An Introduction to the Combinatorics of Symmetric Functions Peter - PowerPoint PPT Presentation

An Introduction to the Combinatorics of Symmetric Functions Peter McNamara Pomona College 4 February 2005 Slides and papers available from www.lacim.uqam.ca/ ∼ mcnamara Combinatorics of Symmetric Functions Peter McNamara 1

What is algebraic combinatorics anyhow? The biggest open problem in combinatorics: Combinatorics of Symmetric Functions Peter McNamara 2

What is algebraic combinatorics anyhow? The biggest open problem in combinatorics: Define combinatorics Combinatorics of Symmetric Functions Peter McNamara 2

What is algebraic combinatorics anyhow? The biggest open problem in combinatorics: Define combinatorics The biggest open problem in algebraic combinatorics: Define algebraic combinatorics Combinatorics that takes its problems, or its tools, from commutative algebra, algebraic geometry, algebraic topology, representation theory, etc. Combinatorics of Symmetric Functions Peter McNamara 2

Outline ◮ Symmetric functions ◮ Schur functions and Littlewood-Richardson coefficients ◮ The Littlewood-Richardson rule ◮ Cylindric skew Schur functions Combinatorics of Symmetric Functions Peter McNamara 3

What are symmetric functions? Definition A symmetric polynomial is a polynomial that is invariant under any permutation of its variables x 1 , x 2 , . . . x n . Examples ◮ x 1 + x 2 + · · · + x n ◮ x 2 1 x 2 + x 2 1 x 3 + x 2 2 x 1 + x 2 2 x 3 + x 2 3 x 1 + x 2 3 x 2 is a symmetric polynomial in x 1 , x 2 , x 3 . Definition A symmetric function is a formal power series that is invariant under any permutation of its (infinite set of) variables x = ( x 1 , x 2 , . . . ) . Examples ◮ � i ≥ 1 x i is a symmetric function, as is � i � = j x 2 i x j . ◮ � i < j x 2 i x j is not symmetric. Combinatorics of Symmetric Functions Peter McNamara 4

A basis for the symmetric functions Fact: The symmetric functions form a vector space. What is a possible basis? Monomial symmetric functions: Start with a monomial: x 7 1 x 4 2 Combinatorics of Symmetric Functions Peter McNamara 5

A basis for the symmetric functions Fact: The symmetric functions form a vector space. What is a possible basis? Monomial symmetric functions: Start with a monomial: x 7 1 x 4 2 + x 4 1 x 7 2 + x 7 1 x 4 3 + x 4 1 x 7 3 + · · · . Given a partition λ = ( λ 1 , . . . , λ ℓ ) , e.g. λ = ( 7 , 4 ) , � x λ 1 i 1 . . . x λ ℓ m λ = i ℓ . i 1 ,..., i ℓ distinct Examples ◮ m ( 3 ) = x 3 1 + x 3 2 + · · · . ◮ m ( 1 , 1 , 1 ) ( x 1 , x 2 , x 3 ) ≡ m 111 ( x 1 , x 2 , x 3 ) = x 1 x 2 x 3 . Combinatorics of Symmetric Functions Peter McNamara 5

Other bases ◮ Elementary symmetric functions, e λ . ◮ Complete homogeneous symmetric functions, h λ . ◮ Power sum symmetric functions, p λ . Typical questions: Prove they are bases, convert from one to another, ... Combinatorics of Symmetric Functions Peter McNamara 6

Schur functions Cauchy, 1815. ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) . ◮ Young diagram. Example: λ = ( 4 , 4 , 3 , 1 ) . Combinatorics of Symmetric Functions Peter McNamara 7

Schur functions Cauchy, 1815. ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) . ◮ Young diagram. 7 Example: λ = ( 4 , 4 , 3 , 1 ) . 5 6 6 ◮ Semistandard Young tableau 4 4 4 9 (SSYT) 1 3 3 4 The Schur function s λ in the variables x = ( x 1 , x 2 , . . . ) is then defined by � x # 1’s in T x # 2’s in T s λ = · · · . 1 2 SSYT T Example s 4431 = x 1 1 x 2 3 x 4 4 x 5 x 2 6 x 7 x 9 + · · · . Combinatorics of Symmetric Functions Peter McNamara 7

Schur functions Example 2 2 3 3 3 3 3 2 1 1 1 2 1 1 1 3 2 2 2 3 1 2 1 3 Hence x 2 1 x 2 + x 1 x 2 2 + x 2 1 x 3 + x 1 x 2 3 + x 2 2 x 3 + x 2 x 2 s 21 ( x 1 , x 2 , x 3 ) = 3 + 2 x 1 x 2 x 3 = m 21 ( x 1 , x 2 , x 3 ) + 2 m 111 ( x 1 , x 2 , x 3 ) . Fact: Schur functions are symmetric functions. Question Why do we care about Schur functions? Combinatorics of Symmetric Functions Peter McNamara 8

Why do we care about Schur functions? ◮ Fact: The Schur functions form a basis for the symmetric functions. ◮ In fact, they form an orthonormal basis: � s λ , s µ � = δ λµ . ◮ Main reason: they arise in many other areas of mathematics. ◮ Representation theory of S n . ◮ Representations of GL ( n , C ) . ◮ Algebraic Geometry: Schubert Calculus. ◮ Linear Algebra: eigenvalues of Hermitian matrices. Combinatorics of Symmetric Functions Peter McNamara 9

Littlewood-Richardson coefficients Note: The symmetric functions form a ring. ( x 2 1 + x 2 2 + x 2 3 + · · · )( x 1 + x 2 + x 3 + · · · ) . � c λ s µ s ν = µν s λ . λ c λ µν : Littlewood-Richardson coefficients Combinatorics of Symmetric Functions Peter McNamara 10

Littlewood-Richardson coefficients Note: The symmetric functions form a ring. ( x 2 1 + x 2 2 + x 2 3 + · · · )( x 1 + x 2 + x 3 + · · · ) . � c λ s µ s ν = µν s λ . λ c λ µν : Littlewood-Richardson coefficients Examples ◮ s 21 s 21 = s 42 + s 411 + s 33 + 2 s 321 + s 3111 + s 222 + s 2211 . ◮ s 32 s 421 = s 44211 + s 54111 + s 4332 + s 4422 + 2 s 4431 + 2 s 5322 + 2 s 5331 + 3 s 5421 + s 52221 + s 5511 + s 62211 + s 6222 + s 43221 + 3 s 6321 + s 43311 + 2 s 6411 + 2 s 53211 + s 63111 + s 444 + 2 s 543 + s 552 + s 633 + 2 s 642 + s 732 + s 741 + s 7221 + s 7311 + s 651 . ◮ c ( 12 , 11 , 10 , 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 ) ( 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 ) , ( 8 , 7 , 6 , 6 , 5 , 4 , 3 , 2 , 1 ) = 7869992. (Maple packages: John Stembridge, Anders Buch.) Combinatorics of Symmetric Functions Peter McNamara 10

Littlewood-Richardson coefficients are non-negative � c λ s µ s ν = µν s λ . λ Theorem For any partitions µ , ν and λ , c λ µν ≥ 0 . (Your take-home fact!) Terminology: We say that s µ s ν = � λ c λ µν s λ is a Schur-positive function. Combinatorics of Symmetric Functions Peter McNamara 11

Littlewood-Richardson coefficients are non-negative � c λ s µ s ν = µν s λ . λ Theorem For any partitions µ , ν and λ , c λ µν ≥ 0 . (Your take-home fact!) Terminology: We say that s µ s ν = � λ c λ µν s λ is a Schur-positive function. Proof 1: Use representation theory of S n . Proof 2: Use representation theory of GL ( n , C ) . Proof 3: Use Schubert Calculus. Want a combinatorial proof: “They must count something simpler!” Combinatorics of Symmetric Functions Peter McNamara 11

Skew Schur functions: a generalization of Schur functions ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) . ◮ Young diagram. Example: λ = ( 4 , 4 , 3 , 1 ) Combinatorics of Symmetric Functions Peter McNamara 12

Skew Schur functions: a generalization of Schur functions ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) . ◮ µ fits inside λ . 7 ◮ Young diagram. Example: 5 6 6 λ/µ = ( 4 , 4 , 3 , 1 ) / ( 3 , 1 ) 4 4 9 4 Combinatorics of Symmetric Functions Peter McNamara 12

Skew Schur functions: a generalization of Schur functions ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) . ◮ µ fits inside λ . 7 ◮ Young diagram. Example: 5 6 6 λ/µ = ( 4 , 4 , 3 , 1 ) / ( 3 , 1 ) 4 4 9 ◮ Semistandard Young tableau 4 (SSYT) The skew Schur function s λ/µ is the variables x = ( x 1 , x 2 , . . . ) is then defined by � x # 1’s in T x # 2’s in T s λ/µ = · · · . 1 2 SSYT T s 4431 / 31 = x 3 4 x 5 x 2 6 x 7 x 9 + · · · . Again, it’s a symmetric function. Remarkable fact: Combinatorics of Symmetric Functions Peter McNamara 12

Skew Schur functions: a generalization of Schur functions ◮ Partition λ = ( λ 1 , λ 2 , . . . , λ ℓ ) . ◮ µ fits inside λ . 7 ◮ Young diagram. Example: 5 6 6 λ/µ = ( 4 , 4 , 3 , 1 ) / ( 3 , 1 ) 4 4 9 ◮ Semistandard Young tableau 4 (SSYT) The skew Schur function s λ/µ is the variables x = ( x 1 , x 2 , . . . ) is then defined by � x # 1’s in T x # 2’s in T s λ/µ = · · · . 1 2 SSYT T s 4431 / 31 = x 3 4 x 5 x 2 6 x 7 x 9 + · · · . Again, it’s a symmetric function. Remarkable fact: � c λ s λ/µ = µν s ν . ν Combinatorics of Symmetric Functions Peter McNamara 12

The Littlewood-Richardson rule Littlewood-Richardson 1934, Schützenberger 1977, Thomas 1974. Theorem c λ µν equals the number of SSYT of shape λ/µ and content ν whose reverse reading word is a ballot sequence. Example λ = ( 5 , 5 , 2 , 1 ) , µ = ( 3 , 2 ) , ν = ( 4 , 3 , 1 ) 3 3 2 1 1 1 2 1 3 2 2 2 1 2 2 1 2 2 1 1 1 1 1 1 11222113 No 11221213 Yes 11221312 Yes to prevent bottom from getting cut off Combinatorics of Symmetric Functions Peter McNamara 13

The Littlewood-Richardson rule Littlewood-Richardson 1934, Schützenberger 1977, Thomas 1974. Theorem c λ µν equals the number of SSYT of shape λ/µ and content ν whose reverse reading word is a ballot sequence. Example λ = ( 5 , 5 , 2 , 1 ) , µ = ( 3 , 2 ) , ν = ( 4 , 3 , 1 ) 3 3 2 1 1 1 2 1 3 2 2 2 1 2 2 1 2 2 1 1 1 1 1 1 11222113 No 11221213 Yes 11221312 Yes to prevent bottom from getting cut off c 5221 32 , 431 = 2. Combinatorics of Symmetric Functions Peter McNamara 13