A compact group A is called an amalgamation basis if, for every way of embedding A into compact groups B and C, there exist a compact group D and embeddings B \tp D and C \to D that agree on the image of A. Bergman in a 1987 paper studied the question of which groups can be amalgamation bases. A fundamental question that is still open is whether the circle group S^1 is an amalgamation basis in the category of compact Lie groups. Further reduction shows that it suffices to take B and C to be the special unitary groups. In our work, we focus on the case when B and C are the special unitary group in dimension three. We reformulate the amalgamation question into an algebraic question of constructing specific Schur-positive symmetric polynomials and use integer linear programming to compute the amalgamation. We conjecture that S^1 is an amalgamation basis based on our data. This is joint work with Michael Joswig, Mario Kummer, and Andreas Thom. Link to the seminar webpage for more information - https://www.wwu.de/AGKramer/index.php?name=TSSS23&menu=teach&lang=de