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# toS -- Schur (s-) basis representation

## Synopsis

• Usage:
fs = toS f
fs = toS(f,S)

## Description

Given a symmetric function f, the function toS yields a representation of f as a linear combination of Schur functions.

If f is an element of a Symmetric ring R and the output Schur ring S is not specified, then the output fs is an element of the Schur ring associated to R (see schurRing).

 i1 : R = symmetricRing(QQ,4); i2 : fs = toS(e_1*h_2+p_3) o2 = 2s + s 3 1,1,1 o2 : schurRing (QQ, s, 4) i3 : S = schurRing(s,2); i4 : toS(fs,S) o4 = 2s 3 o4 : S

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

 i5 : R = symmetricRing(4, EHPVariables => (a,b,c), SVariable => r); i6 : S = symmetricRing(R, 2, EHPVariables => (x,y,z), SVariable => s); i7 : T = symmetricRing(S, 3, SVariable => t); i8 : A = schurRing T; i9 : f = a_3*x_2*e_1 - b_1*z_2*p_3 o9 = - b z p + a x e 1 2 3 3 2 1 o9 : T i10 : toS f o10 = (- r s + r s )t + (r s - r s )t + (- r s + r s )t + 1 2 1 1,1 3 1 2 1 1,1 2,1 1 2 1 1,1 1,1,1 ----------------------------------------------------------------------- r s t 1,1,1 1,1 1 o10 : A