hVariable -- Complete symmetric functions in a Symmetric ring

Description

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

i1 : R = symmetricRing(QQ,2,EHPVariables => (x,y,z));

i2 : R.hVariable 2

o2 = y
      2

o2 : R

Complete symmetric functions translate to one-row Schur functions via h_k = s_k; this is the other half of the e/h-duality.

i3 : R = symmetricRing(QQ,4);

i4 : toS (R.hVariable 3)

o4 = s
      3

o4 : schurRing (QQ, s, 4)

i5 : toS (R.eVariable 2 * R.hVariable 2)

o5 = s    + s
      3,1    2,1,1

o5 : schurRing (QQ, s, 4)

They interact cleanly with power sums: Newton's identities are realized by toE and toP, and here we convert a cube of h_2 into the power-sum basis.

i6 : R = symmetricRing(QQ,3);

i7 : toP ((R.hVariable 2)^2)

     1 4   1 2     1 2
o7 = -p  + -p p  + -p
     4 1   2 1 2   4 2

o7 : R

Complete symmetric functions are available over any coefficient ring.

i8 : Rt = symmetricRing(QQ[t],3);

i9 : toS (Rt.hVariable 2 + t * Rt.eVariable 2)

o9 = s  + t*s
      2      1,1

o9 : schurRing (QQ[t], s, 3)

For the programmer

The object hVariable is a symbol.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:5405:0.

hVariable -- Complete symmetric functions in a Symmetric ring

Description

See also

For the programmer