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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., Comment: 17 pages. To appear in Journal of Topology

Using a result of Gan–Li on F I \mathsf {FI} -hyperhomology and a semisimplicial resolution of configuration spaces due to Randal-Williams, we establish an improved representation stability stable range for configuration spaces of distinct ordered points in a manifold. Our bounds on generation degree improve the best known stability slope by a factor of 5/2 in the most general case. We adapt this result of Gan–Li to apply beyond stability arguments involving highly connected simplicial complexes, and our methods suggest that their result may be widely applicable to improving most stability ranges for F I \mathsf {FI} -modules in the current representation stability literature.

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., Comment: Minor revisions based on referee's comments. Accepted for publication at IMRN

In this paper we study the characters of sequences of representations of any of the three families of classical Weyl groups $$\mathcal {W}_n$$ : the symmetric groups, the signed permutation groups (hyperoctahedral groups), or the even-signed permutation groups. Our results extend work of Church et al. (Duke Math J, 2012. arXiv:1204.4533 ; Geom Topol 18(5):2951–2984, 2014) on the symmetric groups. We use the concept of an $$\text {FI}_{\mathcal {W}}$$ -module, an algebraic object that encodes the data of a sequence of $$\mathcal {W}_n$$ -representations with maps between them, defined in the author’s recent work (J Algebra 420:269–332, 2014). We show that if a sequence $$\{V_n\}$$ of $$\mathcal {W}_n$$ -representations has the structure of a finitely generated $$\text {FI}_{\mathcal {W}}$$ -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 $$\text {FI}_{\mathcal {W}}$$ -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).

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 [7,8] 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.

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., Comment: We have revised Section 4 to correct an error in the previous version. 35 pages, 6 figures. To appear in IMRN

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., Comment: 36 pages. The new version added a citation. Comments welcome!

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., 45 pages, 41 figures. Minor corrections and revisions made throughout to clarify and shorten the paper. For reasons for length the appendix has been cut and will be made into a separate paper. Final version; accepted at Geometry & Topology

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., 24 pages, 3 figures

If $\alpha \in S_n$ is a permutation of $\{1, 2, \ldots, n\}$, the inversion set of $\alpha$ is $\Phi(\alpha) = \{(i, j) \, | \, 1 \leq i < j \leq n, \alpha(i) > \alpha(j)\}$. We describe all $r$-tuples $\alpha_1, \alpha_2, \ldots, \alpha_r \in S_n$ such that $\Delta_n^+ = \{(i, j) \, | \, 1 \leq i < j \leq n\}$ is the disjoint union of $\Phi(\alpha_1), \Phi(\alpha_2), \ldots, \Phi(\alpha_r)$. Using this description we prove that certain faces of the Littlewood-Richardson cone are simplicial and provide an algorithm for writing down their sets of generating rays. We also discuss analogous problems for the Weyl groups of root systems of types $B$, $C$ and $D$ providing solutions for types $B$ and $C$. Finally we provide some enumerative results and introduce a useful tool for visualizing inversion sets., Comment: 43 pages

We present a non-commutative version of the cycle lemma of Dvoretsky and Motzkin that applies to free groups and use this result to solve a number of problems involving cyclic reduction in the free group. We also describe an application to random matrices, in particular the fluctuations of Kesten's Law., 13 pages, minor corrections and improvements