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An Introduction to Symmetric Functions

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An Introduction to Symmetric Functions An Introduction to Symmetric Functions Ira M. Gessel Department of Mathematics Brandeis University Summer School on Algebraic Combinatorics Korea Institute for Advanced Study Seoul, Korea June 13, 2016 They are formal power series in the infinitely many variables x1; x2;::: that are invariant under permutation of the subscripts. In other words, if i1;:::; im are distinct positive integers and α1; : : : ; αm are arbitrary nonnegative integers then the coefficient of xα1 ··· xαm in a symmetric function is the same as i1 im α1 αm the coefficient of x1 ··· xm . Examples: 2 2 I x1 + x2 + ::: P I i≤j xi xj P 2 But not i≤j xi xj What are symmetric functions? Symmetric functions are not functions. In other words, if i1;:::; im are distinct positive integers and α1; : : : ; αm are arbitrary nonnegative integers then the coefficient of xα1 ··· xαm in a symmetric function is the same as i1 im α1 αm the coefficient of x1 ··· xm . Examples: 2 2 I x1 + x2 + ::: P I i≤j xi xj P 2 But not i≤j xi xj What are symmetric functions? Symmetric functions are not functions. They are formal power series in the infinitely many variables x1; x2;::: that are invariant under permutation of the subscripts. Examples: 2 2 I x1 + x2 + ::: P I i≤j xi xj P 2 But not i≤j xi xj What are symmetric functions? Symmetric functions are not functions. They are formal power series in the infinitely many variables x1; x2;::: that are invariant under permutation of the subscripts. In other words, if i1;:::; im are distinct positive integers and α1; : : : ; αm are arbitrary nonnegative integers then the coefficient of xα1 ··· xαm in a symmetric function is the same as i1 im α1 αm the coefficient of x1 ··· xm . P 2 But not i≤j xi xj What are symmetric functions? Symmetric functions are not functions. They are formal power series in the infinitely many variables x1; x2;::: that are invariant under permutation of the subscripts. In other words, if i1;:::; im are distinct positive integers and α1; : : : ; αm are arbitrary nonnegative integers then the coefficient of xα1 ··· xαm in a symmetric function is the same as i1 im α1 αm the coefficient of x1 ··· xm . Examples: 2 2 I x1 + x2 + ::: P I i≤j xi xj What are symmetric functions? Symmetric functions are not functions. They are formal power series in the infinitely many variables x1; x2;::: that are invariant under permutation of the subscripts. In other words, if i1;:::; im are distinct positive integers and α1; : : : ; αm are arbitrary nonnegative integers then the coefficient of xα1 ··· xαm in a symmetric function is the same as i1 im α1 αm the coefficient of x1 ··· xm . Examples: 2 2 I x1 + x2 + ::: P I i≤j xi xj P 2 But not i≤j xi xj I Symmetric functions are useful in counting plane partitions. I Symmetric functions are closely related to representations of symmetric and general linear groups I Symmetric functions are useful in counting unlabeled graphs (Pólya theory). What are symmetric functions good for? I Some combinatorial problems have symmetric function Q generating functions. For example, i<j (1 + xi xj ) counts graphs by the degrees of the vertices. I Symmetric functions are closely related to representations of symmetric and general linear groups I Symmetric functions are useful in counting unlabeled graphs (Pólya theory). What are symmetric functions good for? I Some combinatorial problems have symmetric function Q generating functions. For example, i<j (1 + xi xj ) counts graphs by the degrees of the vertices. I Symmetric functions are useful in counting plane partitions. I Symmetric functions are useful in counting unlabeled graphs (Pólya theory). What are symmetric functions good for? I Some combinatorial problems have symmetric function Q generating functions. For example, i<j (1 + xi xj ) counts graphs by the degrees of the vertices. I Symmetric functions are useful in counting plane partitions. I Symmetric functions are closely related to representations of symmetric and general linear groups What are symmetric functions good for? I Some combinatorial problems have symmetric function Q generating functions. For example, i<j (1 + xi xj ) counts graphs by the degrees of the vertices. I Symmetric functions are useful in counting plane partitions. I Symmetric functions are closely related to representations of symmetric and general linear groups I Symmetric functions are useful in counting unlabeled graphs (Pólya theory). Then the dimension of Λn is p(n), the number of partitions of n. A partition of n is a weakly decreasing sequence of positive integers λ = (λ1; λ2; : : : ; λk ) with sum n. For example, the partitions of4 are (4); (3; 1); (2; 2); (2; 1; 1); and (1; 1; 1; 1). There are several important bases for Λn, all indexed by partitions. The ring of symmetric functions Let Λ denote the ring of symmetric functions, and let Λn be the vector space of symmetric functions homogeneous of degree n. A partition of n is a weakly decreasing sequence of positive integers λ = (λ1; λ2; : : : ; λk ) with sum n. For example, the partitions of4 are (4); (3; 1); (2; 2); (2; 1; 1); and (1; 1; 1; 1). There are several important bases for Λn, all indexed by partitions. The ring of symmetric functions Let Λ denote the ring of symmetric functions, and let Λn be the vector space of symmetric functions homogeneous of degree n. Then the dimension of Λn is p(n), the number of partitions of n. For example, the partitions of4 are (4); (3; 1); (2; 2); (2; 1; 1); and (1; 1; 1; 1). There are several important bases for Λn, all indexed by partitions. The ring of symmetric functions Let Λ denote the ring of symmetric functions, and let Λn be the vector space of symmetric functions homogeneous of degree n. Then the dimension of Λn is p(n), the number of partitions of n. A partition of n is a weakly decreasing sequence of positive integers λ = (λ1; λ2; : : : ; λk ) with sum n. There are several important bases for Λn, all indexed by partitions. The ring of symmetric functions Let Λ denote the ring of symmetric functions, and let Λn be the vector space of symmetric functions homogeneous of degree n. Then the dimension of Λn is p(n), the number of partitions of n. A partition of n is a weakly decreasing sequence of positive integers λ = (λ1; λ2; : : : ; λk ) with sum n. For example, the partitions of4 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 Λn be the vector space of symmetric functions homogeneous of degree n. Then the dimension of Λn is p(n), the number of partitions of n. A partition of n is a weakly decreasing sequence of positive integers λ = (λ1; λ2; : : : ; λk ) with sum n. For example, the partitions of4 are (4); (3; 1); (2; 2); (2; 1; 1); and (1; 1; 1; 1). There are several important bases for Λn, all indexed by partitions. So 2 2 2 m(2;1;1) = x1 x2x3 + x3 x1x4 + x1 x3x5 + ··· : We could write it more formally as X 2 xi xj xk : i6=j; i6=k; j<k Monomial symmetric functions 2 If a symmetric function has a term x1 x2x3 with coefficient 1, 2 then it must contain all terms of the form xi xj xk , 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) = xi xj xk 2 where the sum is over all distinct terms of the form xi xj xk with i, j, and k distinct. We could write it more formally as X 2 xi xj xk : i6=j; i6=k; j<k Monomial symmetric functions 2 If a symmetric function has a term x1 x2x3 with coefficient 1, 2 then it must contain all terms of the form xi xj xk , 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) = xi xj xk 2 where the sum is over all distinct terms of the form xi xj xk with i, j, and k distinct. So 2 2 2 m(2;1;1) = x1 x2x3 + x3 x1x4 + x1 x3x5 + ··· : Monomial symmetric functions 2 If a symmetric function has a term x1 x2x3 with coefficient 1, 2 then it must contain all terms of the form xi xj xk , 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) = xi xj xk 2 where the sum is over all distinct terms of the form xi xj xk with i, j, and k distinct. So 2 2 2 m(2;1;1) = x1 x2x3 + x3 x1x4 + x1 x3x5 + ··· : We could write it more formally as X 2 xi xj xk : i6=j; i6=k; j<k More generally, for any partition λ = (λ1; : : : ; λk ), mλ is the sum of all distinct monomials of the form xλ1 ··· xλk : i1 ik n It’s easy to see that fmλgλ`n is a basis for Λ . If u1; u2;::: are algebraically independent, then fuλgλ`n will be a basis for Λn. Multiplicative bases There are three important multiplicative bases for Λn. Suppose that for each n, un is a symmetric function homogeneous of degree n.
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