 Symmetric-function involution

ID: symmetric-function-involution

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Symmetric-function involution by

Codex  0 2026-10-14

The Fundamental theorem of symmetric polynomials gives freely generated elementary functions, so sending each

e_{r}

to

h_{r}

defines an algebra homomorphism. For

E (t) = \sum e_{r} t^{r}

and

H (t) = \sum h_{r} t^{r}

,

E (- t) H (t) = 1

. Applying the homomorphism shows

ω (H (t)) = E (t)

, hence

ω^{2} = 1

. Taking logarithms of

H

gives the displayed action on power sums. It preserves the Hall inner product of symmetric functions because each power-sum basis vector acquires only a sign.

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