arXiv subjects

Julianna Tymoczko

Publications and source records attributed to Julianna Tymoczko.

At least 19 recordsLinked to original sources

arXiv2026

Stranding $\mathfrak{sl}_n$ webs

Webs are planar directed graphs that encode invariant vectors for tensor products of fundamental $U_q(\mathfrak{sl}_n)$-representations. For $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, web calculus is governed by effective reduction rules and well-understood reduced-web bases, but these features break down in higher rank. In this paper we develop a global combinatorial framework for untagged $\mathfrak{sl}_n$ webs, called strandings, which organizes local labeling data into systems of colored directed paths on the web graph. Our main result is an explicit state-sum formula for the invariant vector of an untagged web as a weighted sum over its valid strandings. The usual labeling-based contruction requires a choice of decomposition into elementary web pieces and a vertex-by-vertex coefficient calculation. Our global reformulation avoids both. We prove that the vectors produced by strandings are $U_q(\mathfrak{sl}_n)$-invariant, compare the untagged theory with the tagged-web framework of Cautis, Kamnitzer, and Morrison, and use this comparison to obtain a complete set of relations for untagged web graphs. We also give applications of strandings to nonvanishing results, basis constructions from tableaux, and connections with Springer-theoretic combinatorics.

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arXiv2026

Universality theorems for generalized splines

We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating set'' for the module of splines over any graph with fixed combinatorial genus. This theorem holds over any Noetherian commutative ring with a chosen finite list of ideals for edge-labels. We then give several applications of this theorem, including showing that a particular generating function associated to splines on trees is algebraic when the base ring satisfies certain finiteness conditions. We illustrate our technical theorems explicitly by giving a classification of splines on graphs with combinatorial genus one and two.

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arXiv2026

Dimension of splines on graphs in the case of degree two and smoothness one in two variables

Continuous spline functions are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data interpolation, to create smooth curves in computer graphics and to find numerical solutions to partial differential equations. Gilbert, Tymoczko, and Viel generalized the classical splines combinatorially and algebraically: a generalized spline is a vertex labeling of a graph $G$ by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the ideal generated by the corresponding edge label. We study the generalized splines on the planar graphs whose edges are labeled by two-variable polynomials of the form $(ax+by+c)^2$ and whose vertices are labeled by polynomials of degree at most two. In this paper we address the upper-bound conjecture for the dimension of degree-2 splines of smoothness 1. The dimension is expressed in terms of the rank of the extended cycle basis matrix. We also provide a combinatorial algorithm on graphs to compute the rank by contracting certain subgraphs.

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arXiv2025

Evacuation of rectangular standard Young tableaux corresponds to reflection of $\mathfrak{sl}_n$ webs

Web graphs form a family of planar directed graphs with boundary that can be used to model quantum $\mathfrak{sl}_n$-invariant vectors. Standard Young tableaux on an $n \times k$ rectangle naturally index a basis for $\mathfrak{sl}_n$ web graphs. We prove that evacuation of the tableau $T$ corresponds to reflection of the associated web graph $w_T$ up to equivalence under a specific set of edge-flip relations. This extends a result of Patrias and Pechenik for the cases $n=2,3$ and mirrors analogous results about rotation of web graphs corresponding to promotion of tableau by Peterson-Pylyavskyy-Rhoades for $n=3$ and Gaetz-Pechenik-Pfannerer-Striker-Swanson for $n=4$. We use an intermediate object called a multicolored noncrossing matching, which is closely related to the notion of strandings recently introduced by Russell and the fourth author.

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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.

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arXiv2025

Cell closures for two-row Springer fibers via noncrossing matchings

Springer fibers are a family of subvarieties of the flag variety parametrized by nilpotent matrices that are important in geometric representation theory and whose geometry encodes deep combinatorics. Two-row Springer fibers, which correspond to nilpotent matrices with two Jordan blocks, also arise in knot theory, in part because their components are indexed by noncrossing matchings. Springer fibers are paved by affines by intersection with appropriately-chosen Schubert cells. In the two-row case, we provide an elementary description of these Springer Schubert cells in terms of standard noncrossing matchings and describe closure relations explicitly. To do this, we define an operation called cutting arcs in a matching, which successively unnests arcs while ``remembering" the arc originally on top. We then prove that the boundary of the Springer Schubert cell corresponding to a matching consists of the affine subsets of cells corresponding to all ways of cutting arcs in that matching.

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arXiv2023

The algebra of splines: duality, group actions and homology

This survey gives an overview of three central algebraic themes related to the study of splines: duality, group actions, and homology. Splines are piecewise polynomial functions of a prescribed order of smoothness on some subdivided domain D in R^k, and appear in applications ranging from approximation theory to geometric modeling to numerical analysis. Alternatively, splines can be interpreted as a collection of polynomials labeling the vertices of a (combinatorial) graph, with adjacent vertex-labels differing by a power of an affine linear form attached to the edge. In most cases of interest, the subdivided domain is essentially dual to the combinatorial graph, and these two characterizations of splines coincide. Properties of splines depend on combinatorics, topology, geometry, and symmetry of a simplicial or polyhedral subdivision of a region D in R^k, and are often quite subtle. We describe how duality, group actions, and homology -- techniques which play a central role in many areas of both pure and applied mathematics -- can be used to illuminate different questions about splines. Our target audience is nonspecialists: we provide a concrete introduction to these methods, and illustrate them with many examples in the context of splines. We also provide a tutorial on computational aspects: all of the objects appearing in this note may be studied using open source computer algebra software.

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arXiv2023

Generalized splines on graphs with two labels and polynomial splines on cycles

Generalized splines are an algebraic combinatorial framework that generalizes and unifies various established concepts across different fields, most notably the classical notion of splines and the topological notion of GKM theory. The former consists of piecewise polynomials on a combinatorial geometric object like a polytope, whose polynomial pieces agree to a specified degree of differentiability. The latter is a graph-theoretic construction of torus-equivariant cohomology that Shareshian and Wachs used to reformulate the well-known Stanley$\unicode{x2013}$Stembridge conjecture, a reformulation that was recently proven to hold by Brosnan and Chow and independently Guay-Paquet. This paper focuses on the theory of generalized splines. A generalized spline on a graph $G$ with each edge labeled by an ideal in a ring $R$ consists of a vertex-labeling by elements of $R$ so that the labels on adjacent vertices $u, v$ differ by an element of the ideal associated to the edge $uv$. We study the $R$-module of generalized splines and produce minimum generating sets for several families of graphs and edge-labelings: $1)$ for all graphs when the set of possible edge-labelings consists of at most two finitely-generated ideals, and $2)$ for cycles when the set of possible edge-labelings consists of principal ideals generated by elements of the form $(ax+by)^2$ in the polynomial ring $\mathbb{C}[x,y]$. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in the theory of classical (analytic) splines.

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arXiv2023

Which Hessenberg varieties are GKM?

Hessenberg varieties $\mathcal{H}(X,H)$ form a class of subvarieties of the flag variety $G/B$, parameterized by an operator $X$ and certain subspaces $H$ of the Lie algebra of $G$. We identify several families of Hessenberg varieties in type $A_{n-1}$ that are $T$-stable subvarieties of $G/B$, as well as families that are invariant under a subtorus $K$ of $T$. In particular, these varieties are candidates for the use of equivariant methods to study their geometry. Indeed, we are able to show that some of these varieties are unions of Schubert varieties, while others cannot be such unions. Among the $T$-stable Hessenberg varieties, we identify several that are {\it GKM spaces}, meaning $T$ acts with isolated fixed points and a finite number of one-dimensional orbits, though we also show that not all Hessenberg varieties with torus actions and finitely many fixed points are GKM. We conclude with a series of open questions about Hessenberg varieties, both in type $A_{n-1}$ and in general Lie type.

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arXiv2022

Toward permutation bases in the equivariant cohomology rings of regular semisimple Hessenberg varieties

Recent work of Shareshian and Wachs, Brosnan and Chow, and Guay-Paquet connects the well-known Stanley-Stembridge conjecture in combinatorics to the dot action of the symmetric group $S_n$ on the cohomology rings $H^*(Hess(S,h))$ of regular semisimple Hessenberg varieties. In particular, in order to prove the Stanley-Stembridge conjecture, it suffices to construct (for any Hessenberg function $h$) a permutation basis of $H^*(Hess(S,h))$ whose elements have stabilizers isomorphic to Young subgroups. In this manuscript we give several results which contribute toward this goal. Specifically, in some special cases, we give a new, purely combinatorial construction of classes in the $T$-equivariant cohomology ring $H^*_T(Hess(S,h))$ which form permutation bases for subrepresentations in $H^*_T(Hess(S,h))$. Moreover, from the definition of our classes it follows that the stabilizers are isomorphic to Young subgroups. Our constructions use a presentation of the $T$-equivariant cohomology rings $H^*_T(Hess(S,h))$ due to Goresky, Kottwitz, and MacPherson. The constructions presented in this manuscript generalize past work of Abe-Horiguchi-Masuda, Chow, and Cho-Hong-Lee.

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arXiv2020

The transition matrix between the Specht and $\mathfrak{sl}_3$ web bases is unitriangular with respect to shadow containment

Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (like, e.g., Kazhdan-Lusztig polynomials). By "flattening" the braiding maps, webs can also be viewed as the basis elements of a symmetric-group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, that measure depths of regions inside the web. As an application, we resolve an open conjecture that the change-of-basis between the so-called Specht basis and web basis of this symmetric-group representation is unitriangular for $\mathfrak{sl}_3$-webs. We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously-studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.

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arXiv2020

A filtration on the cohomology rings of regular nilpotent Hessenberg varieties

Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{\mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $GL(n-1,{\mathbb{C}})/B$, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincaré polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of "Hessenberg Schubert polynomials" in the context of regular nilpotent Hessenberg varieties, and outline several open questions pertaining to them.

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arXiv2020

Hessenberg varieties of parabolic type

This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg variety under the projection of the flag variety to an appropriate partial flag variety and we give three applications of this result. The first application constructs an explicit paving of all Steinberg varieties in Lie type $A$ in terms of semistandard tableaux. As a result, we obtain an elementary proof of a theorem of Steinberg and Shimomura that the well-known Kostka numbers count the maximal-dimensional irreducible components of Steinberg varieties. The second application proves an open conjecture for certain parabolic Hessenberg varieties in Lie type A by showing that their Betti numbers equal those of a specific union of Schubert varieties. The third application proves that the irreducible components of parabolic Hessenberg varieties are in bijection with the irreducible components of the Steinberg variety. All three of these applications extend our geometric understanding of the three varieties at the heart of this paper, a full understanding of which is unknown even for Springer varieties, despite over forty years' worth of work.

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arXiv2020

Representation stability of the cohomology of Springer varieties and some combinatorial consequences

A sequence of $S_n$-representations $\{V_n\}$ is said to be uniformly representation stable if the decomposition of $V_n = \bigoplus_μ c_{μ,n} V(μ)_n$ into irreducible representations is independent of $n$ for each $μ$---that is, the multiplicities $c_{μ,n}$ are eventually independent of $n$ for each $μ$. Church-Ellenberg-Farb proved that the cohomology of flag varieties (the so-called diagonal coinvariant algebra) is uniformly representation stable. We generalize their result from flag varieties to all Springer fibers. More precisely, we show that for any increasing subsequence of Young diagrams, the corresponding sequence of Springer representations form a graded co-FI-module of finite type (in the sense of Church-Ellenberg-Farb). We also explore some combinatorial consequences of this stability.

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arXiv2019

Graphs admitting only constant splines

We study {\em generalized graph splines,} introduced by Gilbert, Viel, and the last author. For a large class of rings, we characterize the graphs that only admit constant splines. To do this, we prove that if a graph has a particular type of cutset (e.g., a bridge), then the space of splines naturally decomposes as a certain direct sum of submodules. As an application, we use these results to describe splines on a triangulation studied by Zhou and Lai, but over a different ring than they used.

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arXiv2019

A formula for the cohomology and $K$-class of a regular Hessenberg variety

Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator $X$ and a nondecreasing function $h$. The family of Hessenberg varieties for regular $X$ is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and $K$-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on $h$. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on $h$ than here.

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arXiv2018

Springer fibers and Schubert points

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.

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arXiv2017

RNA, local moves on plane trees, and transpositions on tableaux

We define a collection of functions $s_i$ on the set of plane trees (or standard Young tableaux). The functions are adapted from transpositions in the representation theory of the symmetric group and almost form a group action. They were motivated by $\textit{local moves}$ in combinatorial biology, which are maps that represent a certain unfolding and refolding of RNA strands. One main result of this study identifies a subset of local moves that we call $s_i$-local moves, and proves that $s_i$-local moves correspond to the maps $s_i$ acting on standard Young tableaux. We also prove that the graph of $s_i$-local moves is a connected, graded poset with unique minimal and maximal elements. We then extend this discussion to functions $s_i^C$ that mimic reflections in the Weyl group of type $C$. The corresponding graph is no longer connected, but we prove it has two connected components, one of symmetric and the other of asymmetric plane trees. We give open questions and possible biological interpretations.

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