arXiv subjects

Esther Banaian

Publications and source records attributed to Esther Banaian.

At least 19 recordsLinked to original sources

arXiv2026

Cluster algebraic interpretation of generalized Markov numbers and their matrixizations

Markov numbers, i.e. positive integers appearing in solutions to $x^2 + y^2 + z^2 = 3xyz$, can be viewed as specializations of cluster variables. The second author and Matsushita gave a generalization of the Markov equation, $x^2 + y^2 + z^2 + k_1yz + k_2xz + k_3xy = (3+k_1+k_2+k_3)xyz$, whose solutions can be viewed as specializations of cluster variables in generalized cluster algebras. We give two families of matrices in $SL(2,\mathbb{Z}[x_1^\pm,x_2^\pm,x_3^\pm])$ associated to these cluster structures. These matrix formulas relate to previous matrices appearing in the context of Markov numbers, including Cohn matrices and generalized Cohn matrices given by the second author, Maruyama, and Sato, as well as matrices appearing in the context of cluster algebras, including matrix formulas given by Kanatarcı Oğuz and Yıldırım. We provide a classification of the two families of matrices and exhibit an explicit family of each. The latter is done by realizing cluster variables in generalized Markov cluster algebras as weight-generating functions of order ideals in certain fence posets which are related to Christoffel words. An interesting observation is that these functions resemble Caldero-Chapoton functions for string modules, and a byproduct of our proofs is a new skein-like formula for such functions.

math.CO↗

arXiv2026

Varieties of chain complexes and mixed dimer covers

A quiver representation consists of a collection of vector spaces along with a set of arrows, which are linear maps between these spaces. In this work, we study quiver representations in equioriented type $A$ which are also chain complexes; that is, in which consecutive arrows compose to zero. We show that orbits of these representations under a change of basis action are in bijection with mixed dimer covers of a $2 \times n$ grid graph. The latter object can be endowed with a partial order which is a distributive lattice, and we show that the degeneration order on the orbits of chain complexes is a coarsening of this partial order. In addition, we use recent matrix formulae of Claussen and Ovenhouse to enumerate these orbits. This also computes the Kostant partition function applied to height-restricted, type $A$ roots. When the dimension vector is uniform, we discuss a correspondence with paths of a beam of light bouncing between glass plates and give an explicit generating function.

math.CO↗

arXiv2026

Type B c-Birkhoff polytopes are order polytopes

In a previous work, we defined (type A) c-Birkhoff polytopes and showed that they were unimodularly equivalent to order polytopes of heap posets. In this note we answer the question: What about type B?

math.CO↗

arXiv2026

Cluster Expansions from Punctured Orbifolds

We provide multiple combinatorial expansion formulas - in terms of snake graphs, labelled posets, matrices, and $T$-walks - for elements in generalized cluster algebras associated to arcs on punctured orbifolds and illustrate their equivalence. This work generalizes and unifies existing work on combinatorial expansion formulas from surfaces and unpunctured orbifolds.

math.CO↗

arXiv2026

Orderings of Generalized k-Markov Numbers

A $k$-Markov number is a positive integer that appears in a positive integral solution to the Diophantine equation $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. This equation was introduced by Gyoda and Matsushita. When $k =0$, this definition recovers that of ordinary Markov numbers. The set of $k$-Markov numbers can be indexed by pairs of coprime positive integers. There is a consistent way to label non-coprime pairs with positive integers as well, yielding a larger set of ``generalized $k$-Markov numbers.'' In this paper, we classify lines along which the generalized $k$-Markov numbers grow monotonically, extending work in the ordinary case by Lee-Li-Rabideau-Schiffler and by the second author. We find that, as $k$ grows, the $k$-Markov numbers are more likely to be monotonic along a random line. This gives evidence that a $k$-version of Frobenius' uniqueness conjecture, which has been proposed by Gyoda and Maruyama, could be true.

math.NT↗

arXiv2026

$c$-Birkhoff polytopes

In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan's question, in this paper we define a pattern-avoiding Birkhoff polytope called a $c$-Birkhoff polytope for each Coxeter element $c$ of the symmetric group. We then show that the $c$-Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the $c$-sorting word of the longest permutation. When $c=s_1s_2\dots s_{n}$, this result recovers an affirmative answer to Davis and Sagan's question. Another consequence of this result is that the normalized volume of the $c$-Birkhoff polytope is the number of the longest chains in the (type A) $c$-Cambrian lattice.

math.CO↗

arXiv2026

Orderings of k-Markov Numbers

The $k$-Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. When $k =0$, this recovers the ordinary Markov numbers. A long-standing question in the theory of Markov numbers is Frobenius's unicity conjecture, concerning whether every Markov number is the maximum in a unique solution triple. Aigner gave a series of weaker, related conjectures which were confirmed to be true by Lee, Li, Rabideau, and Schiffler using techniques from the theory of cluster algebras. We show here that $k$-Markov numbers also satisfy Aigner's conjectures.

math.NT↗

arXiv2026

Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras

We study a twisted version of Fraser, Lam, and Le's higher boundary measurement map, using face weights instead of edge weights, thereby providing Laurent polynomial expansions, in Plücker coordinates, for twisted web immanants for Grassmannians. In some small cases, Fraser, Lam, and Le observe a phenomenon they call "web duality'', where web immanants coincide with web invariants, and they conjecture that this duality corresponds to transposing the standard Young tableaux that index basis webs. We show that this duality continues to hold for a large set of $\text{SL}_3$ and $\text{SL}_4$ webs. Combining this with our twisted higher boundary measurement map, we recover and extend formulas of Elkin-Musiker-Wright for twists of certain cluster variables. We also provide evidence supporting conjectures of Fomin-Pylyavskyy as well as one by Cheung-Dechant-He-Heyes-Hirst-Li concerning classification of cluster variables of low Plücker degree in $\mathbb{C}[\text{Gr}(3,n)]$.

math.CO↗

arXiv2025

The Cyclic Sieving Phenomenon and frieze patterns

We exhibit two instances of the cyclic sieving phenomenon - one on dissections of a polygon of a fixed type and one on triangulations of a once-punctured polygon. We use these results to give refined enumerations of certain families of frieze patterns. We also give an interpretation of finite, positive integral frieze patterns fixed under nontrivial rotations as frieze patterns from a family of orbifolds and show that these are always unitary. Finally, we give a bijection between Holm-Jorgensen frieze patterns and p-Dyck paths, extending a recent construction of Canadas, Espinosa, Gaviria, and Rios, and discuss an induced rotation map on Dyck paths. Several conjectures and questions for future study are highlighted throughout the article.

math.CO↗

arXiv2025

Orbitmesy and promotion on self-dual posets

We introduce the notion of orbitmesy, which is related to homomesy, a central phenomenon in dynamical algebraic combinatorics. An orbit $O$ is said to be orbitmesic with respect to a statistic if the orbit's average statistic value is equal to the global average. We particularly focus on the action of promotion on increasing labelings of certain fence posets called zig-zag posets, and two statistics, the antipodal sum statistic and the total sum statistic. We classify all of the orbitmesic promotion orbits for the zig-zag poset with four elements. Along the way, we investigate how homomesy of one action can be used to find orbitmesic orbits for another action, for the same fixed statistic. We prove several general results which can be used to find infinite families of orbitmesic orbits for any self-dual poset.

math.CO↗

arXiv2025

The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order

For integral weights $λ$ and $μ$ of a classical simple Lie algebra $\mathfrak{g}$, Kostant's weight multiplicity formula gives the multiplicity of the weight $μ$ in the irreducible representation with highest weight $λ$, which we denote by $m(λ,μ)$. Kostant's weight multiplicity formula is an alternating sum over the Weyl group of the Lie algebra whose terms are determined via a vector partition function. The Weyl alternation set $\mathcal{A}(λ,μ)$ is the set of elements of the Weyl group that contribute nontrivially to the multiplicity $m(λ,μ)$. In this article, we prove that Weyl alternation sets are order ideals in the weak Bruhat order of the corresponding Weyl group. Specializing to the Lie algebra $\mathfrak{sl}_{r+1}(\mathbb{C})$, we give a complete characterization of the Weyl alternation sets $\mathcal{A}(\tildeα,μ)$, where $\tildeα$ is the highest root and $μ$ is a negative root, answering a question of Harry posed in 2024. We also provide some enumerative results that pave the way for our future work, where we aim to prove Harry's conjecture that the $q$-analog of Kostant's weight multiplicity formula is $m_q(\tildeα,μ)=q^{r+j-i+1}+q^{r+j-i}-q^{j-i+1}$ when $μ=-(α_i+α_{i+1}+\cdots+α_{j})$ is a negative root of $\mathfrak{sl}_{r+1}(\mathbb{C})$.

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arXiv2024

Skein relations for punctured surfaces

We investigate skein relations in cluster algebras from punctured surfaces, extending the work of Çanakçi-Schiffler and Musiker-Williams on unpunctured surfaces. Using a combinatorial expansion formula by O{ğ}uz-Yıldırım and Pilaud-Reading-Schroll, we provide explicit formulas for these relations. This work demonstrates that the punctured analogues of the bangle and bracelet functions form spanning sets for cluster algebras associated with a punctured surfaces. For surfaces with boundary and closed surfaces of genus 0, we further show that the bangles and bracelets form bases.

math.CO↗

arXiv2024

The $e$-positivity of the chromatic symmetric function for twinned paths and cycles

The operation of twinning a graph at a vertex was introduced by Foley, Hoàng, and Merkel (2019), who conjectured that twinning preserves $e$-positivity of the chromatic symmetric function. A counterexample to this conjecture was given by Li, Li, Wang, and Yang (2021). In this paper, we prove that $e$-positivity is preserved by the twinning operation on cycles, by giving an $e$-positive generating function for the chromatic symmetric function, as well as an $e$-positive recurrence. We derive similar $e$-positive generating functions and recurrences for twins of paths. Our methods make use of the important triple deletion formulas of Orellana and Scott (2014), as well as new symmetric function identities.

math.CO↗

arXiv2024

Friezes over $\mathbb Z[\sqrt{2}]$

A frieze on a polygon is a map from the diagonals of the polygon to an integral domain which respects the Ptolemy relation. Conway and Coxeter previously studied positive friezes over $\mathbb{Z}$ and showed that they are in bijection with triangulations of a polygon. We extend their work by studying friezes over $\mathbb Z[\sqrt{2}]$ and their relationships to dissections of polygons. We largely focus on the characterization of unitary friezes that arise from dissecting a polygon into triangles and quadrilaterals. We identify a family of dissections that give rise to unitary friezes and conjecture that this gives a complete classification of dissections which admit a unitary frieze.

math.CO↗

arXiv2024

An elaborate new proof of Cayley's formula

We construct a bijection between certain Deodhar components of a braid variety constructed from an affine Kac-Moody group of type $A_{n-1}$ and vertex-labeled trees on $n$ vertices. By an argument of Galashin, Lam, and Williams using Opdam's trace formula in the affine Hecke algebra and an identity due to Haglund, we obtain an elaborate new proof for the enumeration of the number of vertex-labeled trees on $n$ vertices.

math.CO↗

arXiv2024

A geometric model for semilinear locally gentle algebras

We consider certain generalizations of gentle algebras that we call semilinear locally gentle algebras. These rings are examples of semilinear clannish algebras as introduced by the second author and Crawley-Boevey. We generalise the notion of a nodal algebra from work of Burban and Drozd and prove that semilinear gentle algebras are nodal by adapting a theorem of Zembyk. We also provide a geometric realization of Zembyk's proof, which involves cutting the surface into simpler pieces in order to endow our locally gentle algebra with a semilinear structure. We then consider this surface glued back together, with the seams in place, and use it to give a geometric model for the finite-dimensional modules over the semilinear locally gentle algebra.

math.RT↗

arXiv2023

Snake Graphs for Graph LP Algebras

Graph LP algebras are a generalization of cluster algebras introduced by Lam and Pylyavskyy. We provide a combinatorial proof of positivity for certain cluster variables in these algebras. This proof uses a hypergraph generalization of snake graphs, a class of planar graphs which were used by Musiker, Schiffler, and Williams to prove positivity for cluster algebras from surfaces. These results extend those given in our previous paper, where we used a related combinatorial object known as a $T$-path.

math.CO↗

arXiv2023

Snake Graphs and Caldero-Chapoton Functions from Triangulated Orbifolds

Generalized cluster algebras from orbifolds were defined by Chekhov and Shapiro to give a combinatorial description of their Teichmüller spaces. One can also assign a gentle algebra to a triangulated orbifold, as in the work of Labardini-Fragoso and Mou. In this work, we show that the Caldero-Chapoton map and the snake graph expansion map agree for arcs in triangulated orbifolds and arc modules, and similarly for closed curves and certain band modules. As a consequence, we have a bijection between some indecomposable modules over a gentle algebra and cluster variables in a generalized cluster algebra where both algebras arise from the same triangulated orbifold.

math.RT↗