arXiv subjects

Elise Catania

Publications and source records attributed to Elise Catania.

4 recordsLinked to original sources

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

Identifying Orbit Lengths for Promotion

In this work we study Schützenberger's promotion operator on standard Young tableaux via a corresponding graphical construction known as $m-$diagrams. In particular, we prove that certain internal structures of SYT are preserved under promotion and correspond to distinct components of $m-$diagrams. By treating these structures as atomic parts of the $m-$diagram, we provide a simple algorithm for computing the promotion orbit length of rectangular SYT. We conclude the paper by applying our results to (column) semi-standard Young tableaux and prove a formula for the promotion orbit lengths of rectangular (column) SSYT.

math.CO↗

arXiv2025

A Toric Analogue for Greene's Rational Function of a Poset

Given a finite poset, Greene introduced a rational function obtained by summing certain rational functions over the linear extensions of the poset. This function has interesting interpretations, and for certain families of posets, it simplifies surprisingly. In particular, Greene evaluated this rational function for strongly planar posets in his work on the Murnaghan-Nakayama formula. In 2012, Develin, Macauley, and Reiner introduced toric posets, which combinatorially are equivalence classes of posets (or rather acyclic quivers) under the operation of flipping maximum elements into minimum elements and vice versa. In this work, we introduce a toric analogue of Greene's rational function for toric posets, and study its properties. In addition, we use toric posets to show that the Kleiss-Kuijf relations, which appear in scattering amplitudes, are equivalent to a specific instance of Greene's evaluation of his rational function for strongly planar posets. Also in this work, we give an algorithm for finding the set of toric total extensions of a toric poset.

math.CO↗

arXiv2021

Dihedral Linking Invariants

A Fox p-colored knot $K$ in $S^3$ gives rise to a corresponding $p$-fold dihedral branched cover $M$ of $S^3$ along $K$. The pre-image of the knot $K$ under the covering map is a $\dfrac{p+1}{2}$-component link $L$ in $M$, and the set of pairwise linking numbers of the components of $L$ is an invariant of $K$. This powerful invariant played a key role in the development of early knot tables, and appears in formulas for many other important knot and manifold invariants. We give an algorithm for computing this invariant for all odd $p$, generalizing an algorithm of Perko. We then extend this algorithm to compute linking numbers of arbitrary curves in a $p$-fold dihedral branched cover of $S^3$ along $K$. As an application, we compute Kjuchukova's ribbon obstruction $\Xi_p$ using a method of the first author and Kjuchukova. We also tabulate the dihedral linking invariant for all $p$-colorings of prime knots of crossing number less than or equal to 13, with $p\geq 3$ prime. Finally, we demonstrate the strength of the dihedral linking invariant by comparing it to several polynomial invariants. For example, the dihedral linking invariant distinguishes more than 98% of the 1183 prime non-mutant knot pairs with the same Fox coloring invariant and the same HOMFLY-PT polynomial through 13 crossings.

math.GT↗