The study of objects from number theory such as metaplectic Whittaker functions has led to surprising applications of combinatorial representation theory. Classical Whittaker functions can be expressed in terms of symmetric polynomials, such as Schur polynomials via the Casselmann-Shalika formula. Tokuyama's theorem is an identity that links Schur polynomials to highest-weight crystals, a symmetric structure that has interesting combinatorial parameterisations. In this talk, we will discuss constructions of metaplectic Iwahori-Whittaker functions inspired by Tokuyama's theorem. These can be related and better understood using representations of the Iwahori-Hecke algebra. This theory carries over to the infinite-dimensional setting, and connects with work on double affine Hecke algebras, where several intriguing open questions remain.