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Symmetric function

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In mathematics, a function of $n$ variables is symmetric if its value is the same no matter the order of its arguments. For example, a function $f\left(x_{1},x_{2}\right)$ of two arguments is a symmetric function if and only if $f\left(x_{1},x_{2}\right)=f\left(x_{2},x_{1}\right)$ for all $x_{1}$ and $x_{2}$ such that $\left(x_{1},x_{2}\right)$ and $\left(x_{2},x_{1}\right)$ are in the domain of $f.$ The most commonly encountered symmetric functions are polynomial functions, which are given by the symmetric polynomials.

A related notion is alternating polynomials, which change sign under an interchange of variables. Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric $k$ -tensors on a vector space $V$ is isomorphic to the space of homogeneous polynomials of degree $k$ on $V.$ Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.

Symmetrization

[edit]

Given any function $f$ in $n$ variables with values in an abelian group, a symmetric function can be constructed by summing values of $f$ over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over even permutations and subtracting the sum over odd permutations. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions $f.$ The only general case where $f$ can be recovered if both its symmetrization and antisymmetrization are known is when $n=2$ and the abelian group admits a division by 2 (inverse of doubling); then $f$ is equal to half the sum of its symmetrization and its antisymmetrization.

Examples

[edit]

Consider the real function $f(x_{1},x_{2},x_{3})=(x-x_{1})(x-x_{2})(x-x_{3}).$ By definition, a symmetric function with $n$ variables has the property that $f(x_{1},x_{2},\ldots ,x_{n})=f(x_{2},x_{1},\ldots ,x_{n})=f(x_{3},x_{1},\ldots ,x_{n},x_{n-1}),\quad {\text{ etc.}}$ In general, the function remains the same for every permutation of its variables. This means that, in this case, $(x-x_{1})(x-x_{2})(x-x_{3})=(x-x_{2})(x-x_{1})(x-x_{3})=(x-x_{3})(x-x_{1})(x-x_{2})$ and so on, for all permutations of $x_{1},x_{2},x_{3}.$
Consider the function $f(x,y)=x^{2}+y^{2}-r^{2}.$ If $x$ and $y$ are interchanged the function becomes $f(y,x)=y^{2}+x^{2}-r^{2},$ which yields exactly the same results as the original $f(x,y).$
Consider now the function $f(x,y)=ax^{2}+by^{2}-r^{2}.$ If $x$ and $y$ are interchanged, the function becomes $f(y,x)=ay^{2}+bx^{2}-r^{2}.$ This function is not the same as the original if $a\neq b,$ which makes it non-symmetric.

Applications

[edit]

U-statistics

[edit]

In statistics, an $n$ -sample statistic (a function in $n$ variables) that is obtained by bootstrapping symmetrization of a $k$ -sample statistic, yielding a symmetric function in $n$ variables, is called a U-statistic. Examples include the sample mean and sample variance.

References

[edit]

F. N. David, M. G. Kendall & D. E. Barton (1966) Symmetric Function and Allied Tables, Cambridge University Press.
Joseph P. S. Kung, Gian-Carlo Rota, & Catherine H. Yan (2009) Combinatorics: The Rota Way, §5.1 Symmetric functions, pp 222–5, Cambridge University Press, ISBN 978-0-521-73794-4.

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In mathematics, a symmetric function is a formal power series in an infinite set of indeterminates

x_1, x_2, \dots

that remains invariant under any permutation of the variables, meaning

f(x_{\sigma(1)}, x_{\sigma(2)}, \dots) = f(x_1, x_2, \dots)

for any permutation

\sigma

of the natural numbers.^[1] This invariance captures the essential symmetry in the arguments, distinguishing symmetric functions from more general multivariate polynomials or series.^[2] The theory originated in the study of symmetric polynomials related to the roots of equations, with early developments traceable to the 17th and 18th centuries through works on algebraic identities, such as those involving power sums and elementary symmetric sums.^[3] The ring of symmetric functions, often denoted

\Lambda

, is generated by the elementary symmetric functions

e_k = \sum_{1 \leq i_1 < \cdots < i_k} x_{i_1} \cdots x_{i_k}

or the complete homogeneous symmetric functions

h_k = \sum_{1 \leq i_1 \leq \cdots \leq i_k} x_{i_1} \cdots x_{i_k}

, which form fundamental bases alongside others like the monomial symmetric functions

m_\lambda

indexed by partitions

\lambda

and the Schur functions

s_\lambda

.^[4] A key result, the fundamental theorem of symmetric functions, states that every symmetric function can be uniquely expressed as a polynomial in the elementary symmetric functions with integer coefficients, providing a complete algebraic structure for the ring.^[3] These bases are interconnected through transition matrices and generating functions, such as the Cauchy identity relating Schur functions to other types, enabling computations and proofs across diverse settings.^[4] Symmetric functions have profound applications in enumerative combinatorics, where they count objects like partitions and Young tableaux via generating functions; in representation theory, as characters of the symmetric group are Schur functions; and in algebraic geometry and physics, modeling invariants in statistical mechanics and quantum groups.^[5] Their study extends to quasisymmetric functions and generalizations, influencing modern areas like Macdonald polynomials and crystal bases.^[6]

Definition and Fundamentals

Formal Definition

In mathematics, particularly in algebraic combinatorics, a symmetric function is a formal power series in countably infinitely many indeterminates

x_1, x_2, \dots

over a commutative ring (typically

\mathbb{Z}

\mathbb{Q}

) that remains invariant under the action of the symmetric group

S_\infty

, consisting of all permutations of the natural numbers that move only finitely many elements. Formally, if

f \in R[[x_1, x_2, \dots ]]

for a commutative ring

R

, then

f

is symmetric if

f(x_{\sigma(1)}, x_{\sigma(2)}, \dots) = f(x_1, x_2, \dots)

for every

\sigma \in S_\infty

.^[4]^[1] This setup views symmetric functions not as evaluating to numerical values but as elements of the graded ring

\Lambda = \bigoplus_{d=0}^\infty \Lambda^d

, where

\Lambda^d

is the space of homogeneous symmetric functions of degree

d

, consisting of finite linear combinations of monomials of total degree

d

invariant under permutation. The ring

\Lambda

is generated by the elementary symmetric functions

e_k

or the complete homogeneous symmetric functions

h_k

, providing a foundational algebraic structure. Symmetric functions are distinguished from alternating functions, which change sign under odd permutations: a formal power series

f

is alternating if

f(x_{\sigma(1)}, x_{\sigma(2)}, \dots) = \operatorname{sgn}(\sigma) f(x_1, x_2, \dots)

for

\sigma \in S_\infty

, implying

f

vanishes if any two variables are equal.^[7] This definition generalizes the classical theory of symmetric polynomials (finite variables) to the stable limit of infinitely many variables, underpinning applications in combinatorics and representation theory.

Historical Context and Motivations

The origins of symmetric functions can be traced to the 17th century, when Isaac Newton developed key ideas around 1665–1666 while investigating methods to solve polynomial equations without explicitly determining their roots. Newton's work focused on relating power sums of roots to elementary symmetric sums, enabling computations like resultants for systems of equations, as detailed in his unpublished manuscripts and later in Arithmetica Universalis (1707). This approach highlighted the utility of expressions invariant under permutations of the roots, providing an early framework for handling symmetric relations in algebraic problems.^[8] In the 19th century, the theory advanced significantly through the contributions of Arthur Cayley and James Joseph Sylvester, who pioneered invariant theory starting in the 1850s. Cayley introduced systematic methods for computing invariants of binary quadratic forms in 1854, while Sylvester, who coined the term "invariant" in 1851, developed algorithmic techniques for higher-degree forms and emphasized their role in classifying algebraic structures up to symmetry. Their collaborative efforts, documented in papers published in the Philosophical Transactions and Cambridge Mathematical Journal, connected symmetric functions to broader questions of algebraic invariance under linear group actions.^[9] Symmetric functions naturally emerge in mathematical contexts requiring invariance under relabeling of variables, such as expressing polynomial coefficients in terms of roots. For instance, power sum symmetric functions correspond to moments in probability distributions. Their study underpins modern algebra, particularly in Galois theory, where the fundamental theorem on symmetric polynomials—intuited by Newton and rigorously established in the 19th century—demonstrates that the fixed field of the symmetric group acting on the roots of a generic polynomial is generated by the elementary symmetric polynomials, linking solvability by radicals to permutation group structure as formalized by Évariste Galois in the 1830s.^[10]

Properties and Structure

Invariance and Basic Properties

Symmetric functions are characterized by their invariance under the action of the symmetric group

S_n

, which permutes the variables of the function while leaving its value unchanged. Specifically, a function

f(x_1, \dots, x_n)

is symmetric if

f(\sigma(x_1), \dots, \sigma(x_n)) = f(x_1, \dots, x_n)

for all permutations

\sigma \in S_n

. This property defines the fixed subspace of the space of all functions under the

S_n

-action, forming a subspace that captures all permutation-invariant behaviors.^[7] The Reynolds operator provides a canonical projection onto this invariant subspace. For a function

f

, it is defined by the formula

P(f) = \frac{1}{n!} \sum_{\sigma \in S_n} f \circ \sigma,

where

f \circ \sigma

denotes the composition of

f

with the permutation

\sigma

acting on the variables. This operator is idempotent, meaning

P(P(f)) = P(f)

, and maps any function to its symmetric counterpart, ensuring that the image lies precisely in the space of symmetric functions.^[11] Any function

f

n

variables decomposes uniquely into a symmetric part and an alternating (antisymmetric) part, given by

f = \frac{f + A(f)}{2} + \frac{f - A(f)}{2},

where the antisymmetrizer

A

is defined as

A(f) = \frac{1}{n!} \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \, f \circ \sigma,

with

\operatorname{sgn}(\sigma)

denoting the sign of the permutation

\sigma

. The symmetric component

\frac{f + A(f)}{2}

is invariant under

S_n

, while the alternating component

\frac{f - A(f)}{2}

changes sign under odd permutations, providing a contrast to symmetric functions by highlighting antisymmetry. This decomposition arises from the averaging over the group action and its signed variant.^[7] For the space of homogeneous polynomials of degree

d

n

variables, the dimension of the subspace of symmetric polynomials equals the number of integer partitions of

d

into at most

n

parts; when

n \geq d

, this simplifies to

p(d)

, the partition function counting all partitions of

d

. This dimension reflects the basis of symmetric monomials indexed by such partitions.^[7]

Algebraic Structure

The ring of symmetric polynomials in

n

variables over the integers, denoted

\Lambda_n = \mathbb{Z}[x_1, \dots, x_n]^{S_n}

, consists of all polynomials invariant under the action of the symmetric group

S_n

by permuting the variables. This ring is freely generated as a polynomial ring by the

n

elementary symmetric polynomials

e_1, \dots, e_n

, where

e_k = \sum_{1 \leq i_1 < \cdots < i_k \leq n} x_{i_1} \cdots x_{i_k}

.^[12]^[10] In the infinite-variable setting, the ring

\Lambda

of symmetric functions is constructed as the direct limit

\Lambda = \varinjlim_{n \to \infty} \Lambda_n

Λ=

limn→∞Λn, comprising formal power series in infinitely many variables $x_1, x_2, \dots$ that are invariant under permutations of finitely many variables. This ring inherits a natural grading $\Lambda = \bigoplus_{d=0}^\infty \Lambda^d$ , where $\Lambda^d$ denotes the homogeneous component of degree $d$ , and the dimension $\dim \Lambda^d$ equals the number of integer partitions of $d$ . The associated Hilbert series is thus $H_\Lambda(t) = \sum_{d=0}^\infty p(d) t^d = \prod_{k=1}^\infty \frac{1}{1 - t^k}$ , where $p(d)$ is the partition function.^[12] As a graded algebra, $\Lambda$ possesses additional structure as a Hopf algebra, equipped with a coproduct $\Delta: \Lambda \to \Lambda \otimes \Lambda$ defined on generators such that $\Delta(e_k) = \sum_{i=0}^k e_i \otimes e_{k-i}$ (with $e_0 = 1$ ), an antipode, and a unit, satisfying the required axioms. This endows $\Lambda$ with a universal property among graded connected commutative Hopf algebras generated by the primitives corresponding to the power sums or elementary symmetric functions, facilitating plethystic operations and compositions central to the theory.^[12] The ring $\Lambda$ also acts as a module over itself in ways that underpin Schur functors, which map a vector space $V$ to its irreducible polynomial representations $S^\lambda V$ parameterized by partitions $\lambda$ , with the Schur function $s_\lambda$ serving as the character; this action encodes the decomposition of tensor powers and symmetric powers into irreducibles without delving into full representation-theoretic details.^[12]

Construction Methods

Symmetrization Techniques

Symmetrization techniques provide fundamental methods for constructing symmetric functions from arbitrary functions on a set of variables, by averaging over the action of the symmetric group

S_n

. The primary tool is the symmetrizer operator, defined as

\operatorname{Sym}(f)(x_1, \dots, x_n) = \frac{1}{n!} \sum_{\sigma \in S_n} f(x_{\sigma(1)}, \dots, x_{\sigma(n)}),

where the symmetric group

S_n

acts by permuting the variables.^[13] This operator, also known as the Reynolds operator in the context of invariant theory, projects any function

f

onto the subspace of symmetric functions invariant under permutations.^[13] The symmetrizer is idempotent, meaning

\operatorname{Sym}^2 = \operatorname{Sym}

, and serves as a projection onto the invariants. To see idempotence, apply

\operatorname{Sym}

\operatorname{Sym}(f)

\operatorname{Sym}(\operatorname{Sym}(f)) = \frac{1}{n!} \sum_{\sigma \in S_n} \operatorname{Sym}(f) \circ \sigma = \frac{1}{n!} \sum_{\sigma \in S_n} \frac{1}{n!} \sum_{\tau \in S_n} f \circ \tau \circ \sigma.

The double sum over

\sigma

and

\tau

can be rewritten by setting

\rho = \tau \sigma

, which runs over all elements of

S_n

exactly

n!

times for each fixed

\rho

, yielding

\operatorname{Sym}(\operatorname{Sym}(f)) = \frac{1}{n!} \sum_{\rho \in S_n} f \circ \rho = \operatorname{Sym}(f).

Thus, applying the symmetrizer twice yields the same result, confirming its projection property.^[13] For polynomials, this constructs the ring of symmetric polynomials as the fixed points under the

S_n

-action. The antisymmetrizer provides a complementary construction for alternating functions, defined by

\operatorname{Alt}(f)(x_1, \dots, x_n) = \frac{1}{n!} \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) f(x_{\sigma(1)}, \dots, x_{\sigma(n)}),

where

\operatorname{sgn}(\sigma)

is the sign of the permutation

\sigma

. Like the symmetrizer, it is idempotent and projects onto the space of alternating functions.^[7] In the context of symmetric functions, the antisymmetrizer relates directly to determinants: applying

\operatorname{Alt}

to the monomial

x_1^{n-1} x_2^{n-2} \cdots x_n^0

yields the Vandermonde determinant

\prod_{1 \leq i < j \leq n} (x_j - x_i)

, up to a scalar factor, which is the prototypical alternating polynomial.^[7] More generally, alternants

a_\lambda = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) x_{\sigma(1)}^{\lambda_1} \cdots x_{\sigma(n)}^{\lambda_n}

form a basis for alternating polynomials, and Schur functions arise as ratios of such alternants.^[7] For multilinear forms, symmetrization can be achieved via the polarization identity, which recovers a symmetric multilinear map from an associated homogeneous polynomial. Given a homogeneous polynomial

P

of degree

n

, the corresponding symmetric

n

-linear form

\phi

satisfies

\phi(v_1, \dots, v_n) = \frac{1}{n! 2^n} \sum_{\epsilon \in \{-1,1\}^n} \left( \prod_{i=1}^n \epsilon_i \right) P\left( \sum_{i=1}^n \epsilon_i v_i \right).

This identity ensures uniqueness and provides an explicit symmetrization procedure for multilinear expressions.^[14] It extends the classical polarization for quadratic forms and is essential for constructing symmetric multilinear functionals from non-symmetric ones. While powerful, symmetrization techniques have limitations. Not all symmetric functions can be obtained this way in non-polynomial settings, such as for general measurable functions, where the projection may not span the full invariant space due to integrability issues.^[15] Computationally, the

n!

summations make these operators inefficient for large

n

, prompting alternative methods like recursive algorithms or basis expansions for practical implementation in symmetric polynomial computations.^[16]

Generating Functions and Bases

Symmetric functions can be generated and represented using several fundamental bases, each with associated generating functions that facilitate computations and transitions between bases. The power sum symmetric polynomials form one such basis, defined for a positive integer

k

p_k = \sum_i x_i^k,

where the sum is over all variables

x_i

. Their ordinary generating function is \sum_{k=1}^\infty p_k t^k = \sum_i \frac{x_i t}{1 - x_i t}.[](https://math.berkeley.edu/~corteel/MATH249/macdonald.pdf) This generating function arises from the [logarithmic derivative](/page/Logarithmic_derivative) of the complete homogeneous generating function and is useful for expressing other bases via Newton identities.[](https://www.ms.uky.edu/~sohum/putnam/enu_comb_stanley.pdf) The elementary symmetric polynomials provide another basis, defined for nonnegative integers $k$ by \[ e_k = \sum_{i_1 < \cdots < i_k} x_{i_1} \cdots x_{i_k}, with

e_0 = 1

. Their generating function is the product

E(t) = \sum_{k \geq 0} e_k t^k = \prod_i (1 + x_i t).[](https://math.berkeley.edu/~corteel/MATH249/macdonald.pdf) This form reflects the inclusion-exclusion principle underlying the sums over distinct indices and is central to the ring structure of symmetric functions.[](https://www.ms.uky.edu/~sohum/putnam/enu_comb_stanley.pdf) The complete homogeneous symmetric polynomials constitute a third basis, defined by \[ h_k = \sum_{i_1 \leq \cdots \leq i_k} x_{i_1} \cdots x_{i_k},

E(t)=k≥0∑ektk=i∏(1+xit).[](https://math.berkeley.edu/ corteel/MATH249/macdonald.pdf)Thisformreflectstheinclusion−exclusionprincipleunderlyingthesumsoverdistinctindicesandiscentraltotheringstructureofsymmetricfunctions.[](https://www.ms.uky.edu/ sohum/putnam/enucombstanley.pdf)Thecompletehomogeneoussymmetricpolynomialsconstituteathirdbasis,definedby\[hk=i1≤⋯≤ik∑xi1⋯xik, summing over nondecreasing sequences of indices. Their generating function is H(t) = \sum_{k \geq 0} h_k t^k = \prod_i (1 - x_i t)^{-1}.[](https://math.berkeley.edu/~corteel/MATH249/macdonald.pdf) This infinite product generates all monomials of degree $k$, allowing expansions in terms of multisets of variables.[](https://www.ms.uky.edu/~sohum/putnam/enu_comb_stanley.pdf) These bases are interconnected through Newton identities, which provide recursive relations between power sums and the other bases. For the elementary symmetric polynomials, the identities state that \[ p_k - e_1 p_{k-1} + e_2 p_{k-2} - \cdots + (-1)^{k-1} e_{k-1} p_1 + (-1)^k k e_k = 0 for

k \geq 1

.^[7] Similar relations hold for the complete homogeneous polynomials, such as

k h_k = \sum_{i=1}^k p_i h_{k-i},

with

h_0 = 1

. These identities enable explicit conversions between bases, essential for algebraic manipulations in the ring of symmetric functions.^[17] Schur functions form an orthogonal basis for symmetric functions, indexed by partitions

\lambda

and defined via the Jacobi-Trudi identity as the determinant

s_\lambda = \det\left( e_{\lambda_i' - i + j} \right)_{i,j \geq 1},

where

\lambda'

denotes the conjugate partition and the determinant is over finitely many rows and columns. Alternatively, they can be expressed in terms of complete homogeneous polynomials: s_\lambda = \det\left( h_{\lambda_i - i + j} \right)_{i,j \geq 1}.[](https://math.berkeley.edu/~corteel/MATH249/macdonald.pdf) Schur functions are positive with respect to the [monomial basis](/page/Monomial_basis) and play a key role in decompositions due to their [orthogonality](/page/Orthogonality) properties.[](https://www.ms.uky.edu/~sohum/putnam/enu_comb_stanley.pdf) ## Specific Classes and Examples ### Symmetric Polynomials Symmetric polynomials are polynomials in several variables that remain unchanged under any permutation of the variables. A central result in their theory is the fundamental theorem of symmetric polynomials, which asserts that every symmetric polynomial in $n$ variables over a field of characteristic zero can be uniquely expressed as a polynomial in the elementary symmetric polynomials $e_1, e_2, \dots, e_n$.[](https://arxiv.org/abs/1301.7116) This theorem provides a complete classification, showing that the ring of symmetric polynomials is freely generated by the elementary symmetric ones. Vieta's formulas exemplify this structure by relating the coefficients of a [monic polynomial](/page/Monic_polynomial) to the elementary symmetric sums of its [roots](/page/The_Roots). For a [monic polynomial](/page/Monic_polynomial) $p(x) = x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = \prod_{i=1}^n (x - r_i)$ with [roots](/page/The_Roots) $r_1, \dots, r_n$, the coefficients are given by $a_{n-k} = (-1)^k e_k(r_1, \dots, r_n)$ for $k = 1, \dots, n$, where $e_k$ is the $k$-th elementary symmetric sum.[](https://mathworld.wolfram.com/VietasFormulas.html) Thus, the coefficients are symmetric polynomials in the [roots](/page/The_Roots), expressible via the elementary symmetric basis. A prominent example of a [symmetric polynomial](/page/Symmetric_polynomial) is the [discriminant](/page/Discriminant) $\Delta = \prod_{1 \leq i < j \leq n} (x_i - x_j)^2$, which vanishes precisely when the variables include repeated values and determines whether a [polynomial](/page/Polynomial) has multiple roots.[](https://ocw.mit.edu/courses/res-18-012-algebra-ii-student-notes-spring-2022/mit18_702s22_lect33.pdf) By the fundamental theorem, $\Delta$ can be expressed as a [polynomial](/page/Polynomial) in the elementary symmetric polynomials, specifically $\Delta_n(e_1, \dots, e_n)$.[](https://ocw.mit.edu/courses/res-18-012-algebra-ii-student-notes-spring-2022/mit18_702s22_lect33.pdf) Power sums offer another concrete class of symmetric polynomials, such as the second power sum $\sum_{i=1}^n x_i^2 = e_1^2 - 2 e_2$, which expresses the sum of squares in terms of the first two elementary symmetric polynomials.[](https://kconrad.math.uconn.edu/blurbs/galoistheory/symmfunction.pdf) Higher power sums $p_k = \sum_{i=1}^n x_i^k$ for $k \geq 1$ are also symmetric and can be recursively expressed via [Newton's identities](/page/Newton's_identities) in the elementary symmetric basis.[](https://kconrad.math.uconn.edu/blurbs/galoistheory/symmfunction.pdf) Cycle index polynomials arise in enumeration under group actions, such as for the [symmetric group](/page/Symmetric_group) $S_n$, where the [cycle index](/page/Cycle_index) $Z_{S_n}(s_1, \dots, s_n) = \frac{1}{n!} \sum_{\sigma \in S_n} \prod_{i=1}^n s_i^{c_i(\sigma)}$ and $c_i(\sigma)$ counts cycles of length $i$ in $\sigma$. In Pólya's enumeration theorem, substituting $s_i = \sum x_j^i$ (power sums in auxiliary variables $x_j$) into the [cycle index](/page/Cycle_index) yields a [symmetric polynomial](/page/Symmetric_polynomial) in the $x_j$, which generates counts of distinct objects up to symmetry.[](https://mathworld.wolfram.com/PolyaEnumerationTheorem.html) ### Other Symmetric Functions Symmetric rational functions are quotients of symmetric polynomials in a set of variables, forming the field of fractions of the ring of symmetric polynomials, and are thus invariant under arbitrary permutations of the variables.[](https://mathworld.wolfram.com/SymmetricFunction.html)[](https://faculty.etsu.edu/gardnerr/5410/notes/V-2-A.pdf) This field, often denoted $\mathbb{Q}(x_1, x_2, \dots)$ where the $x_i$ are indeterminates, is generated by the elementary symmetric polynomials and serves as the quotient field over any base field of characteristic zero.[](https://faculty.etsu.edu/gardnerr/5410/notes/V-2-A.pdf) For instance, in two variables, the function $\frac{x_1 + x_2}{x_1 x_2}$ simplifies to $\frac{1}{x_1} + \frac{1}{x_2}$, which remains unchanged under swapping $x_1$ and $x_2$, illustrating a basic non-polynomial example.[](https://mathworld.wolfram.com/SymmetricFunction.html) Exponential generating functions extend the toolkit for symmetric functions by incorporating factorial scaling, particularly useful for power sum symmetric functions $p_k = \sum_i x_i^k$. The series $\sum_{k=1}^\infty \frac{p_k}{k!} t^k$ provides an exponential generating function that captures the structure of these power sums in a combinatorial context, facilitating connections to labeled structures and plethystic compositions in symmetric function theory.[](https://www.math.uwaterloo.ca/~dgwagner/SF.pdf) This form arises naturally in the exponential formula for symmetric functions, where the logarithm or exponential of such series relates bases like complete homogeneous functions to power sums via $\exp\left( \sum_{k=1}^\infty \frac{p_k}{k} t^k \right)$, though the divided-by-$k!$ variant emphasizes enumerative aspects.[](https://arxiv.org/abs/2401.17687) Symmetric multilinear forms generalize quadratic forms to higher degrees, corresponding to fully symmetric tensors in the tensor space over a [vector space](/page/Vector_space) $V$, where the form is invariant under permutations of its arguments. These forms are obtained via polarization of homogeneous polynomials: for a degree-$d$ homogeneous polynomial $q: V \to K$, the associated symmetric $d$-linear form $\phi: V^d \to K$ satisfies $\phi(v, \dots, v) = q(v)$ and is linear in each slot, with symmetry ensuring $\phi(v_{\sigma(1)}, \dots, v_{\sigma(d)}) = \phi(v_1, \dots, v_d)$ for any [permutation](/page/Permutation) $\sigma$.[](https://arxiv.org/abs/1509.05707) For example, the polarization of a [quadratic form](/page/Quadratic_form) $q(v) = \langle Av, v \rangle$ yields the [symmetric bilinear form](/page/Symmetric_bilinear_form) $\phi(u, v) = \frac{1}{4} (q(u+v) - q(u-v))$, extendable to higher multilinear cases in tensor algebras.[](https://arxiv.org/abs/1509.05707) In statistical mechanics, partition functions for systems of noninteracting indistinguishable particles exemplify symmetric functions, as they remain invariant under relabeling of particle indices due to the underlying permutation symmetry. For instance, the grand canonical partition function for ideal gases of bosons or fermions is a symmetric function in the single-particle energies, computable via group representations of the symmetric group to enforce the symmetry.[](https://link.aps.org/doi/10.1103/PhysRevE.64.066105) This symmetry ensures that the function depends only on the multiset of particle labels, aligning with the broader category of symmetric rational or exponential forms in multi-variable settings.[](https://arxiv.org/abs/cond-mat/0109112) ## Applications ### In Statistics In statistics, symmetric functions play a central role in the construction of [U-statistics](/page/U-statistic), which are unbiased [estimators](/page/Estimator) of [parameters](/page/Parameter) defined as expectations of symmetric kernels applied to independent and identically distributed random variables. A [U-statistic](/page/U-statistic) of degree $m$ based on a sample $X_1, \dots, X_n$ from a distribution $F$ is given by \[ U_n = \frac{1}{\binom{n}{m}} \sum_{1 \leq j_1 < \cdots < j_m \leq n} h(X_{j_1}, \dots, X_{j_m}), where

h: \mathbb{R}^m \to \mathbb{R}

is a symmetric kernel, meaning

h(x_1, \dots, x_m) = h(x_{\pi(1)}, \dots, x_{\pi(m)})

for any permutation

\pi

, and the parameter of interest is

\theta(F) = \mathbb{E}_F[h(X_1, \dots, X_m)]

.^[18] This formulation ensures

U_n

is an unbiased estimator of

\theta(F)

, leveraging the symmetry of

h

to average over all combinations of sample elements.^[18] A classic example is the sample variance, which serves as an unbiased estimator of the population variance

\sigma^2 = \mathbb{E}[(X - \mu)^2]

under finite second moments. For

m=2

, the symmetric kernel is

h(x,y) = \frac{1}{2}(x - y)^2

, yielding

U_n = \frac{1}{\binom{n}{2}} \sum_{1 \leq i < j \leq n} \frac{1}{2}(X_i - X_j)^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2

, the familiar unbiased sample variance.^[18] Other examples include rank correlation statistics like Spearman's rho and the Gini coefficient for inequality measures, both relying on symmetric kernels to produce unbiased estimates.^[18] The asymptotic properties of U-statistics are analyzed through Hoeffding's decomposition, which expresses the centered U-statistic as an orthogonal sum of projection terms. Specifically,

U_n - \theta(F) = \sum_{c=1}^m \binom{m}{c} \left( \frac{1}{n} \sum_{i=1}^n g_c(X_i; F) - \mathbb{E}[g_c(X_1; F)] \right) + R_n,

where

g_c(x_1, \dots, x_c; F) = \mathbb{E}[h(X_1, \dots, X_m) \mid X_1 = x_1, \dots, X_c = x_c]

are the conditional expectation projections (with the first term dominating for large

n

if the kernel is non-degenerate), and

R_n

is a remainder term vanishing in probability.^[18] This decomposition facilitates central limit theorems, showing

\sqrt{n}(U_n - \theta(F))

(Un−θ(F)) converges in distribution to a normal with variance $m^2 \zeta_1$ , where $\zeta_1 = \mathrm{Var}(g_1(X_1; F))$ , under finite variance assumptions.^[18] Resampling methods like the jackknife and bootstrap extend to U-statistics for variance estimation and bias correction of these symmetric function-based estimators. The jackknife, adapted for U-statistics, involves deleting one observation at a time to compute pseudo-values, yielding an unbiased variance estimator that reduces bias in small samples, as shown for kernels of degree up to 2.^[19] Similarly, the bootstrap approximates the distribution of U-statistics by resampling the sample with replacement and recomputing the kernel averages, achieving consistency for the sampling distribution under moment conditions, particularly for non-degenerate cases where the bootstrap replicates converge to the same normal limit as the original statistic.^[20] These techniques are particularly valuable in nonparametric settings, enabling inference for parameters defined via symmetric functions without strong distributional assumptions.^[20]

In Representation Theory and Physics

In representation theory, symmetric functions play a central role in describing the characters of representations of the symmetric group

S_n

. Specifically, the Schur functions

s_\lambda

, indexed by partitions

\lambda

n

, serve as the characters of the irreducible representations of

S_n

, known as Specht modules, via the Schur-Weyl duality that relates actions of

S_n

and

GL(n)

on polynomial rings.^[21] This connection arises because the ring of symmetric functions is the character ring of

S_n

, where the power-sum symmetric functions

p_\mu

correspond to the conjugacy classes labeled by cycle types

\mu

.^[21] The Frobenius formula provides an explicit link between these characters and symmetric functions, expressing the Schur function as

s_\lambda = \frac{1}{n!} \sum_{\sigma \in S_n} \chi^\lambda(\sigma) \, p_{\mathrm{cyc}(\sigma)},

where

\chi^\lambda

is the character value of the irreducible representation corresponding to

\lambda

, and

p_{\mathrm{cyc}(\sigma)}

is the power-sum symmetric function associated with the cycle lengths of the permutation

\sigma

.^[21] This formula, originally developed by Frobenius, allows the computation of Schur functions from group characters and vice versa, facilitating the study of decomposition of representations induced from subgroups.^[21] Plethysm extends this framework to composite representations, particularly tensor products. Defined as a substitution operation on symmetric functions, such as

f

for symmetric functions

f

and

g

, plethysm computes the character of the representation obtained by composing actions, for instance, the

k

-th symmetric power of a tensor product

V \otimes W

, where

V

and

W

have characters

f

and

g

.^[22] In the context of

S_n

-representations, plethysm coefficients determine the multiplicities of irreducibles in such tensor constructions, though explicit formulas remain challenging and are often computed via generating functions or algorithmic methods.^[22] In physics, symmetric functions relate to invariance properties of tensors and states under permutations. In general relativity, the metric tensor

g_{\mu\nu}

is symmetric,

g_{\mu\nu} = g_{\nu\mu}

, ensuring that the spacetime interval

ds^2 = g_{\mu\nu} \, dx^\mu dx^\nu

is invariant under index exchange, which reflects the undirected nature of geodesic distances and preserves the pseudo-Riemannian structure.^[23] This symmetry reduces the number of independent components from 16 to 10 in four dimensions, simplifying the Einstein field equations while maintaining diffeomorphism invariance.^[23] In quantum mechanics, permutation symmetry governs multi-particle states of identical particles. For bosons, which obey Bose-Einstein statistics, the total wave function must be fully symmetric under exchanges of particle labels, ensuring invariance under the action of the symmetric group

S_N

on the

N

-particle Hilbert space. This symmetrization projects states onto the symmetric subspace, leading to phenomena like Bose-Einstein condensation, where the ground state accommodates arbitrary numbers of particles without Pauli exclusion. The symmetric tensor product construction formalizes this, with the bosonic Fock space built from symmetrized products of single-particle states.

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