Standard monomial theory for flag algebras of GL(n) and Sp(2n)

Let G be the general linear group or the symplectic group over the complex numbers, and U be its maximal unipotent subgroup. We study standard monomial theory for the ring of regular functions on G/U, called the flag algebra, using the philosophy of Gröbner bases and SAGBI bases combined with classical invariant theory. From the realization of the flag algebra in a concrete polynomial setting, we obtain explicit standard monomial bases for irreducible representations. We also recreate known combinatorics of Young tableaux and Gelfand–Tsetlin patterns, and toric degenerations of flag varieties from the structure of leading monomials.