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Jennifer C. H. Wilson

Publications and source records attributed to Jennifer C. H. Wilson.

At least 19 recordsLinked to original sources
arXiv2026

FI-hyperhomology and ordered configuration spaces

Using a result of Guès on FI-hyperhomology (a variant of a result of Gan-Li) and a semi-simplicial resolution of configuration spaces due to Kupers-Miller-Tran, we establish an explicit stable range for cohomological representation stability for configuration spaces of distinct ordered points in a manifold. Our bounds on generation degree were a factor of 5/2 better than those previously established at the time this paper was first posted. This updated article replaces an older version, which contained a mistake pointed out to us by Nicolas Guès. We are grateful to Guès for identifying the error and suggesting the resolution.

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arXiv2026

Representation stability in the (co)homology of vertical configuration spaces

In this paper, we study sequences of topological spaces called "vertical configuration spaces" of points in Euclidean space. We apply the theory of FI$_G$-modules, and results of Bianchi-Kranhold, to show that their (co)homology groups are "representation stable" with respect to natural actions of wreath products $S_k \wr S_n$. In particular, we show that in each (co)homological degree, the (co)homology groups (viewed as $S_k \wr S_n$-representations) can be expressed as induced representations of a specific form. Consequently, the characters of their rational (co)homology groups, and the patterns of irreducible $S_k \wr S_n$-representation constituents of these groups, stabilize in a strong sense. In addition, we give a new proof of rational (co)homological stability for unordered vertical configuration spaces, with an improved stable range.

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arXiv2025

Representation stability for ordered Hurwitz spaces

In this paper, we study the topology of ordered Hurwitz space. These are moduli spaces of branched covers with a choice of ordering on the branched points. Answering a question of Ellenberg, we prove that the homology of ordered Hurwitz spaces exhibit representation stability.

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arXiv2025

On rank filtrations of algebraic K-theory and Steinberg modules

Motivated by his work on the stable rank filtration of algebraic K-theory spectra, Rognes defined a simplicial complex called the common basis complex and conjectured that this complex is highly connected for local rings and Euclidean domains. We prove this conjecture in the case of fields. Our methods give a novel description of this common basis complex of a PID as an iterated bar construction on an equivariant monoid built out of Tits buildings. We also identify the Koszul dual of a certain equivariant ring assembled out of Steinberg modules.

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arXiv2024

On the codimension-two cohomology of $\mathrm{SL}_n(\mathbb{Z})$

Borel-Serre proved that $\mathrm{SL}_n(\mathbb{Z})$ is a virtual duality group of dimension $n \choose 2$ and the Steinberg module $\mathrm{St}_n(\mathbb{Q})$ is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to $\mathrm{SL}_n(\mathbb{Q})$. We determine the "relations among the relations" of this Steinberg module. That is, we construct an explicit partial resolution of length two of the $\mathrm{SL}_n(\mathbb{Z})$-module $\mathrm{St}_n(\mathbb{Q})$. We use this partial resolution to show the codimension-2 rational cohomology group $H^{{n \choose 2} -2}(\mathrm{SL}_n(\mathbb{Z});\mathbb{Q})$ of $\mathrm{SL}_n(\mathbb{Z})$ vanishes for $n \geq 3$. This resolves a case of a conjecture of Church-Farb-Putman. We also produce lower bounds for the codimension-1 cohomology of certain congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$.

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arXiv2022

Mapping class group actions on configuration spaces and the Johnson filtration

Let $F_n(Σ_{g,1})$ denote the configuration space of $n$ ordered points on the surface $Σ_{g,1}$ and let $Γ_{g,1}$ denote the mapping class group of $Σ_{g,1}$. We prove that the action of $Γ_{g,1}$ on $H_i(F_n(Σ_{g,1});\mathbb{Z})$ is trivial when restricted to the $i^{th}$ stage of the Johnson filtration $\mathcal{J}(i)\subset Γ_{g,1}$. We give examples showing that $\mathcal{J}(2)$ acts nontrivially on $H_3(F_3(Σ_{g,1}))$ for $g\ge 2$, and provide two new conceptual reinterpretations of a certain group introduced by Moriyama.

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arXiv2021

On the generalized Bykovskii presentation of Steinberg modules

We study presentations of the virtual dualizing modules of special linear groups of number rings, the Steinberg modules. Bykovskii gave a presentation for the Steinberg modules of the integers, and our main result is a generalization of this presentation to the Gaussian integers and the Eisenstein integers. We also show that this generalization does not give a presentation for the Steinberg modules of several Euclidean number rings.

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arXiv2019

Non-integrality of some Steinberg modules

We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartments. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apartments if and only if the ring is Euclidean. We also construct new cohomology classes in the top dimensional cohomology group of the special linear group of some quadratic imaginary number rings.

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arXiv2019

Central stability for the homology of congruence subgroups and the second homology of Torelli groups

We prove a representation stability result for the second homology groups of Torelli subgroups of mapping class groups and automorphism groups of free groups. This strengthens the results of Boldsen-Hauge Dollerup and Day-Putman. We also prove a new representation stability result for the homology of certain congruence subgroups, partially improving upon the work of Putman-Sam. These results follow from a general theorem on syzygies of certain modules with finite polynomial degree.

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arXiv2019

Higher order representation stability and ordered configuration spaces of manifolds

Using the language of twisted skew-commutative algebras, we define \emph{secondary representation stability}, a stability pattern in the {\it unstable} homology of spaces that are representation stable in the sense of Church, Ellenberg, and Farb. We show that the rational homology of configuration spaces of ordered points in noncompact manifolds satisfies secondary representation stability. While representation stability for the homology of configuration spaces involves stabilizing by introducing a point ``near infinity,'' secondary representation stability involves stabilizing by introducing a pair of orbiting points -- an operation that relates homology groups in different homological degrees. This result can be thought of as a representation-theoretic analogue of \emph{secondary homological stability} in the sense of Galatius, Kupers, and Randal-Williams. In the course of the proof we establish some additional results: we give a new characterization of the homology of the complex of injective words, and we give a new proof of integral representation stability for configuration spaces of noncompact manifolds, extending previous results to nonorientable manifolds.

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arXiv2018

Quantitative representation stability over linear groups

We introduce a technique for proving quantitative representation stability theorems for sequences of representations of certain finite linear groups over a field of characteristic zero. In particular, we prove a vanishing result for higher syzygies of VIC- and SI-modules, which can be thought of as a weaker version of a regularity theorem of Church-Ellenberg in the context of FI-modules. We apply these techniques to the rational homology of congruence subgroups of mapping class groups and congruence subgroups of automorphism groups of free groups. This partially resolves a question raised by Church and Putman-Sam. We also prove new homological stability results for mapping class groups and automorphism groups of free groups with twisted coefficients.

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arXiv2017

Convergence criteria for FI$_\mathcal{W}$-algebras and polynomial statistics on maximal tori in type B/C

A result of Lehrer describes a beautiful relationship between topological and combinatorial data on certain families of varieties with actions of finite reflection groups. His formula relates the cohomology of complex varieties to point counts on associated varieties over finite fields. Church, Ellenberg, and Farb use their representation stability results on the cohomology of flag manifolds, together with classical results on the cohomology rings, to prove asymptotic stability for "polynomial" statistics on associated varieties over finite fields. In this paper we investigate the underlying algebraic structure of these families' cohomology rings that makes the formulas convergent. We prove that asymptotic stability holds in general for subquotients of FI$_\mathcal{W}$-algebras finitely generated in degree at most one, a result that is in a sense sharp. As a consequence, we obtain convergence results for polynomial statistics on the set of maximal tori in $\mathrm{Sp}_{2n}(\overline{F_q})$ and $\mathrm{SO}_{2n+1}(\overline{F_q})$ that are invariant under the Frobenius morphism. Our results also give a new proof of the stability theorem for invariant maximal tori in $\mathrm{GL}_n(\overline{F_q})$ due to Church-Ellenberg-Farb.

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arXiv2017

Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields

In this paper we explore a relationship between the topology of the complex hyperplane complements $\mathcal{M}_{BC_n} (\mathbb{C})$ in type B/C and the combinatorics of certain spaces of degree-$n$ polynomials over a finite field $\mathbb{F}_q$. This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras $H^*(\mathcal{M}_{BC_n} (\mathbb{C});\mathbb{C})$, and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over $\mathbb{F}_q$ with nonzero constant term. This result is the type B/C analogue of a theorem due to Church, Ellenberg, and Farb in type A, and we include a new proof of their theorem. To establish these convergence results, we realize the sequences of cohomology algebras of the hyperplane complements as FI$_\mathcal{W}$-algebras finitely generated in FI$_\mathcal{W}$- degree $2$, and we investigate the asymptotic behaviour of general families of algebras with this structure. We prove a negative result implying that this structure alone is not sufficient to prove the necessary convergence conditions. Our proof of convergence for the cohomology algebras involves the combinatorics of their relators.

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arXiv2017

Generating functions and statistics on spaces of maximal tori in classical Lie groups

In this paper we use generating function methods to obtain new asymptotic results about spaces of $F$-stable maximal tori in $GL_n(\overline{F_q})$, $Sp_{2n}(\overline{F_q})$, and $SO_{2n+1}(\overline{F_q})$. We recover stability results of Church--Ellenberg--Farb and Jiménez Rolland--Wilson for "polynomial" statistics on these spaces, and we compute explicit formulas for their stable values. We derive a double generating function for the characters of the cohomology of flag varieties in type B/C, which we use to obtain analogs in type B/C of results of Chen: we recover "twisted homological stability" for the spaces of maximal tori in $Sp_{2n}(\mathbb{C})$ and $SO_{2n+1}(\mathbb{C})$, and we compute a generating function for their "stable twisted Betti numbers". We also give a new proof of a result of Lehrer using symmetric function theory.

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arXiv2015

FI_W-modules and stability criteria for representations of the classical Weyl groups

In this paper we develop machinery for studying sequences of representations of any of the three families of classical Weyl groups, extending work of Church, Ellenberg, Farb, and Nagpal on the symmetric groups S_n to the signed permutation groups B_n and the even-signed permutation groups D_n. For each family W_n, we present an algebraic framework where a sequence V_n of W_n-representations is encoded into a single object we call an FI_W-module. We prove that if an FI_W-module V satisfies a simple finite generation condition then the structure of the sequence is highly constrained. One consequence is that the sequence is uniformly representation stable in the sense of Church-Farb, that is, the pattern of irreducible representations in the decomposition of each V_n eventually stabilizes in a precise sense. Using the theory developed here we obtain new results about the cohomology of generalized flag varieties associated to the classical Weyl groups, and more generally the r-diagonal coinvariant algebras. We analyze the algebraic structure of the category of FI_W-modules, and introduce restriction and induction operations that enable us to study interactions between the three families of groups. We use this theory to prove analogues of Murnaghan's 1938 stability theorem for Kronecker coefficients for the families B_n and D_n. The theory of FI_W-modules gives a conceptual framework for stability results such as these.

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arXiv2015

FI_W-modules and constraints on classical Weyl group characters

In this paper we study the characters of sequences of representations of any of the three families of classical Weyl groups W_n: the symmetric groups, the signed permutation groups (hyperoctahedral groups), or the even-signed permutation groups. Our results extend work of Church, Ellenberg, Farb, and Nagpal on the symmetric groups. We use the concept of an FI_W-module, an algebraic object that encodes the data of a sequence of W_n-representations with maps between them, defined in the author's recent work ArXiv:1309.3817. We show that if a sequence {V_n} of W_n-representations has the structure of a finitely generated FI_W-module, then there are substantial constraints on the growth of the sequence and the structure of the characters: for n large, the dimension of V_n is equal to a polynomial in n, and the characters of V_n are given by a character polynomial in signed-cycle-counting class functions, independent of n. We determine bounds the degrees of these polynomials. We continue to develop the theory of FI_W-modules, and we apply this theory to obtain new results about a number of sequences associated to the classical Weyl groups: the cohomology of complements of classical Coxeter hyperplane arrangements, and the cohomology of the pure string motion groups (the groups of symmetric automorphisms of the free group).

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arXiv2013

Representation stability for the cohomology of the pure string motion groups

The cohomology of the pure string motion group PSigma_n admits a natural action by the hyperoctahedral group W_n. Church and Farb conjectured that for each k > 0, the sequence of degree k rational cohomology groups of PSigma_n is uniformly representation stable with respect to the induced action by W_n, that is, the description of the groups' decompositions into irreducible W_n representations stabilizes for n >> k. We use a characterization of the cohomology groups given by Jensen, McCammond, and Meier to prove this conjecture. Using a transfer argument, we further deduce that the rational cohomology groups of the string motion group vanish in positive degree. We also prove that the subgroup of orientation-preserving string motions, also known as the braid-permutation group, is rationally cohomologically stable in the classical sense.

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