Attachment to

T. DeGrand, Y. Shamir, and B. Svetitsky, SU(4) lattice gauge theory with decuplet fermions:..., Appendix A

Coefficients for U(n) projection (and its derivative): U(3) and U(4)

Set n  = 3 or 4 for SU(n)

n=4;

(Subscript[r, i]) = (Sqrt[(Subscript[g, i])]) -- use instead of (Subscript[g, i]) everywhere

SG=Table[(Subscript[r, i]),{i,0,n-1}]

set of symmetric polynomials -- 3 or 4 of them

SP=If[n==3,{u,v,w},{u,v,w,x}]

Vandermonde matrix ∂(Subscript[c, i])/∂(Subscript[g, j])

VM=Table[(Subscript[r, i])^(2j),{j,0,n-1},{i,0,n-1}];MatrixForm[VM]

Invert it.

VMI=Together[Inverse[VM]];MatrixForm[VMI]

Matrix for Eq. (A4)

M=Transpose[VM];MatrixForm[M]

RHS of Eq. (A4)

W=Table[1/(Subscript[r, i]),{i,0,n-1}]

(Subscript[f, i]), from solving Eq. (A4)

F=Together[Inverse[M].W];

Write (Subscript[f, i]) in terms of the symmetric polynomials. (Use CForm[] to get output that is usable in a C program.)

For[i=1,i≤n,i++,
NUM[i]=Simplify[SymmetricReduction[Numerator[Together[F[[i]]]],SG,SP]]; 
Print[NUM[i][[1]]]
]

DEN=Simplify[SymmetricReduction[Denominator[Together[F[[1]]]],SG,SP][[1]]]

(Check that the denominator is really a common denominator!)
Now for the chain rule: ∂(Subscript[f, i])/∂(Subscript[g, j])

DRV=Table[D[F[[i]],(Subscript[r, j])]/(2(Subscript[r, j])),{i,1,n},{j,0,n-1}];

Multiply by ∂(Subscript[g, j])/∂(Subscript[c, k]), giving the b  coefficients [Eq.(A15)]

B=Together[DRV.VMI];

Print in terms of symmetric polynomials.

For[i=1,i≤n,i++,
For[j=i,j≤n,j++,
NUMB[i,j]=Simplify[SymmetricReduction[Numerator[Together[B[[i]][[j]]]],SG,SP]]; Print[NUMB[i,j][[1]]]
]
]

DENB=SymmetricReduction[Denominator[Together[B[[1]][[1]]]],SG,SP][[1]]

Factor[DENB]