arXiv subjects

Eric Ramos

Publications and source records attributed to Eric Ramos.

33 records · Page 2Linked to original sources

arXiv2019

Spanning subspace configurations and representation stability

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.

math.CO↗

arXiv2019

Functorial invariants of trees and their cones

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.

math.CO↗

arXiv2018

Families of Markov chains with compatible symmetric-group actions

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.

math.CO↗

arXiv2018

An application of the theory of FI-algebras to graph configuration spaces

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 Römer. As an application we prove that the syzygies of the modules in these families exhibit a range of stable behaviors.

math.AT↗

arXiv2018

FI-sets with relations

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.

math.CO↗

arXiv2017

Families of nested graphs with compatible symmetric-group actions

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.

math.CO↗

arXiv2017

Local cohomology and the multi-graded regularity of FI$^m$-modules

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.

math.RT↗

arXiv2017

Asymptotic behaviors in the homology of symmetric group and finite general linear group quandles

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.

math.AT↗

arXiv2017

Configuration spaces of graphs with certain permitted collisions

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.

math.AT↗

arXiv2016

Generalized Representation Stability and FI_d-modules

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.

math.RT↗

arXiv2016

Depth and the Local Cohomology of FI_G-modules

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.

math.RT↗

arXiv2016

Stability phenomena in the homology of tree braid groups

For a tree $G$, we study the changing behaviors in the homology groups $H_i(B_nG)$ as $n$ varies, where $B_nG := π_1($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.

math.AT↗

arXiv2016

On the degree-wise coherence of FI_G-modules

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.

math.KT↗

arXiv2016

Homological Invariants of FI-modules and FI_G-modules

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.

math.RT↗

arXiv2011

The Alexander and Jones Polynomials Through Representations of Rook Algebras

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.

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