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# toS -- Schur (s-) basis representation

- Usage:
    `fs = toS f`
    `fs = toS(f,S)`

## Description

Given a symmetric function f, the function toS yields a representation of f as a linear combination of Schur functions.

If f is an element of a Symmetric ring R and the output Schur ring S is not specified, then the output fs is an element of the Schur ring associated to R (see schurRing).

    i1 : R = symmetricRing(QQ,4);

    i2 : fs = toS(e_1*h_2+p_3)

    o2 = 2s  + s
           3    1,1,1

    o2 : schurRing (QQ, s, 4)

    i3 : S = schurRing(s,2);

    i4 : toS(fs,S)

    o4 = 2s
           3

    o4 : S

This also works over tensor products of Symmetric/Schur rings.

    i5 : R = symmetricRing(4, EHPVariables => (a,b,c), SVariable => r);

    i6 : S = symmetricRing(R, 2, EHPVariables => (x,y,z), SVariable => s);

    i7 : T = symmetricRing(S, 3, SVariable => t);

    i8 : A = schurRing T;

    i9 : f = a_3*x_2*e_1 - b_1*z_2*p_3

    o9 = - b z p  + a x e
            1 2 3    3 2 1

    o9 : T

    i10 : toS f

    o10 = (- r s  + r s   )t  + (r s  - r s   )t    + (- r s  + r s   )t      +
              1 2    1 1,1  3     1 2    1 1,1  2,1       1 2    1 1,1  1,1,1
          -----------------------------------------------------------------------
          r     s   t
           1,1,1 1,1 1

    o10 : A

## See also

- toH -- Complete symmetric (h-) basis representation
- toE -- Elementary symmetric (e-) basis representation
- toP -- Power-sum (p-) basis representation

## Ways to use toS:

- `toS(RingElement)`
- `toS(RingElement,SchurRing)`

## For the programmer

The object toS is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SchurRings.m2:1968:0.