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A nonzero stable Schur function can vanish in too few variables

Statement

For every integer $N \geq 0$ , let $λ = (1^{N + 1})$ be the partition with $N + 1$ parts equal to $1$ . The stable Schur function $s_{λ}$ is nonzero in $Λ$ , but its rank- $N$ specialization is zero. Thus a nonzero stable symmetric function can vanish after specialization to fewer variables than the length of its indexing partition.

Facts & Assumptions

Given: The stable ring's coordinate projections, the partition convention, the stable Schur and elementary sequences, Jacobi–Trudi, the finite elementary polynomial, and the integral Schur basis.

[F1]

Each $Λ^{d}$ is an inverse limit of finite-rank degree- $d$ polynomial spaces, and its rank- $N$ coordinate is specialization to $N$ variables (The stable graded ring of symmetric functions).

[F2]

$λ = (1^{N + 1})$ is a partition of $N + 1$ with length $N + 1$ ; partition size, length, and conjugation use the usual Young-diagram convention (Partitions, English diagrams, and conjugation).

[F3]

The stable $e_{r}$ are compatible sequences obtained from the finite elementary polynomials by setting added variables to zero (Elementary and complete families freely generate the stable ring).

[F4]

In rank $N$ , $e_{k}$ is the sum over $k$ -element subsets of ${1, \dots, N}$ , and $e_{k} = 0$ when $k > N$ (The elementary symmetric polynomials $e_{0}, e_{1}, \dots, e_{n}$ ).

[F5]

The dual Jacobi–Trudi identity expresses $s_{λ}$ as $\det (e_{λ_{i}^{'} - i + j})$ for any allowed determinant size, including the empty-partition convention (Jacobi–Trudi and dual Jacobi–Trudi identities).

[F6]

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

[F7]

Each $s_{λ}$ is the compatible stable sequence of its finite bialternant polynomials (Stable Schur functions from bialternants).

Proof

technique · direct

1.1F2F6F7algebra

Fix $N \geq 0$ and take $λ = (1^{N + 1})$ . By [F2] it is a partition of degree $N + 1$ , and [F7] identifies $s_{λ}$ as its stable Schur element. By [F6], this element is one vector in a $Z$ -basis of $Λ^{N + 1}$ , so $s_{λ} \neq 0$ .

1.2F2F5algebra

The conjugate partition is $λ^{'} = (N + 1)$ , so the dual Jacobi–Trudi identity [F5] with determinant size $c = 1$ gives $s_{(1^{N + 1})} = e_{N + 1}$ in $Λ$ .

1.3F1F3algebra

By [F1] and [F3], the rank- $N$ coordinate of this stable $e_{N + 1}$ is the finite elementary polynomial $e_{N + 1} (x_{1}, \dots, x_{N})$ .

2.1F1F4step 1.2step 1.3algebra∎

By [F4], this polynomial is a sum indexed by the $(N + 1)$ -element subsets of an $N$ -element set; there are no such subsets, including when $N = 0$ , so the sum is zero. Thus every stated rank- $N$ specialization vanishes although the stable element is nonzero, and the coordinate projection has a nontrivial kernel.

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Sources

I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Chapter I §2, equation (2.2), printed pp. 19–20; §3, equations (3.1)–(3.5), printed pp. 40–42 (standard reference, not scraped)
Jeremy L. Martin, Lecture Notes on Algebraic Combinatorics, §9.13, printed pp. 204–207 (standard reference, not scraped)