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## `examples::SymmetricFunctions` -- symmetric functions

#### Introduction

`examples::SymmetricFunctions``(R)` creates a domain for the symmetric functions over the ground ring R.

#### Details

#### Entries

powersum
p

The domain of symmetric functions expanded on the powersums basis

Schur
s

The domain of symmetric functions expanded on the Schur basis

elementary
e

The domain of symmetric functions expanded on the elementary basis

homogeneous
h

The domain of symmetric functions expanded on the complete basis

monomial
m

The domain of symmetric functions expanded on the monomial basis

#### Method antipod: [DocState=beta] antipod of symmetric functions

- `antipod(symmetric function f)`

- Returns the antipod of the symmetric function `f`; the result need not be expressed in the same basis as `f`.

#### Method omega: [DocState=beta] omega operator on symmetric function

- `omega(symmetric function f)`

- Returns the image of the symmetric function `f` by the `omega` operator; the result need not be expressed in the same basis as `f`.

#### Method fromPoly: [DocState=beta] conversion from symmetric polynomials

- `fromPoly(symmetric polynomial (in `DOM_POLY`) f)`

- Express the symmetric polynomial f in the monomial basis.
- The polynomial f is assumed to be symmetric; no check is done.

See the `demonstration` for examples.

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MuPAD Combinat, an open source algebraic combinatorics package