = ( 2 n )! . l � ( 2 p i + l − i )! i = 1

Definition 1 � ( 2 p i − 2 p j − i + j ) · 2 n n ! i < j C p ( Y ) = d p Y p ( Y ) , where d p = ( 2 n )! . l � ( 2 p i + l − i )! i = 1 ◮ Define a linear space V n := { f : f is homogeneous, symmetric, of degree n , or f ≡ 0 }

Definition 1 � ( 2 p i − 2 p j − i + j ) · 2 n n ! i < j C p ( Y ) = d p Y p ( Y ) , where d p = ( 2 n )! . l � ( 2 p i + l − i )! i = 1 ◮ Define a linear space V n := { f : f is homogeneous, symmetric, of degree n , or f ≡ 0 } where f is defined on eigenvalues of matrices. ◮ Basis for V n :

Definition 1 � ( 2 p i − 2 p j − i + j ) · 2 n n ! i < j C p ( Y ) = d p Y p ( Y ) , where d p = ( 2 n )! . l � ( 2 p i + l − i )! i = 1 ◮ Define a linear space V n := { f : f is homogeneous, symmetric, of degree n , or f ≡ 0 } where f is defined on eigenvalues of matrices. ◮ Basis for V n : define the elementary symmetric polynomial � u r ( x 1 , . . . , x m ) := x i 1 · · · x i r . i 1 < ··· < i r Then, for p = ( p 1 , . . . , p l ) ∈ P n · · · u p l − 1 − p l u p l ( − 0 ) U p := u p 1 − p 2 u p 2 − p 3 , 1 2 l − 1 l

Definition 1 � ( 2 p i − 2 p j − i + j ) · 2 n n ! i < j C p ( Y ) = d p Y p ( Y ) , where d p = ( 2 n )! . l � ( 2 p i + l − i )! i = 1 ◮ Define a linear space V n := { f : f is homogeneous, symmetric, of degree n , or f ≡ 0 } where f is defined on eigenvalues of matrices. ◮ Basis for V n : define the elementary symmetric polynomial � u r ( x 1 , . . . , x m ) := x i 1 · · · x i r . i 1 < ··· < i r Then, for p = ( p 1 , . . . , p l ) ∈ P n · · · u p l − 1 − p l u p l ( − 0 ) U p := u p 1 − p 2 u p 2 − p 3 , 1 2 l − 1 l ◮ deg U p =

Definition 1 � ( 2 p i − 2 p j − i + j ) · 2 n n ! i < j C p ( Y ) = d p Y p ( Y ) , where d p = ( 2 n )! . l � ( 2 p i + l − i )! i = 1 ◮ Define a linear space V n := { f : f is homogeneous, symmetric, of degree n , or f ≡ 0 } where f is defined on eigenvalues of matrices. ◮ Basis for V n : define the elementary symmetric polynomial � u r ( x 1 , . . . , x m ) := x i 1 · · · x i r . i 1 < ··· < i r Then, for p = ( p 1 , . . . , p l ) ∈ P n · · · u p l − 1 − p l u p l ( − 0 ) U p := u p 1 − p 2 u p 2 − p 3 , 1 2 l − 1 l ◮ deg U p = p 1 − p 2 + 2 ( p 2 − p 3 ) + · · · + lp l

Definition 1 � ( 2 p i − 2 p j − i + j ) · 2 n n ! i < j C p ( Y ) = d p Y p ( Y ) , where d p = ( 2 n )! . l � ( 2 p i + l − i )! i = 1 ◮ Define a linear space V n := { f : f is homogeneous, symmetric, of degree n , or f ≡ 0 } where f is defined on eigenvalues of matrices. ◮ Basis for V n : define the elementary symmetric polynomial � u r ( x 1 , . . . , x m ) := x i 1 · · · x i r . i 1 < ··· < i r Then, for p = ( p 1 , . . . , p l ) ∈ P n · · · u p l − 1 − p l u p l ( − 0 ) U p := u p 1 − p 2 u p 2 − p 3 , 1 2 l − 1 l ◮ deg U p = p 1 − p 2 + 2 ( p 2 − p 3 ) + · · · + lp l = p 1 + · · · + p l = n .