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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

The quantity of health spending does not equate directly to the quality of healthcare. Social spending and nonmedical determinants of health impact on health outcomes. The value of health spending in the United States is limited by the high cost of goods and labor, and administrative costs, high use of low-level care, and lack of basic healthcare coverage in a substantial minority of the population. There are additional reasons for differences in the value of health spending as revealed by a comparison of Canada and Australia, where similar spending per capita results in better health system performance in Australia than in Canada. Within Canada, there are large differences between provinces in the value of health spending. Overutilization of health resources occurs when an intervention is not indicated, or of low value, or is provided at the wrong time. The Choosing Wisely Initiative is a solution for overutilization of low-value care. Recognition of the reality that effective treatments are often applied inconsistently and inappropriately has spurred the development of a science concerned with improvement of quality of care.

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 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!

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