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How can I create a new basis of Quasisymmetric Functions?

I have defined a function G such that G(alpha) is in QSym = QuasisymmetricFunctions(K) for alpha in Compositions.

Is there a way to declare G as a basis of QSym, so that I can access functions like G(f) to give me the G-expansion of f where f is in QSym?

How can I create a new basis of Quasisymmetric Functions?

I have defined a function G such that G(alpha) is in QSym = QuasisymmetricFunctions(K) QuasiSymmetricFunctions(K) for alpha in Compositions.

Is there a way to declare G as a basis of QSym, so that I can access functions like G(f) to give me the G-expansion of f where f is in QSym? QSym?