arXiv subjects
Publications and source records attributed to Sergey Fomin.
A cyclically ordered quiver is a quiver endowed with an additional structure of a cyclic ordering of its vertices. This structure, which naturally arises in many important applications, gives rise to new powerful mutation invariants.
We show that various classical theorems of real/complex linear incidence geometry, such as the theorems of Pappus, Desargues, M\"obius, and so on, can be interpreted as special cases of a single "master theorem" that involves an arbitrary tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing the known ones.
A mutation cycle is a cycle in a graph whose vertices are labeled by the quivers in a given mutation class and whose edges correspond to single mutations. For any fixed $n\ge 4$, we describe arbitrarily long mutation cycles involving $n$-vertex quivers. Each of these mutation cycles allows for an arbitrary choice of $n \choose 2$ positive integer parameters. None of the mutation cycles we construct can be paved by short mutation cycles.
We describe and investigate a connection between the topology of isolated singularities of plane curves and the mutation equivalence, in the sense of cluster algebra theory, of the quivers associated with their morsifications.
This is a preliminary draft of Chapters 4-5 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. This installment contains: Chapter 4. New patterns from old Chapter 5. Finite type classification
This is a preliminary draft of Chapter 6 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. This installment contains: Chapter 6. Cluster structures in commutative rings
This is a preliminary draft of Chapter 7 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. Chapter 6 has been posted as arXiv:2008.09189. This installment contains: Chapter 7. Plabic graphs
We show that for any positive integer $n$, there exists a quiver $Q$ with $O(n^2)$ vertices and $O(n^2)$ edges such that any quiver on $n$ vertices is a full subquiver of a quiver mutation equivalent to $Q$. We generalize this statement to skew-symmetrizable matrices and obtain other related results. In particular, we show that any quiver is a full subquiver of a quiver mutation equivalent to a quiver of a plabic graph.
Let $G$ be a semisimple simply connected complex algebraic group. Let $U$ be the unipotent radical of a Borel subgroup in $G$. We describe the coordinate rings of $U$ (resp., $G/U$, $G$) in terms of two (resp., four, eight) birational charts introduced in [L94, L19] in connection with the study of total positivity.
We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve $C$ is expressive if (a) each irreducible component of $C$ can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of $C$ in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of $C$ in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)-(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.
Motivated by computational geometry of point configurations on the Euclidean plane, and by the theory of cluster algebras of type A, we introduce and study Heronian friezes, the Euclidean analogues of Coxeter's frieze patterns. We prove that a generic Heronian frieze possesses the glide symmetry (hence is periodic), and establish the appropriate version of the Laurent phenomenon. For a closely related family of Cayley-Menger friezes, we identify an algebraic condition of coherence, which all friezes of geometric origin satisfy. This yields an unambiguous propagation rule for coherent Cayley-Menger friezes, as well as the corresponding periodicity results.
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations, and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from our previous paper, removing unnecessary assumptions on the surface.
Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that when the number of variables is fixed, the semiring complexity of a Schur polynomial $s_λ$ is $O(log(λ_1))$; here $λ_1$ is the largest part of the partition $λ$.
We investigate the following surprisingly widespread phenomenon which we call The Rule of Three: in order for a particular kind of commutation relation to hold for subsequences of elements of a ring labeled by any subset of indices, it is enough that these relations hold for subsets of size one, two, and three.
This is a preliminary draft of Chapters 1-3 of our forthcoming textbook "Introduction to Cluster Algebras." This installment contains: Chapter 1. Total positivity Chapter 2. Mutations of quivers and matrices Chapter 3. Clusters and seeds
The machinery of noncommutative Schur functions provides a general tool for obtaining Schur expansions for combinatorially defined symmetric functions. We extend this approach to a wider class of symmetric functions, explore its strengths and limitations, and obtain new results on Schur positivity. We introduce combinatorial gadgets called switchboards, an adaptation of the D graphs of S. Assaf, and show how symmetric functions associated to them (which include LLT, Macdonald, Stanley, and stable Grothendieck polynomials) fit into the noncommutative Schur functions approach. This extends earlier work by T. Lam, and by C. Greene and the second author, and provides new tools for obtaining combinatorial formulas for Schur expansions of LLT polynomials. This paper can be regarded as a "prequel" to (and, partly, a review of) arXiv:1411.3624, arXiv:1411.3646, and arXiv:1510.00644.
The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. We show that when V is 3-dimensional, each of these rings carries a natural cluster algebra structure (typically, many of them) whose cluster variables include Weyl's generators. We describe and explore these cluster structures using the combinatorial machinery of tensor diagrams. A key role is played by the web bases introduced by G.Kuperberg.
Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division. We use cluster transformations to design efficient subtraction-free algorithms for computing Schur functions and their skew, double, and supersymmetric analogues, thereby generalizing earlier results by P. Koev. We develop such algorithms for computing generating functions of spanning trees, both directed and undirected. A comparison to the lower bound due to M. Jerrum and M. Snir shows that in subtraction-free computations, "division can be exponentially powerful." Finally, we give a simple example where the gap between ordinary and subtraction-free complexity is exponential.