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Some of the most classically relevant Hyperplane arrangements are the Braid Arrangements $B_n$ and their associated compliment spaces $\mathcal{F}_n$. In their recent work, Tsilevich, Vershik, and Yuzvinsky construct what they refer to as the intrinsic hyperplane arrangement within any irreducible representation of the symmetric group that generalize the classical braid arrangements. Through examples it is also shown that the associated compliment spaces to these intrinsic arrangements display behaviors far removed from $\mathcal{F}_n$. In this work we study the intrinsic hyperplane arrangements of irreducible representations of the symmetric group from the perspective of representation stability. This work is both theoretical, proving representation stability theorems for hyperplane complements, as well as statistical, examining the outputs of a number of simulations designed to enumerate flats.

FI-graphs were introduced by the second author and White to capture the idea of a family of nested graphs, each member of which is acted on by a progressively larger symmetric group. That work was built on the newly minted foundations of representation stability theory and FI-modules. Examples of such families include the complete graphs and the Kneser and Johnson graphs, among many others. While it was shown in the originating work how various counting invariants in these families behave very regularly, not much has thus far been proven about the behaviors of the typical extremal graph theoretic invariants such as their independence and clique numbers. In this paper we provide a conjecture on the growth of the independence and clique numbers in these families, and prove this conjecture in one case. We also provide computer code that generates experimental evidence in many other cases. All of this work falls into a growing trend in representation stability theory that displays the regular behaviors of a number of extremal invariants that arise when one looks at FI-algebras and modules., Comment: Associated code can be found here: https://ericgramos.github.io/code.html

Building upon previous works of Proudfoot and Ramos, and using the categorical framework of Sam and Snowden, we extend the weak categorical minor theorem from undirected graphs to quivers. As case of study, we investigate the consequences on the homology of multipath complexes; eg. on its torsion. Further, we prove a comparison result: we show that, when restricted to directed graphs without oriented cycles, multipath complexes and matching complexes yield functors which commute up to a blow-up operation on directed graphs. We use this fact to compute the homotopy type of matching complexes for a certain class of bipartite graphs also known as half-graphs or ladders. We complement the work with a study of the (representation) category of cones, and with analysing related consequences on magnitude cohomology of quivers., Comment: 18 pages, 6 figures, comments are welcome!

We compute the homology of the matching complex $M(\Gamma)$, where $\Gamma$ is the complete hypergraph on $n\geq 2$ vertices, and analyse the $S_n$-representations carried by this homology. These results are achieved using standard techniques in combinatorial topology, such as the theory of shellings. We then broaden the scope to the larger class of fibre-closed families of simplicial complexes and consider these through the lens of representation stability. This allows us to prove a number of results of an asymptotic nature, such as an analysis of the growth of Betti numbers and the kinds of irreducible $S_n$-representations that appear.

We formulate a categorification of Robertson's conjecture analogous to the categorical graph minor conjecture of Miyata--Proudfood--Ramos. We show that these conjectures imply the existence of a finite list of atomic graphs generating the homology of configuration spaces of graphs -- in fixed degree, with a fixed number of particles, under topological embeddings. We explain how the simplest case of our conjecture follows from work of Barter and Miyata--Proudfoot, implying that the category of cographs is Noetherian, a result of potential independent interest., Comment: v2. Various small typographical and exposition improvements. To appear in Selecta Math

For each $n,r \geq 0$, let $KG(n,r)$ denote the Kneser Graph; that whose vertices are labeled by $r$-element subsets of $n$, and whose edges indicate that the corresponding subsets are disjoint. Fixing $r$ and allowing $n$ to vary, one obtains a family of nested graphs, each equipped with a natural action by a symmetric group $\mathfrak{S}_n$, such that these actions are compatible and transitive. Families of graphs of this form were introduced by the authors in [RW], while a systematic study of random walks on these families were considered in [RW2]. In this paper we illustrate that these random walks never exhibit the so-called product condition, and therefore also never display total variation cutoff as defined by Aldous and Diaconis [AD]. In particular, we provide a large family of algebro-combinatorially motivated examples of collections of Markov chains which satisfy some well-known algebraic heuristics for cutoff, while not actually having the property., Comment: arXiv admin note: text overlap with arXiv:1810.08475

We expand upon the notion of equivariant log concavity, and make equivariant log concavity conjectures for Orlik--Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik--Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{sl}_n$, we exploit the theory of representation stability to give computer assisted proofs of these conjectures in low degree.

We study a variety of natural constructions from topological combinatorics, including matching complexes as well as other graph complexes, from the perspective of the graph minor category of \parencite{MiProRa}. We prove that these complexes must have universally bounded torsion in their homology across all graphs of bounded genus. One may think of these results as arising from an algebraic version of the graph minor theorem of Robertson and Seymour \parencite{RSXX,RSXXIII}., Comment: Major Revision: Due to the gap found in the proof of the categorical graph minor theorem, this paper has been rewritten to prove a weaker version of that theorem. arXiv admin note: text overlap with arXiv:2004.05544

We prove that the ith graded pieces of the Orlik-Solomon algebras or Cordovil algebras of resonance arrangements form a finitely generated FS^op-module, thus obtaining information about the growth of their dimensions and restrictions on the irreducible representations of symmetric groups that they contain.

We consider Hilbert series associated to modules over various categories of trees. Using the technology of Sam and Snowden, we show that these Hilbert series must be algebraic. We then apply these technical theorems to prove facts about certain natural generating functions associated to trees.

We define the graph minor category and prove that the category of contravariant representations of the graph minor category over a Noetherian ring is locally Noetherian. This can be regarded as a categorification of the Robertson--Seymour graph minor theorem. In addition, we generalize Sam and Snowden's Gr\"obner theory of categories to the setting of pairs consisting of a category along with a functor to sets, and we apply this theory to the edge functor on the graph minor category. As an application, we study homology groups of unordered configuration spaces of graphs, improving upon various finite generation results in this subject., Comment: A gap has been discovered in the proof the main theorem, and specifically in the justification for Lemma 3.5. For now, the Categorical graph minor theorem will remain a conjecture. One can view the applications of this work to graph configuration spaces as being consequences of this conjecture, as their proofs are unaffected by the gap

We define and study a model of winding for non-colliding particles in finite trees. We prove that the asymptotic behavior of this statistic satisfies a central limiting theorem, analogous to similar results on winding of bounded particles in the plane. We also propose certain natural open questions and conjectures, whose confirmation would provide new insights on configuration spaces of trees.

We study the category whose objects are graphs of fixed genus and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian and we study two families of modules over these categories. The first takes a graph to a graded piece of the homology of its unordered configuration space and the second takes a graph to an intersection homology group whose dimension is given by a Kazhdan-Lusztig coefficient; in both cases we prove that the module is finitely generated. This allows us to draw conclusions about torsion in the homology groups of graph configuration spaces, and about the growth of Betti numbers of graph configuration spaces and Kazhdan-Lusztig coefficients of graphical matroids. We also explore the relationship between our category and outer space, which is used in the study of outer automorphisms of free groups., Comment: arXiv admin note: text overlap with arXiv:1903.10592

Let $V_1, V_2, V_3, \dots $ be a sequence of $\mathbb{Q}$-vector spaces where $V_n$ carries an action of $\mathfrak{S}_n$ for each $n$. {\em Representation stability} and {\em multiplicity stability} are two related notions of when the sequence $V_n$ has a limit. An important source of stability phenomena arises in the case where $V_n$ is the $d^{th}$ homology group (for fixed $d$) of the configuration space of $n$ distinct points in some fixed topological space $X$. We replace these configuration spaces with the variety $X_{n,k}$ of {\em spanning configurations} of $n$-tuples $(\ell_1, \dots, \ell_n)$ of lines in $\mathbb{C}^k$ which satisfy $\ell_1 + \cdots + \ell_n = \mathbb{C}^k$ as vector spaces. We study stability phenomena for the homology groups $H_d(X_{n,k})$ as the parameter $(n,k)$ grows., Comment: 16 pages

We study the category whose objects are trees (with or without roots) and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian, and we study two natural families of modules over these categories. The first takes a tree to a graded piece of the homology of its unordered configuration space, or to the homology of the unordered configuration space of its cone. The second takes a tree to a graded piece of the intersection homology of the reciprocal plane of its cone, which is a vector space whose dimension is given by a Kazhdan-Lusztig coefficient. We prove finite generation results for each of these modules, which allow us to obtain results about the growth of Betti numbers of configuration spaces and of Kazhdan-Lusztig coefficients of graphical matroids.

For each $n,r \geq 0$, let $KG(n,r)$ denote the Kneser Graph; that whose vertices are labeled by $r$-element subsets of $n$, and whose edges indicate that the corresponding subsets are disjoint. Fixing $r$ and allowing $n$ to vary, one obtains a family of nested graphs, each equipped with a natural action by a symmetric group $S_n$, such that these actions are compatible. Collections of graphs of this type are common in algebraic combinatorics and include families such as the Johnson Graphs, Crown Graphs and Rook Graphs. In previous work, the authors systematically studied families of this type using the language of representation stability and FI-modules. In that work, it is shown that such families of graphs exhibit a large variety of asymptotic regular behaviors. The present work applies the theory developed in that previous work, later refined in work of the authors and Speyer, to study random walks on the graphs of such families. We show that the moments of hitting times exhibit rational function behavior asymptotically. By consequence we conclude similar facts about the entries of the discrete Green's functions, as defined by Chung and Yau. Finally, we illustrate how the algebro-combinatorial structure of the graphs in these families give bounds on the mixing times of random walks on those graphs. We suggest some possible directions for future study, including of the appearance, or not, of the cut-off phenomenon, originally presented by Diaconis.

Recent work of An, Drummond-Cole, and Knudsen, as well as the author, has shown that the homology groups of configuration spaces of graphs can be equipped with the structure of a finitely generated graded module over a polynomial ring. In this work we study this module structure in certain families of graphs using the language of FI-algebras recently explored by Nagel and R\"omer. As an application we prove that the syzygies of the modules in these families exhibit a range of stable behaviors.

Let FI denote the category whose objects are the sets $[n] = \{1,\ldots, n\}$, and whose morphisms are injections. We study functors from the category FI into the category of sets. We write $\mathfrak{S}_n$ for the symmetric group on $[n]$. Our first main result is that, if the functor $[n] \mapsto X_n$ is "finitely generated" there there is a finite sequence of integers $m_i$ and a finite sequence of subgroups $H_i$ of $\mathfrak{S}_{m_i}$ such that, for $n$ sufficiently large, $X_n \cong \bigsqcup_i \mathfrak{S}_n/(H_i \times \mathfrak{S}_{n-m_i})$ as a set with $\mathfrak{S}_n$ action. Our second main result is that, if $[n] \mapsto X_n$ and $[n] \mapsto Y_n$ are two such finitely generated functors and $R_n \subset X_n \times Y_n$ is an FI-invariant family of relations, then the $(0,1)$ matrices encoding the relation $R_n$, when written in an appropriate basis, vary polynomially with $n$. In particular, if $R_n$ is an FI-invariant family of relations from $X_n$ to itself, then the eigenvalues of this matrix are algebraic functions of $n$. As an application of this theorem we provide a proof of a result about eigenvalues of adjacency matrices claimed by the first and last author. This result recovers, for instance, that the adjacency matrices of the Kneser graphs have eigenvalues which are algebraic functions of $n$, while also expanding this result to a larger family of graphs.

We develop a local cohomology theory for FI$^m$-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI$^m$-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.

For fixed positive integers $n$ and $k$, the Kneser graph $KG_{n,k}$ has vertices labeled by $k$-element subsets of $\{1,2,\dots,n\}$ and edges between disjoint sets. Keeping $k$ fixed and allowing $n$ to grow, one obtains a family of nested graphs, each of which is acted on by a symmetric group in a way which is compatible with all of the other actions. In this paper, we provide a framework for studying families of this kind using the FI-module theory of Church, Ellenberg, and Farb, and show that this theory has a variety of asymptotic consequences for such families of graphs. These consequences span a range of topics including enumeration, concerning counting occurrences of subgraphs, topology, concerning Hom-complexes and configuration spaces of the graphs, and algebra, concerning the changing behaviors in the graph spectra., Comment: v2. Fixed the statement of Theorem G, as well as various typos throughout

A quandle is an algebraic structure which attempts to generalize group conjugation. These structures have been studied extensively due to their connections with knot theory, algebraic combinatorics, and other fields. In this work, we approach the study of quandles from the perspective of the representation theory of categories. Namely, we look at collections of conjugacy classes of the symmetric groups and the finite general linear groups, and prove that they carry the structure of FI-quandles (resp. VIC$(q)$-quandles). As applications, we prove statements about the homology of these quandles, and construct FI-module and VIC$(q)$-module invariants of links., Comment: v2: Fixed the statement of 2.15 part 3, as well as various typos throughout

If $G$ is a graph with vertex set $V$, let Conf$_n^{\text{sink}}(G,V)$ be the space of $n$-tuples of points on $G$, which are only allowed to overlap on elements of $V$. We think of Conf$_n^{\text{sink}}(G,V)$ as a configuration space of points on $G$, where points are allowed to collide on vertices. In this paper, we attempt to understand these spaces from two separate, but closely related, perspectives. Using techniques of combinatorial topology we compute the fundamental groups and homology groups of Conf$_n^{\text{sink}}(G,V)$ in the case where $G$ is a tree. Next, we use techniques of asymptotic algebra to prove statements about Conf$_n^{\text{sink}}(G,V)$, for general graphs $G$, whenever $n$ is sufficiently large. It is proven that, for general graphs, the homology groups exhibit generalized representation stability in the sense of previous work of the author., Comment: v3: Expanded upon the exposition in the proof of Theorem 3.18

For a tree $G$, we study the changing behaviors in the homology groups $H_i(B_nG)$ as $n$ varies, where $B_nG := \pi_1($UConf$_n(G))$. We prove that the ranks of these homologies can be described by a single polynomial for all $n$, and construct this polynomial explicitly in terms of invariants of the tree $G$. To accomplish this we prove that the group $\bigoplus_n H_i(B_nG)$ can be endowed with the structure of a finitely generated graded module over an integral polynomial ring, and further prove that it naturally decomposes as a direct sum of graded shifts of squarefree monomial ideals. Following this, we spend time considering how our methods might be generalized to braid groups of arbitrary graphs, and make various conjectures in this direction.

In this work we study a kind of coherence condition on FI_G-modules, which generalizes the usual notion of finite generation. We prove that a module is coherent, in the appropriate sense, if and only if its generators, as well as its torsion, appears in only finitely many degrees. Using this technical result, we prove that the category of coherent FI_G-modules is abelian, independent of any assumptions on the group G, or the coefficient ring k. Following this, we consider applications towards the local cohomology theory of FI_G-modules, introduced by Li and the author in previous work.

In this note we consider the complex representation theory of FI_d, a natural generalization of the category FI of finite sets and injections. We prove that finitely generated FI_d-modules exhibit behaviors in the spirit of Church-Farb representation stability theory, generalizing a theorem of Church, Ellenberg, and Farb which connects finite generation of FI-modules to representation stability., Comment: v4: Expanded Theorem B

In this paper we describe a machinery for homological calculations of representations of FI_G, and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the second author coincides with a more classical invariant from commutative algebra, and obtain upper bounds of a few important invariants of FI_G-modules in terms of torsion degrees of their local cohomology groups.

We explore a theory of depth for FI_G-modules which are presented in finite degrees. Using this theory, we prove results about the regularity, and provide novel bounds on stable ranges of FI-modules, making effective a theorem of Nagpal and thereby refining the stable range in results of Church, Ellenberg, and Farb., Comment: 29 pages. v3: Generalized many of the theorems from the earlier versions. No longer require that G be finite, nor that k be a Noetherian ring, for instance

In the 1920's Artin defined the braid group in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of this correspondence, the Jones and Alexander polynomials, two of the most important knot invariants, can be described completely using the braid group. There has been a recent growth of interest in other diagrammatic algebras, whose elements have a similar topological flavor to the braid group. These have wide ranging applications in areas including representation theory and quantum computation. We consider representations of the braid group when passed through another diagrammatic algebra, the planar rook algebra. By studying traces of these matrices, we recover both the Jones and Alexander polynomials.