symmetric function
C2/AcademicTechnical/Specialist
Definition
Meaning
A function of several variables that remains unchanged under any permutation of its variables.
In mathematics, a function whose value is the same for any ordering of its arguments; fundamental in algebra, combinatorics, and invariant theory.
Linguistics
Semantic Notes
Primarily used in pure mathematics (algebra, combinatorics) and theoretical computer science. Not to be confused with 'symmetric relation' or 'even/odd function'.
Dialectal Variation
British vs American Usage
Differences
No significant difference in meaning; 'symmetric' is consistently preferred over 'symmetrical' in this compound in both variants.
Connotations
Highly technical, implies formal mathematical training.
Frequency
Extremely rare outside academic mathematics; common in graduate-level texts.
Vocabulary
Collocations
Grammar
Valency Patterns
The symmetric function f(x₁, x₂, ..., xₙ) is invariant under permutation.One can expand the polynomial in terms of elementary symmetric functions.Vocabulary
Synonyms
Neutral
Vocabulary
Antonyms
Usage
Context Usage
Academic
The fundamental theorem of symmetric functions states that every symmetric polynomial can be expressed uniquely as a polynomial in the elementary symmetric functions.
Technical
Schur functions form an important basis for the ring of symmetric functions and are indexed by integer partitions.
Examples
By Part of Speech
noun
British English
- The monograph explores symmetric functions in relation to representation theory.
- Newton's identities provide relations between power sum symmetric functions.
American English
- The researcher's thesis focused on symmetric functions and Macdonald polynomials.
- We need a basis for the space of symmetric functions in three variables.
Examples
By CEFR Level
- In algebra, a symmetric function does not change when you swap its input variables.
- The sum x+y+z is a simple example of a symmetric function.
- The ring of symmetric functions has several classical bases, including the monomial symmetric functions and the Schur functions.
- One application of symmetric functions is in describing the eigenvalues of a matrix without specifying their order.
Learning
Memory Aids
Mnemonic
Think of a symmetric function as a referee who treats all players (variables) equally—swapping them around doesn't change the final score (function value).
Conceptual Metaphor
A perfectly fair committee where every member's vote (variable) is treated with identical weight regardless of seating order.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Do not translate as 'симметричная функция' when referring to geometric symmetry of a graph; the term specifically concerns variable permutation invariance.
- Avoid confusion with 'чётная/нечётная функция' (even/odd function), which concerns sign changes, not permutations.
Common Mistakes
- Using 'symmetrical function' (less common in mathematics).
- Confusing with 'symmetric relation' in set theory.
- Assuming it means the graph of the function is symmetrical.
Practice
Quiz
Which of the following is an elementary symmetric function in variables x₁, x₂, x₃?
FAQ
Frequently Asked Questions
No, but the theory most commonly deals with symmetric polynomials. More generally, symmetric functions can be formal power series invariant under variable permutations.
An even function satisfies f(-x) = f(x) for a single variable, dealing with sign change. A symmetric function deals with multiple variables and their permutations, not signs.
They are central to algebraic combinatorics, representation theory of symmetric groups, and mathematical physics (e.g., integrable systems, random matrix theory).
Yes, the number of variables is independent of the symmetry property. The function must be invariant under any permutation of those specific variables.