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# scalarProduct(RingElement,RingElement) -- Standard scalar product of symmetric functions

- Function: scalarProduct
- Usage:
    `sp = scalarProduct(f1,f2)`
- Inputs:
  - f1, a ring element, element of a Symmetric Ring
  - f2, a ring element, element of a Symmetric Ring
- Outputs:
  - sp, a rational number,

## Description

Given symmetric functions f1 and f2, the method computes the standard pairing between f1 and f2.

    i1 : R = symmetricRing(QQ,5);

    i2 : S = schurRing R

    o2 = S

    o2 : SchurRing

    i3 : scalarProduct(h_5,p_5)

    o3 = 1

    o3 : QQ

    i4 : scalarProduct(S_{4,1},p_5)

    o4 = -1

    o4 : QQ

Indeed, the coefficients of s_5 and s_{4,1} in the s-basis expansion of h_5 are as computed above:

    i5 : R = symmetricRing(QQ,5);

    i6 : toS p_5

    o6 = s  - s    + s      - s        + s
          5    4,1    3,1,1    2,1,1,1    1,1,1,1,1

    o6 : schurRing (QQ, s, 5)

## Ways to use this method:

- scalarProduct(RingElement,RingElement) -- Standard scalar product of symmetric functions

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SchurRings.m2:2904:0.