Congruence-simplicity of Steinberg algebras of non-Hausdorff ample groupoids over semifields

We investigate the algebra of an ample groupoid, introduced by Steinberg, over a semifield S. In particular, we obtain a complete characterization of congruence-simpleness for Steinberg algebras of second-countable ample groupoids, extending the well-known characterizations when S is a field. We apply our congruence-simplicity results to tight groupoids of inverse semigroup representations associated to self-similar graphs.     

The Steinberg variety and representations of reductive groups

We give an overview of some of the main results in geometric representation theory that have been proved by means of the Steinberg variety. Steinberg's insight was to use such a variety of triples in order to prove a conjectured formula by Grothendieck. The Steinberg variety was later used to give an alternative approach to Springer's representations and played a central role in the proof of the Deligne–Langlands conjecture for Hecke algebras by Kazhdan and Lusztig.     

A method of proving non-unitarity of representations of p-adic groups I

In this paper we exhibit a new method of proving non-unitarity of representations, based on semi simplicity of unitarizable representations. Non-unitarity is proved for a half of all irreducible representations of classical p-adic groups whose infinitesimal character is the same as the infinitesimal character of a generalized Steinberg representation (as defined in Tadić, Am J Math 120:159–210, 1998). Only the Steinberg representation and its Aubert dual are expected to be unitary here. In this way we partially generalize a result of Casselman to the case of classical groups. Our argument is completely different from Casselman’s argument (which is hard to extend to this case). It requires a very limited knowledge of the inducing cuspidal representation.     

Some infinite dimensional representations of reductive groups with Frobenius maps

In this paper,we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps.In particular,a few classical results of Steinberg and Deligne&Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.     

On the depth <Emphasis Type="Italic">r</Emphasis> Bernstein projector

In this paper we prove an explicit formula for the Bernstein projector to representations of depth \(\le r\). As a consequence, we show that the depth zero Bernstein projector is supported on topologically unipotent elements and it is equal to the restriction of the character of the Steinberg representation. As another application, we deduce that the depth r Bernstein projector is stable. Moreover, for integral depths our proof is purely local.     

Simplicité abstraite des groupes de Kac–Moody non affines

We announce the simplicity of non-affine Kac–Moody lattices (modulo center). The groups under consideration are minimal Kac–Moody groups. They were defined by Jacques Tits by means of a presentation à la Steinberg. The ground field is finite, assumed to be of cardinality greater than the rank of the buildings these groups naturally act upon. We work in the general combinatorial context of twin root data. To cite this article: P.-E. Caprace, B. Rémy, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

The Steinberg formula for orbit groups

The aim of this note is to give a direct proof of the fact that the Steinberg formula holds for the Vaserstein symbol and the universal weak Mennicke symbol. We also give an alternate proof of the fact that the Steinberg formula holds for the Vaserstein symbol.     

On folded coverings of quasi-manifolds with boundary

LetM andN be quasi-manifolds possibly with boundary, we call afolded-covering p: N→M to a non-degenerate simplicial map, such thatp: N/p ^?1(?M)→M/?M is a branched covering. The purpose of this paper will be to classify the folded coverings of a given quasi-manifold by using the representations of a particular group on permutation groups, similar to the monodromy representations for coverings and in such a way that includes these ones a particular case. We shall apply our result to the theory of orbifolds to obtain a presentation (alternative to the one in [H-Q]) of the fundamental group for an orbifold defined on a quasi-manifold.     

Partition Complexes, Tits Buildings and Symmetric Products

We construct a homological approximation to the partition complex,and identify it as the Tits building. This gives a homologicalrelationship between the symmetric group and the affine group,leads to a geometric tie between symmetric powers of spheresand the Steinberg idempotent, and allows us to use the self-dualityof the Steinberg module to study layers in the Goodwillie towerof the identity functor. 2000 Mathematics Subject Classification:55N25, 55S15, 20B30, 55P25.     

The principal bundles over an inverse semigroup

This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra (Theory Appl Categ 24(7):117–147, 2010). For the topos of sets, we show that torsion-free functors on Loganathan’s category L(S) of an inverse semigroup S are equivalent to a special class of non-strict representations of S, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of S. We describe the correspondence between directed and pullback preserving functors on L(S) and transitive and effective representations of S, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation, or, equivalently, an S-torsor, of an inverse semigroup S in the topos of sheaves \({\mathsf {Sh}}(X)\) on a topological space X. We prove that the category of filtered functors from L(S) to the topos \({\mathsf {Sh}}(X)\) is equivalent to the category of universal representations of S in \({\mathsf {Sh}}(X)\). We finally propose a definition of an inverse semigroup action in an arbitrary Grothendieck topos, which arises from a functor on L(S).     

Norm Series for Lubin–Tate Formal Groups

The Steinberg relation for the classical Hilbert symbol is generalized to the Lubin–Tate formal groups. Necessary and sufficient conditions for the series satisfying the generalized Steinberg relation are obtained (in the paper such series are called norm series). Bibliography: 5 titles.     

Jucys–Murphy elements and a presentation for partition algebras

We give a new presentation for the partition algebras. This presentation was discovered in the course of establishing an inductive formula for the partition algebra Jucys–Murphy elements defined by Halverson and Ram (Eur. J. Comb. 26:869–921, 2005). Using Schur–Weyl duality we show that our recursive formula and the original definition of Jucys–Murphy elements given by Halverson and Ram are equivalent. The new presentation and inductive formula for the partition algebra Jucys–Murphy elements given in this paper are used to construct the seminormal representations for the partition algebras in a separate paper.     

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space which is totally disconnected, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{?}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.     

Knuth’s Coherent Presentations of Plactic Monoids of Type A

We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation and we reduce it to a Tietze equivalent one having Knuth’s generators.     

Hermitian Groups and K-Theory

This article provides the fundamental constructions and results for Hermitian groups which are necessary for an algebraic understanding of the functors K₁ and K₂ of Hermitian forms. These include a definition of the elementary subgroup EH of a general Hermitian group GH, the Hermitian Whitehead lemma for stabilized GH, a definition of the Hermitian Steinberg group, and the theorem that a stabilized Steinberg group is the universal central extension of its associated stabilized elementary group.     

Characterizing prospective secondary teachers’ foundation and contingency knowledge for definitions of transformations

One promising approach for connecting undergraduate content coursework to secondary teaching is using teacher-created representations of practice. Using these representations effectively requires seeing teachers' use of mathematical knowledge in the work of teaching. We argue that the dimensions of Rowland's (2013) Knowledge Quartet, especially Foundation and Contingency, form a fruitful framework for this purpose. We contribute an analytic framework to characterize the quality of mathematical knowledge observed in the Foundation and Contingency dimensions, developed using a purposive sampling from over 300 representations. These representations all featured geometry teaching. We showcase the framework with examples of "high" and "developing" Foundation and Contingency.Then, we compare our coding along these dimensions with performance on a measure of mathematical knowledge for teaching geometry. Finally, we describe the potential for generalizing this framework to other domains, such as algebra and mathematical modeling.     

Exterior square gamma factors for cuspidal representations of $$\mathrm {GL}_n$$ GL n : simple supercuspidal representations

We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations.