symmetricFunction -- Converts class function to symmetric function

Usage:

f = symmetricFunction(ch,S)
Inputs:
- ch, a Class function,
- S, a ring, a Symmetric or Schur ring
Outputs:
- f, a ring element, element of a Symmetric or Schur ring

Description

Given a virtual character cF of a symmetric group, and given a Symmetric ring S, the method computes the corresponding symmetric function as an element of S. The conversion uses the Frobenius characteristic map: the regular representation (n! on the identity, zero elsewhere) maps to n! e_1^n in the e-basis, equivalently to sum_\lambda f^\lambda s_\lambda in the s-basis.

i1 : S = symmetricRing(QQ,4);

i2 : cF = new ClassFunction from {{1,1,1,1}=>24};

i3 : symmetricFunction(cF,S)

      4
o3 = p
      1

o3 : S

i4 : symmetricFunction(cF,schurRing S)

o4 = s  + 3s    + 2s    + 3s      + s
      4     3,1     2,2     2,1,1    1,1,1,1

o4 : schurRing (QQ, s, 4)

The standard representation of S_4 has character given by classFunction(s_{3,1}); passing it through symmetricFunction recovers s_{3,1} on the SchurRing side:

i5 : R = symmetricRing(QQ,4);

i6 : Sch = schurRing R;

i7 : stdCh = classFunction(jacobiTrudi({3,1},R));

i8 : symmetricFunction(stdCh, Sch)

o8 = s
      3,1

o8 : Sch

Composing classFunction with symmetricFunction is the identity on the S_n-side of the Frobenius correspondence -- roundtripping a class function through the symmetric function ring returns it unchanged:

i9 : sF = symmetricFunction(stdCh, R);

i10 : classFunction sF == stdCh

o10 = true

The trivial character of S_5 has Frobenius image h_5; the sign character maps to e_5:

i11 : R5 = symmetricRing(QQ,5);

i12 : symmetricFunction(classFunction{5}, R5)

       1  5    1 3     1   2   1 2     1       1       1
o12 = ---p  + --p p  + -p p  + -p p  + -p p  + -p p  + -p
      120 1   12 1 2   8 1 2   6 1 3   6 2 3   4 1 4   5 5

o12 : R5

i13 : symmetricFunction(classFunction{1,1,1,1,1}, R5)

       1  5    1 3     1   2   1 2     1       1       1
o13 = ---p  - --p p  + -p p  + -p p  - -p p  - -p p  + -p
      120 1   12 1 2   8 1 2   6 1 3   6 2 3   4 1 4   5 5

o13 : R5

Ways to use symmetricFunction:

symmetricFunction(ClassFunction,Ring)

For the programmer

The object symmetricFunction is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:7149:0.

symmetricFunction -- Converts class function to symmetric function

Description

See also

Ways to use symmetricFunction:

For the programmer