1,744 research outputs found
Computing the canonical representation of constructible sets
Constructible sets are needed in many algorithms of Computer Algebra, particularly in the GröbnerCover and other algorithms for parametric polynomial systems. In this paper we review the canonical form ofconstructible sets and give algorithms for computing it.Peer ReviewedPostprint (author's final draft
On the K-theory of crossed products by automorphic semigroup actions
Let P be a semigroup that admits an embedding into a group G. Assume that the
embedding satisfies a certain Toeplitz condition and that the Baum-Connes
conjecture holds for G. We prove a formula describing the K- theory of the
reduced crossed product A \rtimes{\alpha},r P by any automorphic action of P.
This formula is obtained as a consequence of a result on the K-theory of
crossed products for special actions of G on totally disconnected spaces. We
apply our result to various examples including left Ore semigroups and
quasi-lattice ordered semigroups. We also use the results to show that for
certain semigroups P, including the ax + b-semigroup for a Dedekind domain R,
the K-theory of the left and right regular semigroup C*-algebras of P coincide,
although the structure of these algebras can be very different
Polynomial Bounds for Invariant Functions Separating Orbits
Consider the representations of an algebraic group G. In general, polynomial
invariant functions may fail to separate orbits. The invariant subring may not
be finitely generated, or the number and complexity of the generators may grow
rapidly with the size of the representation. We instead study "constructible"
functions defined by straight line programs in the polynomial ring, with a new
"quasi-inverse" that computes the inverse of a function where defined. We write
straight line programs defining constructible functions that separate the
orbits of G. The number of these programs and their length have polynomial
bounds in the parameters of the representation.Comment: Clarified proofs, algorithms, and notation. Corrected typo
Geometry and categorification
We describe a number of geometric contexts where categorification appears
naturally: coherent sheaves, constructible sheaves and sheaves of modules over
quantizations. In each case, we discuss how "index formulas" allow us to easily
perform categorical calculations, and readily relate classical constructions of
geometric representation theory to categorical ones.Comment: 23 pages. an expository article to appear in "Perspectives on
Categorification.
Constructible characters and canonical bases
We give closed formulas for all vectors of the canonical basis of a level 2
irreducible integrable representation of . These formulas
coincide at v=1 with Lusztig's formulas for the constructible characters of the
Iwahori-Hecke algebras of type B and D.Comment: 16 page
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