Homomorphism from braid groups to Steinberg groups defined.
problem Understanding the relationship between braid groups and Steinberg groups.
method Construction of a homomorphism from braid groups to Steinberg groups.
result Description of the image and kernel of the homomorphism.
Study shows Steinberg representation's multiplicity in cohomology of congruence subgroups.
problem Analyzing multiplicity of Steinberg representation in cohomology of congruence subgroups.
method Computation of cohomology of S-arithmetic groups outside a linear range of degrees. result Multiplicity of Steinberg representation is 1 in top-degree cohomology.
Projective resolves symplectic Steinberg module for number rings.
problem Constructing a projective resolution for symplectic Steinberg module.
method Similar to special linear group, but more complex construction.
result Computed top degree cohomology of congruence subgroups.
The abstract defines and studies a Tits building for commutative rings and proves a Solomon-Tits theorem under certain conditions.
problem Defining and studying a Tits building for commutative rings.
method Proving a Solomon-Tits theorem for commutative rings under specific conditions, defining Steinberg modules, and computing ranks and lengths.
result Proves a Solomon-Tits theorem for commutative rings satisfying certain conditions.
Generalized Steinberg module presentation for Gaussian and Eisenstein integers.
problem Presenting Steinberg modules for specific number rings.
method Generalization of Bykovskii's presentation to Gaussian and Eisenstein integers.
result Generalization does not yield a presentation for all Euclidean number rings.
We prove that H^{d-1}(SL_n Z; Q) = 0, where d = n-choose-2 is the cohomological dimension of SL_n Z, and similarly for GL_n Z. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group GL_n. These theorems are derived from a presentation of the Stein…
Proves conjecture simplifying mapping class group action on Steinberg module.
problem Characterizing action of mapping class group on Steinberg module.
method Simple characterisation of mapping class group action on Harvey's complex of curves.
result Kernel of mapping class group action on Steinberg module is trivial.
Let W⋉L be an irreducible affine Weyl group with Coxeter complex Σ, where W denotes the associated finite Weyl group and L the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of Σ by the lattice L. We show that the ordinary and flag h-polynomial…
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 ri…
We find a presentation of symplectic Steinberg modules and show vanishing cohomology for certain groups.
problem Cohomology vanishing for specific groups and modules.
method Presented a symplectic Steinberg module and used it to prove cohomology vanishing.
result Cohomology of Sp2n(Z) vanishes in a specific degree for n≥2. We introduce the notions of overcommutation and overcommutation length in groups, and show that these concepts are closely related to representations of the fundamental groups of 3-manifold and their Heegaard genus. We give many examples including translations in the affine group of the line and provide upper bounds fo…
Explicitly found generators of cohomology for SL_n(Z) using sharbly cycles and cosharbly cocycles.
problem Finding explicit generators for the cohomology of SL_n(Z).
method Using sharbly cycles and cosharbly cocycles, and applying Borel-Serre duality.
result Explicitly found generators of H_t(SL_n(Z),St) in terms of sharbly cycles and cosharbly cocycles.
Study on cohomology of SL_n(Z) for n>=3, proving vanishing of certain cohomology groups.
problem Determine the cohomology of SL_n(Z) for n>=3.
method Construct a partial resolution of the Steinberg module to show vanishing of specific cohomology groups.
result Vanishing of codimension-2 rational cohomology group H^{{n \choose 2} -2} for n >= 3.
New identities link Frobenius elements to Jones-Wenzl projectors at roots of unity.
problem Understanding relationships between Frobenius elements and Jones-Wenzl projectors at roots of unity.
method Obtained skein identities relating Frobenius elements to Jones-Wenzl projectors in the Kauffman bracket skein module.
result Skein identities provide new proofs of the existence of the Chebyshev-Frobenius homomorphism.
By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the …
Shows natural quasi-Poisson structure on multiplicative Grothendieck-Springer space.
problem Exploring new structures in algebraic geometry.
method Reduction along Dirac realizations.
result Natural quasi-Poisson structure exists on multiplicative Grothendieck-Springer space.
For a number ring O, Borel and Serre proved that SLn(O) is a virtual duality group whose dualizing module is the Steinberg module. They also proved that GLn(O) is a virtual duality group. In contrast to SLn(O), we prove that the dualizing module of…
Study of symplectic groupoids from tt*-Toda equations.
problem Geometry of meromorphic connections with irregular singularities.
method Holomorphic symplectic groupoid structure over Steinberg cross section.
result Proves the space of tt*-Toda connections is a symplectic Lie groupoid.
These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal …
We propose a Lie-theoretic definition of the tt*-Toda equations for any complex simple Lie algebra g, based on the concept of topological-antitopological fusion which was introduced by Cecotti and Vafa. Our main result concerns the Stokes data of a certain meromorphic connection, whose isomonodromic deform…
We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this int…
We show that, at the prime p=2, the spectrum Σ−nD(n) splits off the Madsen-Tillmann spectrum MTO(n)=BO(n)−γn which is compatible with the classic splitting of M(n) off BO(n)+. For n=2, together with our previous splitting result on Madsen-Tillmann spectra, this shows that MTO(2) is homotopy equiva…
We give a Lie-theoretic explanation for the convex polytope which parametrizes the globally smooth solutions of the topological-antitopological fusion equations of Toda type (tt∗-Toda equations) which were introduced by Cecotti and Vafa. It is known from [GL] [GIL1] [M1] [M2] that these solutions can be parametrized…
Via a computer search, Altshuler and Steinberg found that there are 1296 +1 combinatorial 3-manifolds on nine vertices, of which only one is non-sphere. This exceptional 3-manifold K93 triangulates the twisted S2-bundle over S1. It was first constructed by Walkup. In this paper, we present a computer-…
Adaptive OMD reduces variance in learning optimal strategies for imperfect information games.
problem High variance in learning optimal strategies for imperfect information games.
method Fixed sampling approach with locally applied Online Mirror Descent (OMD) algorithm.
result Convergence rate of ildeO(T−1/2) with high probability. New representation theory for closed geodesic subflows.
problem Classifying representations with good geometric properties.
method Restricting to invariant closed geodesic subflows.
result Equivalent characterizations and properties of new representations.
Proves EGF representations in specific geometric contexts.
problem Understanding representations of groups with hyperbolic properties.
method Analyzes projectively convex cocompact manifolds and convex projective manifolds with generalized cusps.
result Holonomy representations of specific geometric manifolds are EGF representations.
The paper establishes isomorphisms and constructs colored versions of Lawrence representations.
problem Understanding isomorphisms and colored versions of Lawrence representations.
method Explicit isomorphisms and construction of colored versions.
result Matrices for colored versions of BKL and Lawrence representations provided.
Study k-positive surface group representations and their degenerations.
problem Understanding the behavior of surface group representations under degenerations.
method Introduced k-positive representations and studied their degenerations using a limit theorem for positively ratioed representations.
result Degenerations of k-positive representations can lead to limits that are at least (k-3)-positive and irreducible limits are (k-1)-positive.
Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.
problem Understanding geometrically finite Fuchsian groups and their representations.
method Theory of Anosov representations, type-preserving deformations, limit maps, relative Anosov and dominated representations.
result Cusped Hitchin representations are Borel Anosov, stable under deformations, and limit maps vary analytically.
New representations defined for groups and graphs, with applications to stable representations.
problem Defining and constructing new types of representations for groups and graphs.
method Introducing (R,Λ)-directed Anosov representations and using Fock-Goncharov positivity to construct them. result Constructs large families of primitive stable representations from F2 to PGL(V), including non-discrete and non-faithful examples. Collar lemma proven for certain surface group representations.
problem Proving a collar lemma for specific surface group representations.
method Using partial hyperconvexity properties and Anosov representations.
result 'Positivity properties' hold for partially hyperconvex representations.
Study subgroup actions on mapping class groups using Heisenberg representations.
problem Untwisting representations of mapping class groups on Heisenberg subgroups.
method Restrict and analyze twisted representations of mapping class groups to Heisenberg subgroups.
result Untwisting representations on Torelli group for any Heisenberg representation.
Researchers describe unitary representations of mixed braid groups.
problem Understanding unitary representations of mixed braid groups.
method Explicitly describe unitary representations on cohomology of Abelian branched covers.
result Image of the representation is generated by complex reflections and related to the multivariate Burau representation.
This paper addresses law invariant coherent risk measures and their Kusuoka representations. By elaborating the existence of a minimal representation we show that every Kusuoka representation can be reduced to its minimal representation. Uniqueness -- in a sense specified in the paper -- of the risk measure's Kusuoka r…
A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group B_n. In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this pape…
Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain …
In this paper, we introduce a study of prolongations of representations of Lie groups. We obtain a faithful (one-to-one) representation of TG where G is a finite-dimensional Lie group and TG is the tangent bundle of G, by using (not necessarily faithful) representations of G. We show that tangent functions of Lie group…
Polynomial representations found in surface braid and mapping class groups.
problem Homological representations of surface braid and mapping class groups.
method Study of homological representation functors and short exact sequences.
result Many homological representation functors are polynomial.
Paper addresses the disparity between sampled and mean representations in disentangled learning.
problem Disparity between sampled and mean representations in disentangled learning.
method Proposes a method to eliminate the disparity by proving and utilizing the relationship between total correlation of sampled and mean representations for multivariate normal distributions.
result Demonstrates that a factorized mean representation can have lower total correlation than the sampled representation.
New Θ-positive representations of surface groups discovered.
problem Generalizing Lusztig's total positivity to surface groups.
method Introducing Θ-positivity and proving properties of Θ-positive representations. result Discrete and faithful Θ-positive representations exist and form open sets in representation varieties. New representations for surface groups expand known Anosov classes.
problem Understanding new types of representations for surface groups.
method Introducing and studying simple Anosov representations of closed hyperbolic surface groups.
result Simple Anosov representations strictly contain Anosov representations.
New findings on cusped Borel Anosov representations and their properties.
problem Characterizing and understanding cusped Borel Anosov representations.
method Analyzing representations of lattices in PGL2(R) to PGLd(R). result Cusped Borel Anosov representations with specific properties are Hitchin representations.
The paper formalizes criteria for non-spurious and disentangled representations using causal methods.
problem Formalizing criteria for non-spurious and disentangled representations in representation learning.
method Causal perspective, counterfactual quantities, observable consequences of causal assertions.
result Computable metrics for assessing representation learning based on observed data.
Characterizes Anosov reducible representations in terms of eigenvalues.
problem Understanding Anosov representations in reducible settings.
method Characterizes Anosov representations using eigenvalue magnitudes of irreducible block factors.
result Connected components of character varieties do not contain reducible representations for many non-elementary hyperbolic groups.
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermit…
Study local structure of knot group representations into SL(n,C).
problem Understanding the local structure of knot group representations.
method Analysis of tangent cone and use of Luna's slice theorem.
result Local structure of representation variety at diagonal representations.