1,377 research outputs found
Buildings, spiders, and geometric Satake
Let G be a simple algebraic group. Labelled trivalent graphs called webs can
be used to product invariants in tensor products of minuscule representations.
For each web, we construct a configuration space of points in the affine
Grassmannian. Via the geometric Satake correspondence, we relate these
configuration spaces to the invariant vectors coming from webs. In the case G =
SL(3), non-elliptic webs yield a basis for the invariant spaces. The
non-elliptic condition, which is equivalent to the condition that the dual
diskoid of the web is CAT(0), is explained by the fact that affine buildings
are CAT(0).Comment: 49 pages; revised and to appear in Compositio Mathematic
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A design representation model for high-level synthesis
Design tools share and exchange various types of information pertaining to the design. The identification of a uniform design representation to capture this information is essential for the development of a successful design environment. We have done an extensive study on the representation needs of existing database tools in the UCI CADLAB; examples of which are graph compilers for high-level hardware specifications, state schedulers, hardware allocators, and microarchitecture optimizers. The result of this study is the development of a design representation model that will serve as a common internal representation (DDM) for all system and behavioral synthesis tools. DDM thus builds the foundation for a CAD Framework in which design tools can communicate via operating on this common representation. The design information is composed of three separate graph models: the conceptual model, the behavioral model and the structural model. The conceptual model (represented by a Design Entity Graph) captures the overall organization of the design information, such as, versions and configurations. The behavioral model (represented by an Augmented Control/Data Flow Graph) describes the design behavior. The structural model (represented by an Annotated Component Graph) captures the hierarchical data path structure and its geometric information. In this paper, we define the last two graph models. They both capture the actual design data of the application domain. Since VHDL has gained increasing popularity as hardware description language for synthesis, we give numerous examples throughout this report that show how the proposed design representation model can be used to represent VHDL specifications
Path isomorphisms between quiver Hecke and diagrammatic Bott-Samelson endomorphism algebras
We construct an explicit isomorphism between (truncations of) quiver Hecke
algebras and Elias-Williamson's diagrammatic endomorphism algebras of
Bott-Samelson bimodules. As a corollary, we deduce that the decomposition
numbers of these algebras (including as examples the symmetric groups and
generalised blob algebras) are tautologically equal to the associated
-Kazhdan-Lusztig polynomials, provided that the characteristic is greater
than the Coxeter number. We hence give an elementary and more explicit proof of
the main theorem of Riche-Williamson's recent monograph and extend their
categorical equivalence to cyclotomic Hecke algebras, thus solving
Libedinsky-Plaza's categorical blob conjecture
Auxetic regions in large deformations of periodic frameworks
In materials science, auxetic behavior refers to lateral widening upon
stretching. We investigate the problem of finding domains of auxeticity in
global deformation spaces of periodic frameworks. Case studies include planar
periodic mechanisms constructed from quadrilaterals with diagonals as periods
and other frameworks with two vertex orbits. We relate several geometric and
kinematic descriptions.Comment: Presented at the International Conference on "Interdisciplinary
Applications of Kinematics" (IAK18), Lima, Peru, March 201
Path Similarity Analysis: a Method for Quantifying Macromolecular Pathways
Diverse classes of proteins function through large-scale conformational
changes; sophisticated enhanced sampling methods have been proposed to generate
these macromolecular transition paths. As such paths are curves in a
high-dimensional space, they have been difficult to compare quantitatively, a
prerequisite to, for instance, assess the quality of different sampling
algorithms. The Path Similarity Analysis (PSA) approach alleviates these
difficulties by utilizing the full information in 3N-dimensional trajectories
in configuration space. PSA employs the Hausdorff or Fr\'echet path
metrics---adopted from computational geometry---enabling us to quantify path
(dis)similarity, while the new concept of a Hausdorff-pair map permits the
extraction of atomic-scale determinants responsible for path differences.
Combined with clustering techniques, PSA facilitates the comparison of many
paths, including collections of transition ensembles. We use the closed-to-open
transition of the enzyme adenylate kinase (AdK)---a commonly used testbed for
the assessment enhanced sampling algorithms---to examine multiple microsecond
equilibrium molecular dynamics (MD) transitions of AdK in its substrate-free
form alongside transition ensembles from the MD-based dynamic importance
sampling (DIMS-MD) and targeted MD (TMD) methods, and a geometrical targeting
algorithm (FRODA). A Hausdorff pairs analysis of these ensembles revealed, for
instance, that differences in DIMS-MD and FRODA paths were mediated by a set of
conserved salt bridges whose charge-charge interactions are fully modeled in
DIMS-MD but not in FRODA. We also demonstrate how existing trajectory analysis
methods relying on pre-defined collective variables, such as native contacts or
geometric quantities, can be used synergistically with PSA, as well as the
application of PSA to more complex systems such as membrane transporter
proteins.Comment: 9 figures, 3 tables in the main manuscript; supplementary information
includes 7 texts (S1 Text - S7 Text) and 11 figures (S1 Fig - S11 Fig) (also
available from journal site
Diagrams for perverse sheaves on isotropic Grassmannians and the supergroup SOSP(m|2n)
We describe diagrammatically a positively graded Koszul algebra \mathbb{D}_k
such that the category of finite dimensional \mathbb{D}_k-modules is equivalent
to the category of perverse sheaves on the isotropic Grassmannian of type D_k
constructible with respect to the Schubert stratification. The connection is
given by an explicit isomorphism to the endomorphism algebra of a projective
generator described in by Braden. The algebra is obtained by a "folding"
procedure from the generalized Khovanov arc algebras. We relate this algebra to
the category of finite dimensional representations of the orthosymplectic
supergroups. The proposed equivalence of categories gives a concrete
description of the categories of finite dimensional SOSP(m|2n)-modules
Vertex elimination orderings for hereditary graph classes
We provide a general method to prove the existence and compute efficiently
elimination orderings in graphs. Our method relies on several tools that were
known before, but that were not put together so far: the algorithm LexBFS due
to Rose, Tarjan and Lueker, one of its properties discovered by Berry and
Bordat, and a local decomposition property of graphs discovered by Maffray,
Trotignon and Vu\vskovi\'c. We use this method to prove the existence of
elimination orderings in several classes of graphs, and to compute them in
linear time. Some of the classes have already been studied, namely
even-hole-free graphs, square-theta-free Berge graphs, universally signable
graphs and wheel-free graphs. Some other classes are new. It turns out that all
the classes that we study in this paper can be defined by excluding some of the
so-called Truemper configurations. For several classes of graphs, we obtain
directly bounds on the chromatic number, or fast algorithms for the maximum
clique problem or the coloring problem
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