TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Monomial symmetric polynomials form an $R$ -basis of the symmetric-polynomial ring

Statement

As $λ$ ranges over partitions of length at most $n$ , the polynomials $m_{λ}$ form an $R$ -basis of $R [x_{1}, \dots, x_{n}]^{{Sym}_{n}}$ .

Facts & Assumptions

Given: A commutative ring $R$ and a natural number $n$ .

[L1]

The polynomial $m_{λ}$ is the sum of the distinct monomials whose exponent tuples lie in the permutation orbit of $λ$ (Monomial symmetric polynomials indexed by partitions).

[L2]

A polynomial is symmetric exactly when every permutation of its variables fixes it (Symmetric polynomials as the invariants of variable permutations).

Proof

technique · direct

1.1L1algebra

Variable permutations partition the monomials into disjoint orbits, and every orbit contains exactly one weakly decreasing exponent tuple $λ$ .

1.2L1L2

If $f$ is symmetric, the coefficients of two monomials in the same orbit are equal, because a variable permutation carries either monomial to the other and fixes $f$ . Since $f$ has finite support, it is therefore a finite $R$ -linear combination of the corresponding orbit sums $m_{λ}$ .

1.3L1algebra

Distinct $m_{λ}$ have disjoint monomial supports. Hence a finite relation $\sum_{λ} c_{λ} m_{λ} = 0$ has $c_{λ} = 0$ for every $λ$ , by comparing the coefficient of any monomial in the orbit of $λ$ .

2.1step 1.2step 1.3∎

Steps 1.2 and 1.3 give spanning and linear independence, respectively, so the $m_{λ}$ form an $R$ -basis.

Depends on

Used by

Power sums fail to span integrally in degree two Counterexample
Cauchy kernel through bidegree three Example
The five standard symmetric-function bases in degree three Example
Elementary and complete families freely generate the stable ring Theorem
Power-sum, complete, and Schur expansions of the Cauchy kernel Theorem
The monomial symmetric functions form the integral stable basis Theorem

Dependency tree · two levels

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Sources

D. Grinberg, An Introduction to Algebraic Combinatorics, Theorem 7.2.7 (standard reference, not scraped)

Monomial symmetric polynomials form an RR-basis of the symmetric-polynomial ring