pVariable -- Power-sum symmetric functions in a Symmetric ring

Description

For a Symmetric ring R of dimension n, R.pVariable is a function which assigns to each index 1\leq i\leq n the i-th power-sum symmetric function. If i is outside the given bounds, an error is returned.

i1 : R = symmetricRing(QQ,4);

i2 : R.pVariable 2

o2 = p
      2

o2 : R

Power sums are an algebraically independent generating set for the ring of symmetric functions over QQ, and products of p_i's expand into the Schur basis using the character table of the symmetric group.

i3 : R = symmetricRing(QQ,5);

i4 : toS ((R.pVariable 2)^3)

o4 = s  - s    + 3s    - 2s      - 3s    + 2s        + 3s      - 3s        +
      6    5,1     4,2     4,1,1     3,3     3,1,1,1     2,2,2     2,2,1,1  
     ------------------------------------------------------------------------
     s
      2,1,1,1,1

o4 : schurRing (QQ, s, 5)

The same variable in a Symmetric ring with GroupActing => "Sn" represents a virtual character of a symmetric group, and conversion to elementary and complete generators is handled by toE and toH.

i5 : Rsn = symmetricRing(QQ,4,GroupActing => "Sn");

i6 : toE (Rsn.pVariable 2)

      2
o6 = e  - 2e
      1     2

o6 : Rsn

i7 : toH (Rsn.pVariable 3)

      3
o7 = h  - 3h h  + 3h
      1     1 2     3

o7 : Rsn

Power sums are also useful as the natural input to plethysm.

i8 : R = symmetricRing(QQ,4);

i9 : toS plethysm(R.pVariable 2, R.hVariable 2)

o9 = s  - s    + s
      4    3,1    2,2

o9 : schurRing (QQ, s, 4)

For the programmer

The object pVariable is a symbol.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:5457:0.

pVariable -- Power-sum symmetric functions in a Symmetric ring

Description

See also

For the programmer