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# toP -- Power-sum (p-) basis representation

## Synopsis

• Usage:
fp = toP f
• Inputs:
• f, , element of a Symmetric or Schur ring
• Outputs:
• fp, , element of a Symmetric ring

## Description

Given a symmetric function f, the function toP yields a representation of f as a polynomial in the power-sum symmetric functions.

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

 i1 : R = symmetricRing 7; i2 : toP(h_3*e_3) 1 6 1 2 2 1 3 1 2 o2 = --p - -p p + -p p + -p 36 1 4 1 2 9 1 3 9 3 o2 : R i3 : S = schurRing(s,4) o3 = S o3 : SchurRing i4 : toP S_{3,2,1} 1 6 1 4 1 2 2 5 3 1 1 2 1 2 o4 = --p + --p p - -p p - --p p + -p p p - -p + -p p 72 1 12 1 2 8 1 2 18 1 3 6 1 2 3 9 3 4 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 : toP f 2 2 2 o10 = p + (2z + 2c 1)p + z + 2c z + c 1 1 1 1 1 1 1 1 o10 : A