arXiv subjects

Jesse Kim

Publications and source records attributed to Jesse Kim.

11 recordsLinked to original sources

arXiv2026

Cyclic Sieving for Staircase Plane Partitions via Crystals and Electrical Networks

We prove a cyclic sieving result for the action of promotion on the staircase plane partitions of height two. Our proof has two major algebraic inputs: an interpretation of this promotion action in terms of tensor powers of the spin crystal that was recently studied by Pappe--Pfannerer--Schilling--Simone, and the bush basis of the degree two part of the coordinate ring of the space of electrical networks that was recently introduced by Gao--Lam--Xu. Moreover, we explain how the existence of an electrical canonical basis in all degrees would yield cyclic sieving for promotion of staircase plane partitions of all heights.

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arXiv2025

Odd Shifted Parking Functions

Stanley recently introduced the shifted parking function symmetric function $SH_n$, which is the shiftification of Haiman's parking function symmetric function $PF_n$. The function $SH_n$ lives in the subalgebra of symmetric functions generated by odd power sums. Stanley showed how to expand $SH_n$ into the $V-$basis of this algebra, which is indexed by partitions with all parts odd and is analogous to the complete homogeneous (or elementary) basis of symmetric functions. We introduce odd shifted parking functions to give combinatorial and representation-theoretic realizations of the $V-$expansion of $SH_n$, resolving the main open problem in his paper. Further, we present two representation-theoretic realizations of shiftification allowing us to interpret $SH_n$ as the spin character of a projective representation. We conclude with further directions, including a relationship between $SH_n$ and Haglund's $(q,t)-$Schr\"oder theorem.

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arXiv2024

Rotation invariant webs for three row flamingo Specht modules

We introduce a new rotation-invariant web basis for a family of Specht modules $S^{(d^3, 1^{n-3d})}$, indexed by normal plabic graphs satisfying a degree condition and resembling $A_2$ webs. We show that the $\mathfrak{S}_n$ action on our basis can be understood combinatorially via a set of skein relations. From this basis, we obtain a cyclic sieving result for a $q$-analog of the hook length formula for $\lambda$. Our construction extends the jellyfish invariants of Fraser, Patrias, Pechenik, and Striker and is closely related to the weblike subgraphs of Lam.

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arXiv2023

The combinatorics of supertorus sheaf cohomology

Affine superspace $\mathbb{C}^{1 \mid n}$ has a single bosonic coordinate $z$ and $n$ fermionic coordinates $\theta_1, \dots, \theta_n$. Let $M$ be the supertorus obtained by quotienting $\mathbb{C}^{1 \mid n}$ by the abelian group generated by the maps $S: (z,\theta_1, \dots, \theta_n) \mapsto (z + 1, \theta_1, \dots, \theta_n)$ and $T: (z, \theta_1, \dots, \theta_n) \mapsto (z + t, \theta_1 + \alpha_1, \dots, \theta_n + \alpha_n)$ where $t \in \mathbb{C}$ has positive imaginary part and $\alpha_1, \dots, \alpha_n$ are independent fermionic parameters. We compute the zeroth and first cohomology groups of the structure sheaf $\mathcal{O}$ of $M$ as doubly graded $\mathfrak{S}_n$-modules, exhibiting an instance of Serre duality between these groups. We use skein relations and noncrossing matchings to give a combinatorial presentation of $H^0(M,\mathcal{O})$ in terms of generators and relations.

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arXiv2022

An embedding of the skein action on set partitions into the skein action on matchings

Rhoades defined a skein action of the symmetric group on noncrossing set partitions which generalized an action of the symmetric group on matchings. The $\mathfrak{S}_n$-action on matchings is made possible via the Ptolemy relation, while the action on set partitions is defined in terms of a set of skein relations that generalize the Ptolemy relation. The skein action on noncrossing set partitions has seen applications to coinvariant theory and coordinate rings of partial flag varieties. In this paper, we will show how Rhoades' $\mathfrak{S}_n$-module can be embedded into the $\mathfrak{S}_n$-module generated by matchings, thereby explaining how Rhoades' generalized skein relations all arise from the Ptolemy relation.

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arXiv2022

A pentagonal number theorem for tribone tilings

Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we study a two-parameter family of roughly hexagonal regions in the hexagonal grid and show that a tiling by tribones exists if and only if the two parameters associated with the region are the paired pentagonal numbers $k(3k \pm 1)/2$.

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arXiv2022

A combinatorial model for the fermionic diagonal coinvariant ring

Let $\Theta_n = (\theta_1, \dots, \theta_n)$ and $\Xi_n = (\xi_1, \dots, \xi_n)$ be two lists of $n$ variables and consider the diagonal action of $\mathfrak{S}_n$ on the exterior algebra $\wedge \{ \Theta_n, \Xi_n \}$ generated by these variables. Jongwon Kim and Rhoades defined and studied the fermionic diagonal coinvariant ring $FDR_n$ obtained from $\wedge \{ \Theta_n, \Xi_n \}$ by modding out by the $\mathfrak{S}_n$-invariants with vanishing constant term. In joint work with Rhoades we gave a basis for the maximal degree components of this ring where the action of $\mathfrak{S}_n$ could be interpreted combinatorially via noncrossing set partitions. This paper will do similarly for the entire ring, although the combinatorial interpretation will be limited to the action of $\mathfrak{S}_{n-1} \subset \mathfrak{S}_n$. The basis will be indexed by a certain class of noncrossing partitions.

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arXiv2021

Set partitions, fermions, and skein relations

Let $\Theta_n = (\theta_1, \dots, \theta_n)$ and $\Xi_n = (\xi_1, \dots, \xi_n)$ be two lists of $n$ variables and consider the diagonal action of $\mathfrak{S}_n$ on the exterior algebra $\wedge \{ \Theta_n, \Xi_n \}$ generated by these variables. Jongwon Kim and the second author defined and studied the fermionic diagonal coinvariant ring $FDR_n$ obtained from $\wedge \{ \Theta_n, \Xi_n \}$ by modding out by the $\mathfrak{S}_n$-invariants with vanishing constant term. On the other hand, the second author described an action of $\mathfrak{S}_n$ on the vector space with basis given by noncrossing set partitions of $\{1,\dots,n\}$ using a novel family of skein relations which resolve crossings in set partitions. We give an isomorphism between a natural Catalan-dimensional submodule of $FDR_n$ and the skein representation. To do this, we show that set partition skein relations arise naturally in the context of exterior algebras. Our approach yields an $\mathfrak{S}_n$-equivariant way to resolve crossings in set partitions. We use fermions to clarify, sharpen, and extend the theory of set partition crossing resolution.

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arXiv2019

Simplicial Dollar Game

The dollar game is a chip-firing game introduced by Baker and Norine (2007) as a context in which to formulate and prove the Riemann-Roch theorem for graphs. A divisor on a graph is a formal integer sum of vertices. Each determines a dollar game, the goal of which is to transform the given divisor into one that is effective (nonnegative) using chip-firing moves. We use Duval, Klivans, and Martin's theory of chip-firing on simplicial complexes to generalize the dollar game and results related to the Riemann-Roch theorem for graphs to higher dimensions. In particular, we extend the notion of the degree of a divisor on a graph to a (multi)degree of a chain on a simplicial complex and use it to establish two main results. The first of these is Theorem 18, generalizing the fact that if a divisor on a graph has large enough degree (at least as large as the genus of the graph), it is winnable; and the second is Corollary 34, generalizing the fact that trees (graphs of genus 0) are exactly the graphs on which every divisor of degree 0, interpreted as an instance of the dollar game, is winnable.

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arXiv2019

Many associated primes of powers of primes

We construct families of prime ideals in polynomial rings for which the number of associated primes of the second power (or higher powers) is exponential in the number of variables in the ring. We give a lower bound on the Ananyan-Hochster constant for the number of associated primes.

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arXiv2016

Jacobi-Trudi determinants over finite fields

In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field $\mathbb{F}_q$, what is the probability that the Schur function $s_λ$ maps to zero? We show that this probability is always at least $1/q$ and is asymptotically $1/q$. Moreover, we give a complete classification of all shapes that can achieve probability $1/q$. In addition, we identify certain families of shapes where the corresponding Schur functions being sent to zero are independent events, and we look into the probability that a Schur functions is mapped to nonzero values in $\mathbb{F}_q$.

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