Laplace Expansion of Schur Functions Helen Riedtmann University of - PowerPoint PPT Presentation

Laplace Expansion of Schur Functions Helen Riedtmann University of Zurich March 29th, 2017 S´ eminaire Lotharingien de Combinatoire 1 / 22

Outline Background and Notation 1 Sequences and Partitions Schur Functions Laplace Expansion 2 Concatenation of Partitions Two Concatenation Identities for Schur Functions Visual Interpretation of Concatenation 3 Application 4 2 / 22

Sequences A sequence is a finite list of elements. length subsequence (not necessarily consecutive) addition (componentwise) union 3 / 22

Sequences A sequence is a finite list of elements. length S = (5 , 3) subsequence (not T = (4 , 4 , 0) necessarily consecutive) addition (componentwise) S ∪ T = (5 , 3 , 4 , 4 , 0) union 3 / 22

Two ∆-Functions Let X = ( x 1 , . . . , x n ) and Y = ( y 1 , . . . , y m ) be sequences. � ∆( X ) = ( x i − x j ) 1 ≤ i < j ≤ n � � ∆( X ; Y ) = ( x i − y j ) 1 ≤ i ≤ n 1 ≤ j ≤ m 4 / 22

Partitions A partition is a non-increasing sequence λ = ( λ 1 , . . . , λ n ) of non-negative integers. The length of a partition is the number of its positive parts. We freely think of partitions as Young diagrams. 5 / 22

Partitions A partition is a non-increasing sequence λ = ( λ 1 , . . . , λ n ) of non-negative integers. The length of a partition is the number of its positive parts. We freely think of partitions as Young diagrams. ρ n = ( n − 1 , . . . , 1 , 0) � m n � = ( m , . . . , m ) � �� � n 5 / 22

Schur Functions Definition Let X be a set of variables of length n and λ a partition. If l ( λ ) > n , then s λ ( X ) = 0; otherwise, � � x λ j + n − j det i 1 ≤ i , j ≤ n s λ ( X ) = . ∆( X ) The Schur function s λ ( X ) is a symmetric homogeneous polynomial of degree | λ | . 6 / 22

Laplace Expansion of Matrices   a 11 a 12 a 13 a 14 a 15 a 21 a 22 a 23 a 24 a 25     a 31 a 32 a 33 a 34 a 35     a 41 a 42 a 43 a 44 a 45   a 51 a 52 a 53 a 54 a 55 7 / 22

Laplace Expansion of Matrices   a 11 a 12 a 13 a 14 a 15 a 21 a 22 a 23 a 24 a 25     a 31 a 32 a 33 a 34 a 35     a 41 a 42 a 43 a 44 a 45   a 51 a 52 a 53 a 54 a 55 7 / 22

Laplace Expansion of Matrices   a 11 a 12 a 13 a 14 a 15 a 21 a 22 a 23 a 24 a 25     a 31 a 32 a 33 a 34 a 35     a 41 a 42 a 43 a 44 a 45   a 51 a 52 a 53 a 54 a 55 7 / 22

Laplace Expansion of Matrices   a 11 a 12 a 13 a 14 a 15 a 21 a 22 a 23 a 24 a 25     a 31 a 32 a 33 a 34 a 35     a 41 a 42 a 43 a 44 a 45   a 51 a 52 a 53 a 54 a 55 7 / 22

Laplace Expansion of Matrices (formal statement) Let A be an n × n matrix. For any subsequence K ⊂ [ n ], � � � 1 det( A ) = ε ( sort ( K , J )) det ( A KJ ) det A [ n ] \ K [ n ] \ J J ⊂ [ n ]: l ( J )= l ( K ) 8 / 22

Laplace Expansion of Matrices (formal statement) Let A be an n × n matrix. For any subsequence K ⊂ [ n ], � � � 1 det( A ) = ε ( sort ( K , J )) det ( A KJ ) det A [ n ] \ K [ n ] \ J J ⊂ [ n ]: l ( J )= l ( K ) � � � 2 det( A ) = ε ( sort ( I , K )) det ( A IK ) det A [ n ] \ I [ n ] \ K I ⊂ [ n ]: l ( I )= l ( K ) 8 / 22

Laplace Expansion of Schur Functions (Dehaye ’12) (1/2)   x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5 1 1 1 1 1 x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5    2 2 2 2 2  x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5     3 3 3 3 3   x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5   4 4 4 4 4 x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5 5 5 5 5 5 9 / 22

Laplace Expansion of Schur Functions (Dehaye ’12) (1/2)   x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5 1 1 1 1 1 x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5    2 2 2 2 2  x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5     3 3 3 3 3   x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5   4 4 4 4 4 x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5 5 5 5 5 5 9 / 22

Laplace Expansion of Schur Functions (Dehaye ’12) (1/2)   x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5 1 1 1 1 1 x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5    2 2 2 2 2  x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5     3 3 3 3 3   x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5   4 4 4 4 4 x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5 5 5 5 5 5 sort λ + ρ 5 = ( µ + ρ 3 ) ∪ ( ν + ρ 2 ) 9 / 22

Laplace Expansion of Schur Functions (Dehaye ’12) (1/2)   x µ 1 +3 − 1 x ν 1 +2 − 1 x µ 2 +3 − 2 x µ 3 +3 − 3 x ν 2 +2 − 2 1 1 1 1 1 x µ 1 +3 − 1 x µ 2 +3 − 2 x µ 3 +3 − 3 x ν 1 +2 − 1 x ν 2 +2 − 2   2 2 2 2 2   x µ 1 +3 − 1 x ν 1 +2 − 1 x µ 2 +3 − 2 x µ 3 +3 − 3 x ν 2 +2 − 2    3 3 3 3 3    x µ 1 +3 − 1 x ν 1 +2 − 1 x µ 2 +3 − 2 x µ 3 +3 − 3 x ν 2 +2 − 2   4 4 4 4 4 x µ 1 +3 − 1 x µ 2 +3 − 2 x µ 3 +3 − 3 x ν 1 +2 − 1 x ν 2 +2 − 2 5 5 5 5 5 sort λ + ρ 5 = ( µ + ρ 3 ) ∪ ( ν + ρ 2 ) 9 / 22

Laplace Expansion of Schur Functions (Dehaye ’12) (1/2)   x µ 1 +3 − 1 x ν 1 +2 − 1 x µ 2 +3 − 2 x µ 3 +3 − 3 x ν 2 +2 − 2 1 1 1 1 1 x µ 1 +3 − 1 x µ 2 +3 − 2 x µ 3 +3 − 3 x ν 1 +2 − 1 x ν 2 +2 − 2   2 2 2 2 2   x µ 1 +3 − 1 x ν 1 +2 − 1 x µ 2 +3 − 2 x µ 3 +3 − 3 x ν 2 +2 − 2    3 3 3 3 3    x µ 1 +3 − 1 x ν 1 +2 − 1 x µ 2 +3 − 2 x µ 3 +3 − 3 x ν 2 +2 − 2   4 4 4 4 4 x µ 1 +3 − 1 x µ 2 +3 − 2 x µ 3 +3 − 3 x ν 1 +2 − 1 x ν 2 +2 − 2 5 5 5 5 5 ε ( sort ) s µ ( S ) s ν ( T )∆( S )∆( T ) � s λ ( X ) = ∆( X ) S , T ⊂X : S∪ 3 , 2 T sort = X 9 / 22

Laplace Expansion of Schur Functions (Dehaye ’12) (2/2) sort λ + ρ 5 = ( µ + ρ 3 ) ∪ ( ν + ρ 2 ) ε ( sort ) s µ ( S ) s ν ( T )∆( S )∆( T ) � s λ ( X ) = ∆( X ) S , T ⊂X : S∪ 3 , 2 T sort = X ε ( sort ) s µ ( S ) s ν ( T ) � = ∆( S ; T ) S , T ⊂X : S∪ 3 , 2 T sort = X 10 / 22

Concatenation of Partitions Definition Let µ and ν be two partitions of length at most m and n , respectively. The ( m , n )-concatenation of µ and ν , denoted µ ⋆ m , n ν , is the partition that satisfies sort µ ⋆ m , n ν + ρ m + n = ( µ + ρ m ) ∪ ( ν + ρ n ) if it exists; otherwise, we set µ ⋆ m , n ν = ∞ . Here, ∞ is just a symbol with the property that s ∞ ( X ) = 0 for any set of variables X . The sign of the concatenation is given by ε ( µ, ν ) = ε ( sort ) . 11 / 22

Concatenation of Partitions Definition Let µ and ν be two partitions of length at most m and n , respectively. The ( m , n )-concatenation of µ and ν , denoted µ ⋆ m , n ν , is the partition that satisfies sort µ ⋆ m , n ν + ρ m + n = ( µ + ρ m ) ∪ ( ν + ρ n ) if it exists; otherwise, we set µ ⋆ m , n ν = ∞ . Here, ∞ is just a symbol with the property that s ∞ ( X ) = 0 for any set of variables X . The sign of the concatenation is given by ε ( µ, ν ) = ε ( sort ) . (5 , 1) ⋆ 2 , 4 (3 , 3) = ∞ 11 / 22

Concatenation of Partitions Definition Let µ and ν be two partitions of length at most m and n , respectively. The ( m , n )-concatenation of µ and ν , denoted µ ⋆ m , n ν , is the partition that satisfies sort µ ⋆ m , n ν + ρ m + n = ( µ + ρ m ) ∪ ( ν + ρ n ) if it exists; otherwise, we set µ ⋆ m , n ν = ∞ . Here, ∞ is just a symbol with the property that s ∞ ( X ) = 0 for any set of variables X . The sign of the concatenation is given by ε ( µ, ν ) = ε ( sort ) . (5 , 1) ⋆ 3 , 2 (3 , 3) = (7 , 4 , 3 , 2 , 0) − ρ 5 = (3 , 1 , 1 , 1 , 0) 11 / 22

First Concatenation Identity for Schur Functions Lemma (Dehaye ’12) Let the set X consist of m + n variables. For any pair of partitions µ and ν with at most m and n parts, respectively, it holds that ε ( µ, ν ) s µ ( S ) s ν ( T ) � s µ⋆ m , n ν ( X ) = . ∆( S ; T ) S , T ⊂X : S∪ m , n T sort = X 12 / 22

Laplace Expansion of Schur Functions (Transposed)   x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5 1 1 1 1 1 x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5    2 2 2 2 2  x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5    3 3 3 3 3    x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5   4 4 4 4 4 x λ 1 +5 − 1 x λ 2 +5 − 2 x λ 3 +5 − 3 x λ 4 +5 − 4 x λ 5 +5 − 5 5 5 5 5 5 13 / 22