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The five standard symmetric-function bases in degree three

Example

Write every $m$ , $e$ , $h$ , $p$ and $s$ element in $Λ^{3}$ in the ordered monomial basis $(m_{3}, m_{21}, m_{111})$ , with their transition matrices and integral versus rational behavior.

Facts & Assumptions

Given: The degreewise stable ring, partition indexing, the finite rank-three orbit-sum conventions, and the stable integral and rational basis results.

[F1]

The degree- $d$ stable component is the inverse limit of the finite-rank degree- $d$ components under specialization of added variables to zero (The stable graded ring of symmetric functions).

[F2]

The partitions of $3$ are the finite weakly decreasing positive sequences of sum $3$ ; the empty partition has degree zero (Partitions, English diagrams, and conjugation).

[F3]

In finite rank, $e_{k}$ is the sum of products indexed by $k$ -element subsets of variables (The elementary symmetric polynomials $e_{0}, e_{1}, \dots, e_{n}$ ).

[F4]

In finite rank, $p_{k}$ is the sum of the $k$ th powers of the variables (Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$ ).

[F5]

In finite rank, $h_{k}$ is the sum of all monomials of total degree $k$ (Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$ ).

[F6]

The finite monomial symmetric polynomial is the sum of the distinct monomials whose exponent tuples lie in the variable-permutation orbit of its partition (Monomial symmetric polynomials indexed by partitions).

[F7]

In rank $N$ , the $m_{λ}$ indexed by partitions of length at most $N$ form a $Z$ -basis of the finite symmetric polynomials (Monomial symmetric polynomials form an $R$ -basis of the symmetric-polynomial ring).

[F8]

In each degree $d$ , the stable orbit sums $m_{λ}$ indexed by $λ ⊢ d$ form a $Z$ -basis of $Λ^{d}$ and project to their finite orbit sums (The monomial symmetric functions form the integral stable basis).

[F9]

In each degree, both ${e_{λ} : λ ⊢ d}$ and ${h_{λ} : λ ⊢ d}$ are $Z$ -bases of $Λ^{d}$ (Elementary and complete families freely generate the stable ring).

[F10]

For every integer $d \geq 0$ , the $p_{λ}$ indexed by $λ ⊢ d$ form a $Q$ -basis of $Λ_{Q}^{d}$ (Power sums form a rational but not integral stable basis).

[F11]

The Jacobi–Trudi and dual Jacobi–Trudi determinants express $s_{λ}$ using the $h$ and $e$ functions, with zero padding and the conventions $h_{0} = e_{0} = 1$ and $h_{k} = e_{k} = 0$ for $k < 0$ (Jacobi–Trudi and dual Jacobi–Trudi identities).

[F12]

For every $d$ , the stable Schur functions indexed by partitions of $d$ form a $Z$ -basis of $Λ^{d}$ (Schur functions form an orthonormal integral basis).

Verification

technique · direct

1.1F1F2F7F8

The partitions of $3$ are $(3), (2, 1), (1, 1, 1)$ , all of length at most $3$ . For every $N \geq 3$ , [F7] gives the rank- $N$ finite orbit-sum basis and [F8] identifies its labels with the stable monomial basis; the specialization maps preserve each labelled orbit sum. Thus projection to rank $3$ is an isomorphism in degree $3$ , so the finite calculations below determine the stable coordinates.

2.1F6step 1.1algebra

In three variables, the orbit sums are $m_{3} = \sum_{i} x_{i}^{3}$ , $m_{21} = \sum_{i \neq j} x_{i}^{2} x_{j}$ , and $m_{111} = x_{1} x_{2} x_{3}$ . Each monomial of type $(3)$ or $(2, 1)$ occurs once in $m_{2} m_{1}$ ; $m_{11} m_{1}$ counts each type- $(2, 1)$ monomial once and the all-distinct monomial three times; $m_{1}^{3}$ counts the patterns $(3), (2, 1), (1, 1, 1)$ with multiplicities $1, 3, 6$ .

$m_{2} m_{1} = m_{3} + m_{21}, m_{11} m_{1} = m_{21} + 3 m_{111}, m_{1}^{3} = m_{3} + 3 m_{21} + 6 m_{111} .$

3.1F3F9step 2.1algebra

The finite subset definition gives $e_{1} = m_{1}$ , $e_{2} = m_{11}$ , and $e_{3} = m_{111}$ . Multiplication using step 2.1 then gives the three degree-three elementary products.

$e_{3} = m_{111}, e_{21} = e_{2} e_{1} = m_{21} + 3 m_{111}, e_{111} = e_{1}^{3} = m_{3} + 3 m_{21} + 6 m_{111} .$

3.2F4F5F9step 2.1algebra

The complete functions list all degree- $k$ monomials, while the power sums are the sums of pure $k$ th powers. Thus $h_{1} = m_{1}$ , $h_{2} = m_{2} + m_{11}$ , and $h_{3} = m_{3} + m_{21} + m_{111}$ ; multiplying and using the orbit counts yields the remaining complete and power-sum products.

$h_{3} = m_{3} + m_{21} + m_{111}, h_{21} = h_{2} h_{1} = m_{3} + 2 m_{21} + 3 m_{111}, h_{111} = h_{1}^{3} = m_{3} + 3 m_{21} + 6 m_{111} .$

$p_{3} = m_{3}, p_{21} = p_{2} p_{1} = m_{3} + m_{21}, p_{111} = p_{1}^{3} = m_{3} + 3 m_{21} + 6 m_{111} .$

4.1F11step 3.1step 3.2algebra

Jacobi–Trudi at sizes one and two gives $s_{3} = h_{3}$ and $s_{21} = h_{2} h_{1} - h_{3}$ ; dual Jacobi–Trudi at size one gives $s_{111} = e_{3}$ . Substitution from steps 3.1 and 3.2 yields the three monomial coordinates.

$s_{3} = m_{3} + m_{21} + m_{111}, s_{21} = m_{21} + 2 m_{111}, s_{111} = m_{111} .$

5.1F8F9F10F12step 3.1step 3.2step 4.1algebra

With rows labelled $(3), (2, 1), (1, 1, 1)$ and columns ordered $(m_{3}, m_{21}, m_{111})$ , the rows of each matrix are the displayed coordinates from steps 3.1, 3.2, and 4.1. Their determinants show that the $m, e, h, s$ matrices are unimodular, while the $p$ matrix has nonzero determinant $6$ . The $p$ rows therefore form a rational basis; an explicit nonintegral coordinate for $h_{3}$ also shows failure of integral spanning.

$M^{(m)} = (\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}), E = (\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 3 \\ 1 & 3 & 6 \end{array}), H = (\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{array}) .$

$P = (\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 3 & 6 \end{array}), S = (\begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{array}), (\det M^{(m)}, \det E, \det H, \det P, \det S) = (1, - 1, 1, 6, 1) .$

$h_{3} = \frac{1}{3} p_{3} + \frac{1}{2} p_{21} + \frac{1}{6} p_{111} .$

6.1F1F2F6F7step 1.1step 5.1algebra∎

The degree-three claim has no empty-partition row because $\emptyset$ has degree zero; zero has the all-zero coordinate vector. The one-part label $(3)$ is the first row of every matrix, and repeated parts occur in the $(1, 1, 1)$ row. Degree $3$ and rank $3$ are the claimed degree endpoint and threshold rank. All orbit and product counts use finite sets, so no choice is made; no iff assertion occurs.

Depends on

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Sources

I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Chapter I §2, equations (2.1)–(2.8) and (2.10)–(2.14′), printed pp. 17–25; §3, equation (3.4), printed p. 41 (standard reference, not scraped)
Jeremy L. Martin, Lecture Notes on Algebraic Combinatorics, §§9.4–9.6, pp. 179–183, and §9.8, pp. 187–190 (standard reference, not scraped)