# toH -- Complete symmetric (h-) basis representation

## Synopsis

• Usage:
fh = toH f
• Inputs:
• f, , element of a Symmetric or Schur ring
• Outputs:
• fh, , element of a Symmetric ring

## Description

Given a symmetric function f, the function toH yields a representation of f as a polynomial in the complete symmetric functions.

If f is an element of a Schur ring S then the output fh is an element of the Symmetric ring associated to S (see symmetricRing).

 i1 : R = symmetricRing 7; i2 : toH(h_3*e_3) 3 2 o2 = h h - 2h h h + h 1 3 1 2 3 3 o2 : R i3 : S = schurRing(s,4) o3 = S o3 : SchurRing i4 : toH S_{3,2,1} 6 4 2 2 3 2 2 o4 = - h + 4h h - 3h h - 3h h + 3h h h - h + h h 1 1 2 1 2 1 3 1 2 3 3 1 4 o4 : QQ[e ..e , p ..p , h ..h ] 1 4 1 4 1 4

This also works over tensor products of Symmetric/Schur rings.

 i5 : R = schurRing(r, 4, EHPVariables => (a,b,c)); i6 : S = schurRing(R, s, 2, EHPVariables => (x,y,z)); i7 : T = schurRing(S, t, 3); i8 : A = symmetricRing T; i9 : f = (r_1+s_1+t_1)^2 o9 = t + t + (2r s + 2r s )t + (s + s + 2r s + (r + 2 1,1 () 1 1 () 1 2 1,1 1 1 2 ------------------------------------------------------------------------ r )s )t 1,1 () () o9 : T i10 : toH f 2 2 2 o10 = h + (2y + 2b )h + y + 2b y + b 1 1 1 1 1 1 1 1 o10 : A