UC Berkeley

The study of integrable systems - also known as vertex models, ice models, or multiline queues - is a classical subject. Recently they have enjoyed an advent into the world of (non)symmetric polynomials, and have been generalized to colored vertex models and polyqueue tableaux. In this thesis, we study three topics in algebraic combinatorics - LLT polynomials, super ribbon functions, and domino tilings of the Aztec diamond - using several Yang-Baxter integrable colored vertex models.

Employing the ``white" vertices, we construct a certain class of partition functions that we show are equal to the LLT polynomials of Lascoux, Leclerc, and Thibon. We then give alternate proofs of many properties of these polynomials, including symmetry and a Cauchy identity, using the vertex model formalism.

Employing the ``white" and ``purple" vertices, we construct a certain class of partition functions that we show are essentially equal to the super ribbon functions of Lam. This construction generalizes our construction of partition functions equal to the LLT polynomials which employ just the ``white" vertices. We then give proofs of many properties of these polynomials, namely a Cauchy identity and generalizations of known identities for supersymmetric Schur polynomials, using the vertex model formalism.

Finally, we study k-tilings (k-tuples of domino tilings) of the Aztec diamond. We describe two models - one based on the ``purple" and ``gray" vertices and one based on the ``white" and ``pink" vertices - for assigning a weight to each k-tiling, depending on the number of dominos of certain types and the number of ``interactions" between the tilings. We compute the generating polynomials of the k-tilings in both models, as well as the arctic curves of the k-tilings in certain limits of the interaction strength.