23,493 research outputs found
Thick Soergel calculus in type A
Let R be the polynomial ring in n variables, acted on by the symmetric group
S_n. Soergel constructed a full monoidal subcategory of R-bimodules which
categorifies the Hecke algebra, whose objects are now known as Soergel
bimodules. Soergel bimodules can be described as summands of Bott-Samelson
bimodules (attached to sequences of simple reflections), or as summands of
generalized Bott-Samelson bimodules (attached to sequences of parabolic
subgroups). A diagrammatic presentation of the category of Bott-Samelson
bimodules was given by the author and Khovanov in previous work. In this paper,
we extend it to a presentation of the category of generalized Bott-Samelson
bimodules. We also diagrammatically categorify the representations of the Hecke
algebra which are induced from trivial representations of parabolic subgroups.
The main tool is an explicit description of the idempotent which picks out a
generalized Bott-Samelson bimodule as a summand inside a Bott-Samelson
bimodule. This description uses a detailed analysis of the reduced expression
graph of the longest element of S_n, and the semi-orientation on this graph
given by the higher Bruhat order of Manin and Schechtman.Comment: Changed title. Expanded the exposition of the main proof. This paper
relies extensively on color figure
The role of concurrency in an evolutionary view of programming abstractions
In this paper we examine how concurrency has been embodied in mainstream
programming languages. In particular, we rely on the evolutionary talking
borrowed from biology to discuss major historical landmarks and crucial
concepts that shaped the development of programming languages. We examine the
general development process, occasionally deepening into some language, trying
to uncover evolutionary lineages related to specific programming traits. We
mainly focus on concurrency, discussing the different abstraction levels
involved in present-day concurrent programming and emphasizing the fact that
they correspond to different levels of explanation. We then comment on the role
of theoretical research on the quest for suitable programming abstractions,
recalling the importance of changing the working framework and the way of
looking every so often. This paper is not meant to be a survey of modern
mainstream programming languages: it would be very incomplete in that sense. It
aims instead at pointing out a number of remarks and connect them under an
evolutionary perspective, in order to grasp a unifying, but not simplistic,
view of the programming languages development process
PLACES'10: The 3rd Workshop on Programmng Language Approaches to concurrency and Communication-Centric Software
Paphos, Cyprus. March 201
- …