# Category: Symmetric functions

Giambelli's formula
In mathematics, Giambelli's formula, named after Giovanni Giambelli, expresses Schubert classes in terms of special Schubert classes, or Schur functions in terms of complete symmetric functions. It st
Quasisymmetric function
In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring wit
Hall algebra
In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-groups. It was first discussed by but forgotten until it was rediscover
Symmetric function
In mathematics, a function of variables is symmetric if its value is the same no matter the order of its arguments. For example, a function of two arguments is a symmetric function if and only if for
Young's lattice
In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative substitutional analysis, developed the
In mathematics, an Adams operation, denoted ψk for natural numbers k, is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic const
Newton's inequalities
In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1, a2, ..., an are real numbers and let denote the kth elementary symmetric polynomial in a1, a2, ..., an. Then the eleme
Littlewood–Richardson rule
In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur fu
Stanley symmetric function
In mathematics and especially in algebraic combinatorics, the Stanley symmetric functions are a family of symmetric functions introduced by Richard Stanley in his study of the symmetric group of permu
Newton's identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials
Bender–Knuth involution
In algebraic combinatorics, a Bender–Knuth involution is an involution on the set of semistandard tableaux, introduced by , pp. 46–47) in their study of plane partitions.
Kostka number
In mathematics, the Kostka number Kλμ (depending on two integer partitions λ and μ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape λ and weight μ. They w
Symmetric polynomial
In mathematics, a symmetric polynomial is a polynomial P(X1, X2, …, Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric
Alternating polynomial
In algebra, an alternating polynomial is a polynomial such that if one switches any two of the variables, the polynomial changes sign: Equivalently, if one permutes the variables, the polynomial chang
LLT polynomial
In mathematics, an LLT polynomial is one of a family of symmetric functions introduced by Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon (1997) as q-analogues of products of Schur functions. J.
Complete homogeneous symmetric polynomial
In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial
Schubert polynomial
In mathematics, Schubert polynomials are generalizations of Schur polynomials that represent cohomology classes of Schubert cycles in flag varieties. They were introduced by and are named after Herman
Jucys–Murphy element
In mathematics, the Jucys–Murphy elements in the group algebra of the symmetric group, named after Algimantas Adolfas Jucys and G. E. Murphy, are defined as a sum of transpositions by the formula: The
Symmetrization
In mathematics, symmetrization is a process that converts any function in variables to a symmetric function in variables.Similarly, antisymmetrization converts any function in variables into an antisy
Maclaurin's inequality
In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let a1, a2, ..., an be positive real numbers, and for k = 1, 2
Plethysm
In algebra, plethysm is an operation on symmetric functions introduced by Dudley E. Littlewood, who denoted it by {λ} ⊗ {μ}. The word "plethysm" for this operation (after the Greek word πληθυσμός mean
Elementary symmetric polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can
Young symmetrizer
In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vec
Ring of symmetric functions
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring
Hall–Littlewood polynomials
In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t is 0 and monomial symmetric functions when t is 1
Power sum symmetric polynomial
In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with
Plethystic exponential
In mathematics, the plethystic exponential is a certain operator defined on (formal) power series which, like the usual exponential function, translates addition into multiplication. This exponential
Robinson–Schensted–Knuth correspondence
In mathematics, the Robinson–Schensted–Knuth correspondence, also referred to as the RSK correspondence or RSK algorithm, is a combinatorial bijection between matrices A with non-negative integer entr
Young tableau
In mathematics, a Young tableau (/tæˈbloʊ, ˈtæbloʊ/; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the gro
Pieri's formula
In mathematics, Pieri's formula, named after Mario Pieri, describes the product of a Schubert cycle by a special Schubert cycle in the Schubert calculus, or the product of a Schur polynomial by a comp
Ringel–Hall algebra
In mathematics, a Ringel–Hall algebra is a generalization of the Hall algebra, studied by Claus Michael Ringel. It has a basis of equivalence classes of objects of an abelian category, and the structu
Polykay
In statistics, a polykay, or generalised k-statistic, (denoted ) is a statistic defined as a linear combination of sample moments.
Monk's formula
In mathematics, Monk's formula, found by , is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product
Zonal polynomial
In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials. They appear as zonal spherical function
Kronecker coefficient
In mathematics, Kronecker coefficients gλμν describe the decomposition of the tensor product (= Kronecker product) of two irreducible representations of a symmetric group into irreducible representati
Plethystic substitution
Plethystic substitution is a shorthand notation for a common kind of substitution in the algebra of symmetric functions and that of symmetric polynomials. It is essentially basic substitution of varia
Jack function
In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal
Symmetric product of an algebraic curve
In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian product C × C × ... × C or Cn by the group action of the symmetric group Sn on n lett
Newton polygon
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the local field of inte
Vandermonde polynomial
In algebra, the Vandermonde polynomial of an ordered set of n variables , named after Alexandre-Théophile Vandermonde, is the polynomial: (Some sources use the opposite order , which changes the sign
Kostka polynomial
In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and
Schur polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the comple