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Cluster algebras, first defined by Fomin and Zelevinsky have proven to be a fascinating mathematical object, with connections in a wide range of mathematical fields as well as in physics. Many cluster algebras with desirable properties arise from triangulated surfaces. The goal of this thesis is to explore various generalizations of the narrative of cluster algebras from surfaces and to see which nice properties continue to hold. One fruitful direction is working with generalized cluster algebras from triangulated orbifolds. We give a combinatorial proof of positivity for such generalized cluster algebras by generalizing the snake graphs to the orbifold setting. We also show that our snake graphs give good expansions for generalized arcs and closed curves in an orbifold by showing the expansion formula satisfies the skein relations. We then turn to some applications of this snake graph expansion formula. The first is a generalization of the Markov numbers. Markov numbers can be seen as cluster variables from a certain cluster algebra with each initial cluster variable set to 1. We take a natural generalization of this cluster algebra and again specialize the initial variables to 1 to produce this new family of numbers. Since this generalized cluster algebra comes from an orbifold, we can explicitly describe the shape of the snake graphs corresponding to these numbers. The second application of the snake graph expansion formula is algebraic. Caldero and Chapoton gave a map which takes a module over an algebra and produces a Laurent polynomial; in some cases, the image of this map will be in a cluster algebra related to the original algebra. Using the snake graph expansion formula, we study the result of this map on a class of algebras determined by certain orbifolds. We also look to a different generalization by studying friezes and frieze patterns on surfaces with dissections, following work by Holm and Jorgensen. Caldero and Chapoton showed a connection between frieze patterns and cluster algebras. We study the space of all frieze patterns from dissections and give an algorithm to determine when an arbitrary frieze pattern comes from a dissection. We also give a combinatorial interpretation of the entries.

Cluster algebras are a class of commutative rings with a remarkable combinatorial structure, introduced by Fomin and Zelevinsky. A cluster algebra has a distinguished set of generators, called cluster variables, which are grouped together into overlapping subsets called seeds. This dissertation is concerned with the cluster algebra structure of coordinate rings of open positroid varieties in the Grassmannian. Open positroid varieties are projections of open Richardson varieties from the full flag variety to the Grassmannian. They were studied first by Lusztig and Rietsch in the context of total positivity, and then by Knutson--Lam--Speyer, who connected them to the combinatorics of the totally nonnegative Grassmannian developed by Postnikov. Open positroid varieties are smooth, irreducible, and stratify the Grassmannian; open Schubert varieties are a special case. Seminal work of Scott established that the homogeneous coordinate ring of the Grassmannian is a cluster algebra, and moreover that Postnikov's plabic graphs for the Grassmannian give seeds for this cluster algebra. Postnikov defined plabic graphs not just for the Grassmannian but for all positroid varieties. Accordingly, experts long believed that the coordinate ring of any open positroid variety is also a cluster algebra, with seeds given by plabic graphs. In Chapter 3, which is joint work with Khrystyna Serhiyenko and Lauren Williams, we prove this in the case of open Schubert varieties in the Grassmannian. Work of Leclerc on Richardson varieties in the full flag variety implies that the coordinate rings of these varieties are cluster algebras, but does not give any explicit descriptions of seeds. We show that Postnikov's plabic graphs give seeds in this cluster algebra. For skew Schubert varieties, we show that Leclerc's cluster algebra is given by relabeled plabic graphs, whose boundary vertices are permuted. Shortly following my work with Serhiyenko and Williams, Galashin--Lam showed that Postnikov's graphs give a cluster algebra structure on coordinate rings of arbitrary positroid varieties using similar methods. In Chapter 4, which is joint work with Chris Fraser, we expand on this result to show that positroid varieties admit a number of different cluster structures, with seeds given by relabeled plabic graphs. Along the way, we show that many positroid varieties are isomorphic, using a permuted version of the Muller--Speyer twist map. We conjecture that all of these distinct cluster structures differ only by rescaling, and prove this conjecture for open Schubert varieties. This enlarges the class of combinatorially well-understood seeds for positroid varieties, which provides additional tools to further study the cluster structure on positroid varieties.

The theory of total positivity was classically concerned with totally nonnegative matrices (matrices with all nonnegative minors). These matrices appear in many varied areas of mathematics including probability, asymptotic representation theory, algebraic and enumerative combinatorics, and linear algebra. However, motivated by surprising positivity properties of Lusztig's canonical bases for quantum groups, the field of total positivity has more recently grown to include other totally nonnegative varieties. We first discuss results regarding immanants on the space of k-positive matrices (matrices where all minors of size up to k are positive). Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan--Lustzig immanants, defined by Rhoades and Skandera, are of particular interest, as elements of the dual canonical basis of the coordinate ring of GL_n(C) can be expressed as Kazhdan--Lustzig immanants. Results of Stembridge, Haiman, and Rhoades--Skandera show that Kazhdan--Lustzig immanants are nonnegative on totally nonnegative matrices. Here, we give conditions on v in S_n so that the Kazhdan-Lusztig immanant corresponding to v is positive on k-positive matrices. We then consider a space that arises from the study of totally nonnegative Grassmannians. Postnikov's plabic graphs in a disk are used to parametrize these spaces. In recent years plabic graphs have found numerous applications in math and physics. One of the key features of the theory is the fact that if a plabic graph is reduced, the face weights can be uniquely recovered from boundary measurements. On surfaces more complicated than a disk this property is lost. In this thesis, we undertake a comprehensive study of a certain semi-local transformation of weights for plabic networks on a cylinder that preserve boundary measurements. We call this a plabic R-matrix. We show that plabic R-matrices have underlying cluster algebra structure, generalizing work of Inoue--Lam--Pylyavskyy. Special cases of transformations we consider include geometric R-matrices appearing in Berenstein--Kazhdan theory of geometric crystals, and also certain transformations appearing in a recent work of Goncharov--Shen.

We describe the action of the admissible groups on the scattering diagrams for cluster varieties associated to positive log Calabi-Yau surfaces. By introducing the technique of folding to scattering diagrams, we give a functorial construction of locally closed sub-cluster varieties. In cases of universal families of positive log Calabi-Yau surfaces, the folding procedure allows us to study the full theta bases and scattering diagrams of degenerate subfamilies, overcoming the technical difficulty of infinite wall-crossings that appear when we restrict to degenerate loci. As a byproduct, we construct examples of cluster varieties with non-equivalent cluster structures.

This thesis shows that the coordinate ring of the open Richardson variety in type A has the structure of an upper cluster algebra. We begin by deriving in- equalities related to northwest rank conditions on a general collection of columns of a matrix. In the special case where w is the unipeak expression for

The hexahedron recurrence was introduced by R. Kenyon and R. Pemantle in the study of the double-dimer model in statistical mechanics. It describes a relationship among certain minors of a square matrix. This recurrence is closely related to the Kashaev equation, which has its roots in the Ising model and in the study of relations among principal minors of a symmetric matrix. Certain solutions of the hexahedron recurrence restrict to solutions of the Kashaev equation. We characterize the solutions of the Kashaev equation that can be obtained by such a restriction. This characterization leads to new results about principal minors of symmetric matrices. We describe and study other recurrences whose behavior is similar to that of the Kashaev equation and hexahedron recurrence. These include equations that appear in the study of s-holomorphicity, as well as other recurrences which, like the hexahedron recurrence, can be related to cluster algebras.

In this thesis we address several questions involving quantum groups, quantum cluster algebras, and integrable systems, and provide some novel examples of the very useful interplay between these subjects. In the Chapter 2, we introduce the classical reflection equation (CRE), and give a construction of integrable Hamiltonian systems on $G/K$, where $G$ is a quasitriangular Poisson Lie group and $K$ is a Lie subgroup arising as the fixed point set of a group automorphism $\sigma$ of $G$ satisfying the CRE. As an application, we provide a detailed treatment of the algebraic integrability of the XXZ spin chain with reflecting boundary conditions. In Chapter 3, we study doubles of Hopf algebras and dual pairs of quantum moment maps. For any Hopf algebra $A$, we construct a natural generalization of the (quantized) Grothendieck-Springer resolution; the standard resolution corresponds to taking $A$ a quantum Borel subalgebra. In this latter case, we apply the general construction to yield an algebra embedding of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ into a quantum torus algebra which is a central extension of the quantum coordinate ring of the reduced big double Bruhat cell in the corresponding simply-connected group $G$. Chapter 4 gives an alternative geometric description of this quantum torus embedding. Namely, we construct an embedding of $U_q(\mathfrak{sl}_n)$ into a quantum cluster chart on a quantum character variety associated to a marked punctured disk. We obtain a description of the coproduct of $U_q(\mathfrak{sl}_{n})$ in terms of a quantum character variety associated to the marked twice punctured disk, and express the action of the $R$-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the algebra automorphism of $U_q(\mathfrak{sl}_{n})^{\otimes 2}$ given by conjugation by the $R$-matrix as an explicit sequence of cluster mutations, and derive a refined factorization of the $R$-matrix into quantum dilogarithms of cluster monomials. We conclude by mentioning some applications of our cluster realization of quantum groups to the decomposition of tensor products of positive representations, and the construction of a modular functor from quantum higher Teichmuller theory.

We construct a periodic infinite frieze using a class of peripheral elements of a cluster algebra of type D or affine A. We discover new symmetries and formulas relating the entries of this frieze and bracelet elements. We also present a correspondence between Broline, Crowe and Isaacs’s classical matching tuples and various recent interpretations of elements of cluster algebras from surfaces. We extend a T-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type D. We further generalize our work and present T-path formulas for tagged arcs with one or two notchings on a marked surface with punctures.

Let V be a k-dimensional complex vector space. The Plucker ring of polynomial SL(V) invariants of a collection of n vectors in V can be alternatively described as the homogeneous coordinate ring of the Grassmannian Gr(k,n). In 2003, using combinatorial tools developed by A. Postnikov, J. Scott showed that the Plucker ring carries a cluster algebra structure. Over the ensuing decade, this has become one of the central examples of cluster algebra theory. In the 1930s, H. Weyl described the structure of the "mixed" Plucker ring, the ring of polynomial SL(V) invariants of a collection of n vectors in V and m covectors in V*. In this thesis, we generalize Scott's construction and Postnikov's combinatorics to this more general setting. In particular, we show that each mixed Plucker ring carries a natural cluster algebra structure, which was previously established by S. Fomin and P. Pylyavskyy only in the case k=3. We also introduce mixed weak separation as a combinatorial condition for compatibility of cluster variables in this cluster structure and prove that maximal collections of weakly separated mixed subsets satisfy a purity result, a property proved in the Grassmannian case by Oh, Postnikov, and Speyer.