Generalized Cauchy determinant and Schur Pfaffian, and Their - PowerPoint PPT Presentation

Generalized Cauchy determinant and Schur Pfaffian, and Their Applications Soichi OKADA (Nagoya University) Lattice Models: Exact Methods and Combinatorics Firenze, May 21, 2015

Cauchy determinants ∏ 1 ≤ i<j ≤ n ( x j − x i ) ∏ ( ) 1 ≤ i<j ≤ n ( y j − y i ) 1 ∏ n det = , 1 − x i y j i, j =1 (1 − x i y j ) 1 ≤ i, j ≤ n ∏ 1 ≤ i<j ≤ n ( x j − x i ) ∏ ( ) 1 ≤ i<j ≤ n ( y j − y i ) 1 ∏ n . det = x i + y j i, j =1 ( x i + y j ) 1 ≤ i, j ≤ n Schur Pfaffians ( x j − x i ) ∏ x j − x i Pf = , x j + x i x j + x i 1 ≤ i, j ≤ n 1 ≤ i<j ≤ n ( x j − x i ) ∏ x j − x i Pf = . 1 − x i x j 1 − x i x j 1 ≤ i, j ≤ n 1 ≤ i<j ≤ n

A generalization of Cauchy determinant ( a i − b j ) det x i − y j 1 ≤ i, j ≤ n   1 · · · x n − 1 1 · · · a 1 x n − 1 1 x 1 x 2 a 1 a 1 x 1 a 1 x 2 1 1   . . . . . . . . . . . . . . . .   . . . . . . . .     1 x n x 2 n · · · x n − 1 a n a n x n a n x 2 n · · · a n x n − 1 ( − 1) n ( n − 1) / 2   n n ∏ n = i, j =1 ( x i − y j ) det .   1 · · · y n − 1 1 · · · b 1 y n − 1 1 y 1 y 2 b 1 b 1 y 1 b 1 y 2    1 1  . . . . . . . .   . . . . . . . . . . . . . . . .   1 y n y 2 n · · · y n − 1 b n b n y n b n y 2 n · · · b n y n − 1 n n If we replace x i by x 2 y i by y 2 i , i , a i by x i , b i by y i , or x i by x i , y i by − y i , a i by 1 , b i by 0 , then this generalization reduces to the original Cauchy determinant.

A generalization of Cauchy determinant ( a i − b j ) det x i − y j 1 ≤ i, j ≤ n   1 x 1 x 2 1 · · · x n − 1 a 1 a 1 x 1 a 1 x 2 1 · · · a 1 x n − 1 1 1   . . . . . . . .  . . . . . . . .  . . . . . . . .     1 x n x 2 n · · · x n − 1 a n a n x n a n x 2 n · · · a n x n − 1 ( − 1) n ( n − 1) / 2   n n ∏ n = i, j =1 ( x i − y j ) det .   1 y 1 y 2 1 · · · y n − 1 b 1 b 1 y 1 b 1 y 2 1 · · · b 1 y n − 1   1 1   . . . . . . . .   . . . . . . . . . . . . . . . .   1 y n y 2 n · · · y n − 1 b n b n y n b n y 2 n · · · b n y n − 1 n n By replacing x i by x 6 y i by y 6 a i by x 2 b i by y 2 i , i , i , i , this generalization can be used to evaluate the Izergin–Korepin determi- nant in the enumeration problem of alternating sign matrices.

Plan • Cauchy determinant and Cauchy formula for Schur functions • A generalization of Cauchy determinant and restricted Cauchy formula • Schur Pfaffian and Littlewood formula for Schur functions • A generalization of Schur Pfaffian and restricted Littlewood formulae • Application of generalized Schur Pfaffian to Schur’s P -functions

Cauchy Determinant and Cauchy Formula for Schur Functions

Partitions and Schur functions A partition is a weakly decreasing sequence of nonnegative integers λ = ( λ 1 , λ 2 , λ 3 , . . . ) , λ 1 ≥ λ 2 ≥ λ 3 ≥ · · · ≥ 0 with finitely many nonzero entries. We put ∑ | λ | = λ i , l ( λ ) = # { i : λ i > 0 } . i ≥ 1 Let n be a positive integer and x = ( x 1 , · · · , x n ) be a sequence of n indeterminates. For a partition λ of length ≤ n , the Schur function s λ ( x 1 , · · · , x n ) corresponding to λ is defined by ( ) x λ j + n − j det i 1 ≤ i, j ≤ n ( ) s λ ( x ) = s λ ( x 1 , · · · , x n ) = . x n − j det i 1 ≤ i, j ≤ n Remark If l ( λ ) > n , then we define s λ ( x 1 , · · · , x n ) = 0 .

Cauchy formula for Schur functions Theorem For x = ( x 1 , · · · , x n ) and y = ( y 1 , · · · , y n ) , we have ∑ 1 ∏ n ∏ n s λ ( x ) s λ ( y ) = j =1 (1 − x i y j ) , i =1 λ where λ runs over all partitions. This theorem can be proved in several ways. For example, it follows from • Representation thoeretical proof (irreducible decomposition of GL n × GL n -module S ( M n ) ); • Combinatorical proof (Robinson–Schensted–Knuth correspondence) • Linear algebraic proof

Liear algebraic proof uses • Cauchy–Binet formula: For two n × N matrices X and Y , ( ) ∑ X t Y det X ( I ) · det Y ( I ) = det , I where I = { i 1 < · · · < i n } runs over all n -element subsets of column ( ) ( ) indices, and X ( I ) = x p,i q 1 ≤ p, q ≤ n , Y ( I ) = x p,i q 1 ≤ p, q ≤ n . • Cauchy determinant: ( ) 1 ∆( x )∆( y ) det = ∏ n ∏ n j =1 (1 − x i y j ) , 1 − x i y j i =1 1 ≤ i, j ≤ n where ∏ ∏ ∆( x ) = ( x j − x i ) , ∆( y ) = ( y j − y i ) . 1 ≤ i<j ≤ n 1 ≤ i<j ≤ n

Proof of the Cauchy formula First we apply the Cauchy–Binet formula (with N = ∞ ) to the matri- ces 0 1 2 3 · · · 0 1 2 3 · · ·     x 2 x 3 y 2 y 3 1 x 1 · · · 1 y 1 · · · 1 1 1 1  x 2 x 3   y 2 y 3  1 x 2 · · · 1 y 2 · · ·     2 2 2 2 X =  , Y =  . . . . . . . . .   . . . . . . . . . . . . . . . . x 2 x 3 y 2 y 3 1 x n · · · 1 y n · · · n n n n To a partitions of length ≤ n , we associate an n -element subsets of N given by I n ( λ ) = { λ 1 + n − 1 , λ 2 + n − 2 , · · · , λ n − 1 + 1 , λ n } . Then the correspondence λ �→ I n ( λ ) is a bijection and s λ ( x ) = det X ( I n ( λ )) s λ ( y ) = det Y ( I n ( λ )) , . ∆( x ) ∆( y )

By applying the Cauchy–Binet formula, we have ∑ ∑ 1 s λ ( x ) s λ ( y ) = det X ( I ) · det Y ( I ) ∆( x )∆( y ) λ I ( ) 1 X t Y = ∆( x )∆( y ) det ( ) 1 1 = ∆( x )∆( y ) det . 1 − x i y j 1 ≤ i, j ≤ n Now we can use the Cauchy determinant to obtain ∑ 1 ∆( x )∆( y ) ∏ n s λ ( x ) s λ ( y ) = ∆( x )∆( y ) · i, j =1 (1 − x i y j ) λ 1 ∏ n = i, j =1 (1 − x i y j ) .

Generalized Cauchy Determinant and Column-length Restricted Cauchy Formula

Theorem (Cauchy formula) For x = ( x 1 , · · · , x n ) and y = ( y 1 , · · · , y n ) , we have ∑ 1 ∏ n ∏ n s λ ( x ) s λ ( y ) = j =1 (1 − x i y j ) , i =1 λ where λ runs over all partitions. Problem Fix a nonnegative integer l . For x = ( x 1 , · · · , x n ) and y = ( y 1 , · · · , y n ) , find a formula for ∑ s λ ( x ) s λ ( y ) , l ( λ ) ≤ l where λ runs over all partitions of length l ( λ ) ≤ l .

Let l be a nonnegative integer. To a nonnegative integer r and two partitions α , β with length ≤ r , we associate a partition , t β 1 , t β 2 , · · · ) . Λ( r, α, β ) = ( r + α 1 , · · · , r + α r , r, · · · , r � �� � l α r r r l t β

Let l be a nonnegative integer. To a nonnegative integer r and two partitions α , β with length ≤ r , we associate a partition , t β 1 , t β 2 , · · · ) . Λ( r, α, β ) = ( r + α 1 , · · · , r + α r , r, · · · , r � �� � l We denote r by p (Λ( r, α, β )) . We put C l = the set of such partitions Λ( r, α, β ) . Let Λ �→ Λ ∗ be the involution on C l defined by Λ( r, α, β ) ∗ = Λ( r, β, α ) . Note that, if l = 0 , then C 0 = the set of all partitions , Λ ∗ = t Λ (the conjugate partition) .

Theorem (Column-length restricted Cauchy formula; King) For x = ( x 1 , . . . , x m ) and y = ( y 1 , . . . , y n ) , we have ∑ µ ∈C l ( − 1) | µ | + lp ( µ ) s µ ( x ) s µ ∗ ( y ) ∑ s λ ( x ) s λ ( y ) = ∏ m ∏ n . j =1 (1 − x i y j ) i =1 l ( λ ) ≤ l Two extreme cases: • If l ≥ min( m, n ) , then we recover the Cauchy formula: ∑ 1 ∏ m ∏ n s λ ( x ) s λ ( y ) = j =1 (1 − x i y j ) i =1 λ • If l = 0 , then we have the dual Cauchy formula: m n ∑ ∏ ∏ ( − 1) | µ | s µ ( x ) s t µ ( y ) = (1 − x i y j ) . µ i =1 j =1

Recall the bijection λ ← → I n ( λ ) = { λ 1 + n − 1 , λ 2 + n − 2 , · · · , λ n − 1 + 1 , λ n } . Then we have l ( λ ) ≤ l ⇐ ⇒ [0 , n − l − 1] ⊂ I n ( λ ) . In this case, we have   x λ l + n − l · · · x λ 1 + n − 1 1 x 1 · · · x n − l − 1 1 1 1   x λ l + n − l · · · x λ 1 + n − 1   1 x 2 · · · x n − l − 1 1   s λ ( x ) = ∆( x ) det 2 2 2 .   . . . . . . . . . . . . . . .   x λ l + n − l · · · x λ 1 + n − 1 1 x n · · · x n − l − 1 n n n

Proof of the restricted Cauchy formula We prove the formula by using • generalized Cauchy–Binet formula: ∑ det X ( { 1 , . . . , m − l } ∪ { i 1 + ( m − l ) , . . . , i l + ( m − l ) } ) I × det Y ( { 1 , . . . , n − l } ∪ { i 1 + ( n − l ) , . . . , i l + ( n − l ) } ) • generalized Cauchy determinant:   ( a i − b j ) ( ) i , · · · , x q − 1 1 , x i , x 2   i   x i − y j 1 ≤ i ≤ m  1 ≤ i ≤ m, 1 ≤ j ≤ n  det   − t ( )   j , · · · , y p − 1 1 , y j , y 2 O j 1 ≤ j ≤ n