An Introduction to Symmetric Functions Ira M. Gessel Department of - PowerPoint PPT Presentation

An Introduction to Symmetric Functions Ira M. Gessel Department of Mathematics Brandeis University Brandeis Combinatorics Seminar November 1, 2016

What are symmetric functions? Symmetric functions are not functions.

What are symmetric functions? Symmetric functions are not functions. They are formal power series in the infinitely many variables x 1 , x 2 , . . . that are invariant under permutation of the subscripts.

What are symmetric functions? Symmetric functions are not functions. They are formal power series in the infinitely many variables x 1 , x 2 , . . . that are invariant under permutation of the subscripts. In other words, if i 1 , . . . , i m are distinct positive integers and α 1 , . . . , α m are arbitrary nonnegative integers then the coefficient of x α 1 i 1 · · · x α m in a symmetric function is the same as i m the coefficient of x α 1 1 · · · x α m m .

What are symmetric functions? Symmetric functions are not functions. They are formal power series in the infinitely many variables x 1 , x 2 , . . . that are invariant under permutation of the subscripts. In other words, if i 1 , . . . , i m are distinct positive integers and α 1 , . . . , α m are arbitrary nonnegative integers then the coefficient of x α 1 i 1 · · · x α m in a symmetric function is the same as i m the coefficient of x α 1 1 · · · x α m m . Examples: ◮ x 2 1 + x 2 2 + . . . ◮ � i ≤ j x i x j

What are symmetric functions? Symmetric functions are not functions. They are formal power series in the infinitely many variables x 1 , x 2 , . . . that are invariant under permutation of the subscripts. In other words, if i 1 , . . . , i m are distinct positive integers and α 1 , . . . , α m are arbitrary nonnegative integers then the coefficient of x α 1 i 1 · · · x α m in a symmetric function is the same as i m the coefficient of x α 1 1 · · · x α m m . Examples: ◮ x 2 1 + x 2 2 + . . . ◮ � i ≤ j x i x j i ≤ j x i x 2 But not � j

What are symmetric functions good for? ◮ Some combinatorial problems have symmetric function generating functions. For example, � i < j ( 1 + x i x j ) counts graphs by the degrees of the vertices.

What are symmetric functions good for? ◮ Some combinatorial problems have symmetric function generating functions. For example, � i < j ( 1 + x i x j ) counts graphs by the degrees of the vertices. ◮ Symmetric functions are useful in counting plane partitions.

What are symmetric functions good for? ◮ Some combinatorial problems have symmetric function generating functions. For example, � i < j ( 1 + x i x j ) counts graphs by the degrees of the vertices. ◮ Symmetric functions are useful in counting plane partitions. ◮ Symmetric functions are closely related to representations of symmetric and general linear groups

What are symmetric functions good for? ◮ Some combinatorial problems have symmetric function generating functions. For example, � i < j ( 1 + x i x j ) counts graphs by the degrees of the vertices. ◮ Symmetric functions are useful in counting plane partitions. ◮ Symmetric functions are closely related to representations of symmetric and general linear groups ◮ Symmetric functions are useful in counting unlabeled graphs (Pólya theory).

The ring of symmetric functions Let Λ denote the ring of symmetric functions, and let Λ n be the vector space of symmetric functions homogeneous of degree n .

The ring of symmetric functions Let Λ denote the ring of symmetric functions, and let Λ n be the vector space of symmetric functions homogeneous of degree n . Then the dimension of Λ n is p ( n ) , the number of partitions of n .

The ring of symmetric functions Let Λ denote the ring of symmetric functions, and let Λ n be the vector space of symmetric functions homogeneous of degree n . Then the dimension of Λ n is p ( n ) , the number of partitions of n . A partition of n is a weakly decreasing sequence of positive integers λ = ( λ 1 , λ 2 , . . . , λ k ) with sum n .

The ring of symmetric functions Let Λ denote the ring of symmetric functions, and let Λ n be the vector space of symmetric functions homogeneous of degree n . Then the dimension of Λ n is p ( n ) , the number of partitions of n . A partition of n is a weakly decreasing sequence of positive integers λ = ( λ 1 , λ 2 , . . . , λ k ) with sum n . For example, the partitions of 4 are ( 4 ) , ( 3 , 1 ) , ( 2 , 2 ) , ( 2 , 1 , 1 ) , and ( 1 , 1 , 1 , 1 ) .

The ring of symmetric functions Let Λ denote the ring of symmetric functions, and let Λ n be the vector space of symmetric functions homogeneous of degree n . Then the dimension of Λ n is p ( n ) , the number of partitions of n . A partition of n is a weakly decreasing sequence of positive integers λ = ( λ 1 , λ 2 , . . . , λ k ) with sum n . For example, the partitions of 4 are ( 4 ) , ( 3 , 1 ) , ( 2 , 2 ) , ( 2 , 1 , 1 ) , and ( 1 , 1 , 1 , 1 ) . There are several important bases for Λ n , all indexed by partitions.

Monomial symmetric functions If a symmetric function has a term x 2 1 x 2 x 3 with coefficient 1, then it must contain all terms of the form x 2 i x j x k , with i , j , and k distinct, with coefficient 1. If we add up all of these terms, we get the monomial symmetric function � x 2 m ( 2 , 1 , 1 ) = i x j x k where the sum is over all distinct terms of the form x 2 i x j x k with i , j , and k distinct.

Monomial symmetric functions If a symmetric function has a term x 2 1 x 2 x 3 with coefficient 1, then it must contain all terms of the form x 2 i x j x k , with i , j , and k distinct, with coefficient 1. If we add up all of these terms, we get the monomial symmetric function � x 2 m ( 2 , 1 , 1 ) = i x j x k where the sum is over all distinct terms of the form x 2 i x j x k with i , j , and k distinct. So m ( 2 , 1 , 1 ) = x 2 1 x 2 x 3 + x 2 3 x 1 x 4 + x 2 1 x 3 x 5 + · · · .

Monomial symmetric functions If a symmetric function has a term x 2 1 x 2 x 3 with coefficient 1, then it must contain all terms of the form x 2 i x j x k , with i , j , and k distinct, with coefficient 1. If we add up all of these terms, we get the monomial symmetric function � x 2 m ( 2 , 1 , 1 ) = i x j x k where the sum is over all distinct terms of the form x 2 i x j x k with i , j , and k distinct. So m ( 2 , 1 , 1 ) = x 2 1 x 2 x 3 + x 2 3 x 1 x 4 + x 2 1 x 3 x 5 + · · · . We could write it more formally as � x 2 i x j x k . i � = j , i � = k , j < k

More generally, for any partition λ = ( λ 1 , . . . , λ k ) , m λ is the sum of all distinct monomials of the form x λ 1 i 1 · · · x λ k i k . It’s easy to see that { m λ } λ ⊢ n is a basis for Λ n .

Multiplicative bases There are three important multiplicative bases for Λ n . Suppose that for each n , u n is a symmetric function homogeneous of degree n . Then for any partition λ = ( λ 1 , . . . , λ k ) , we may define u λ to be u λ 1 · · · u λ k .

Multiplicative bases There are three important multiplicative bases for Λ n . Suppose that for each n , u n is a symmetric function homogeneous of degree n . Then for any partition λ = ( λ 1 , . . . , λ k ) , we may define u λ to be u λ 1 · · · u λ k . If u 1 , u 2 , . . . are algebraically independent, then { u λ } λ ⊢ n will be a basis for Λ n .

We define the n th elementary symmetric function e n by � e n = x i 1 · · · x i n , i 1 < ··· < i n so e n = m ( 1 n ) .

We define the n th elementary symmetric function e n by � e n = x i 1 · · · x i n , i 1 < ··· < i n so e n = m ( 1 n ) . The n th complete symmetric function is � h n = x i 1 · · · x i n , i 1 ≤···≤ i n so h n is the sum of all distinct monomials of degree n .

We define the n th elementary symmetric function e n by � e n = x i 1 · · · x i n , i 1 < ··· < i n so e n = m ( 1 n ) . The n th complete symmetric function is � h n = x i 1 · · · x i n , i 1 ≤···≤ i n so h n is the sum of all distinct monomials of degree n . The n th power sum symmetric function is ∞ � x n p n = i , i = 1 so p n = m ( n ) .

Theorem. Each of { h λ } λ ⊢ n , { e λ } λ ⊢ n , and { p λ } λ ⊢ n is a basis for Λ n .

Some generating functions We have ∞ ∞ e n t n = � � ( 1 + x i t ) n = 0 i = 1 and ∞ ∞ h n t n = i t 2 + · · · ) � � ( 1 + x i t + x 2 n = 0 i = 1 ∞ 1 � = 1 − x i t . i = 1 (Note that t is extraneous, since if we set t = 1 we can get it back by replacing each x i with x i t .)

Some generating functions We have ∞ ∞ e n t n = � � ( 1 + x i t ) n = 0 i = 1 and ∞ ∞ h n t n = i t 2 + · · · ) � � ( 1 + x i t + x 2 n = 0 i = 1 ∞ 1 � = 1 − x i t . i = 1 (Note that t is extraneous, since if we set t = 1 we can get it back by replacing each x i with x i t .) It follows that � ∞ ∞ � − 1 h n t n = � � ( − 1 ) n e n t n . n = 0 n = 0

Also ∞ ∞ 1 1 � � log 1 − x i t = log 1 − x i t i = 1 i = 1 ∞ ∞ t n � � x n = i n i = 1 n = 1 ∞ p n � n t n . = n = 1 Therefore � ∞ ∞ � p n h n t n = exp � � n t n . n = 0 n = 1

Also ∞ ∞ 1 1 � � log 1 − x i t = log 1 − x i t i = 1 i = 1 ∞ ∞ t n � � x n = i n i = 1 n = 1 ∞ p n � n t n . = n = 1 Therefore � ∞ ∞ � p n h n t n = exp � � n t n . n = 0 n = 1 If we expand the right side and equate coefficients of t n