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

Synopsis

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,

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)