In this paper, we study various tropical analogues of objects from algebraic geometry. In particular, a tropical curve is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. The complete linear system |D| of a divisor D on a tropical curve analogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, |D| defines a map from the tropical curve to a tropical projective space, and the image can be extended to a parameterized tropical curve of degree equal to deg(D). The tropical convex hull of the image realizes the linear system |D| as an embedded polyhedral complex. We also show that curves for which the canonical divisor is not very ample are hyperelliptic. Also, we show that the Picard group of a rational tropical curve is a direct limit of critical groups of finite graphs converging to the curve.
We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type.
We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings a certain graph that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laruent polynomial expansions in terms of subgraphs of the constructed graph.
Let q be a power of a prime, and E be an elliptic curve defined over F_q. Such curves have a classical group structure, and one can form an infinite tower of groups by considering E over field extensions F_q^k for all k greater than or equal to 1. The critical group of a graph may be defined as the cokernel of L(G), the Laplacian matrix of G. In this paper, we compare elliptic curve groups with the critical groups of a certain family of graphs. This collection of critical groups also decomposes into towers of subgroups, and we highlight additional comparisons by using the Frobenius map of E over F_q.
This work lies across lies across three areas of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied the three above areas together. This link consists of a single formal power series with a multifaced interpretation. The deeper exploration of this link yielded results as well as methods for solving some numerical problems including the computation of certain Kronecker coefficients and enumeration of solutions to a Diophantine system corresponding to the hyperoctahedral group.
In this paper, we give a graph theoretic combinatorial interpretation for the cluster variables in most cluster algebras of finite type with a bipartite seed. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the A_n case, while providing a novel interpretation for the B_n, C_n, and D_n cases.
In 2002, Feigin and Veselov defined the space of m-quasiinvariants for any Coxeter group, building on earlier work of Chalykh and Veselov. Many properties of these spaces were proven in subsequent papers from this definition, however, an explicit computation of a basis was only done in the cases of dihedral groups and the symmetric group S_3. Felder and Veslov also computed the non-symmetric m-quasiinvariants of lowest degree for general S_n. In this paper, we provide a new characterization of the m-quasiinvariants of S_n, and use this to provide a basis for the isotypic component indexed by the partition [n-1,1]. This builds on a previous paper by the authors, in which we computed a basis for S_3 via combinatorial methods.
My Ph.D. Thesis is mostly a combination of the published papers ''Combinatorial aspects of elliptic curves'' and ''The critical groups of a family of graphs and elliptic curves over finite fields''.
Chapter 6 also contains the following; Additionally, the theory of critical groups allows us to re-interpret the group elements as the set of admissible words for a
primitive circuit in a specific deterministic finite automaton. As an application, we will then compare the zeta function of an elliptic curve and the zeta function of the corresponding cyclic
language.
Given an elliptic curve C, we study here N_k = #C(F_q^k), the number of points of C over the finite field F_q^k. This sequence of numbers, as k runs over positive integers, has numerous remarkable properties of a combinatorial flavor in addition to the usual number theoretical interpretations. In particular, we prove that N_k = -W_k(q,-N_1), where W_k(q,t) is a (q,t)-analogue of the number of spanning trees of the wheel graph. Additionally we develop a determinantal formula for Nk, where the eigenvalues can be explicitly written in terms of q, N_1, and roots of unity. We also discuss here a new sequence of bivariate polynomials related to the factorization of N_k, which we refer to as elliptic cyclotomic polynomials because of their various properties.
Fomin and Zelevinsky show that a certain two-parameter family of
rational recurrence relations, here called the (b,c) family,
possesses the Laurentness property: for all b,c, each term of the
(b,c) sequence can be expressed as a Laurent polynomial in the two
initial terms. In the case where the positive integers b,c satisfy
bc<4, the recurrence is related to the root systems of
finite-dimensional rank 2 Lie algebras; when bc>4, the recurrence
is related to Kac-Moody rank 2 Lie algebras of general type. Here
we investigate the borderline cases bc=4, corresponding to Kac-Moody
Lie algebras of affine type. In these cases, we show that the Laurent
polynomials arising from the recurence can be viewed as generating
functions that enumerate the perfect matchings of certain graphs. By
providing combinatorial interpretations of the individual coefficients of these Laurent polynomials, we establish their positivity.
Let s_ij represent a transposition in S_n. A polynomial P in Q[x_1, x_2, ..., x_n] is said to be m-quasiinvariant with respect to S_n if (x_i - x_j)^(2m+1) divides (1 - s_ij)P for all i and j inclusively between
1 and n. We describe a method for constructing a basis for the quotient of the m-quasiinvariants of S_3 by the ring of symmetric functions in 3 variables. This leads to the evaluation of certain binomial determinants that are interesting in their own right.
In this thesis, we will investigate the theory of cluster algebras, a recently created combinatorial theory
that is still developing. Cluster algebras are not only intrinsically interesting, but have useful applications to
the theory of Somos sequences and Laurent polynomials, generalized associahedra and many other fields. We will
concentrate on an axiomatic development of cluster algebras, motivating them by their aforementioned
applications. We will end with several open problems and conjectures. This exposition will utilize semisimple
Lie algebras and root systems; however, the necessary results from these mathematical areas will be
presented here and developed as needed. This should be accessible to anyone familiar with graph theory and
recurrence relations.