21 research outputs found
Mask formulas for cograssmannian Kazhdan-Lusztig polynomials
We give two contructions of sets of masks on cograssmannian permutations that
can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the
Iwahori-Hecke algebra. The constructions are respectively based on a formula of
Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky. The
first construction relies on a basis of the Hecke algebra constructed from
principal lower order ideals in Bruhat order and a translation of this basis
into sets of masks. The second construction relies on an interpretation of
masks as cells of the Bott-Samelson resolution. These constructions give
distinct answers to a question of Deodhar.Comment: 43 page
Mirabolic affine Grassmannian and character sheaves
We compute the Frobenius trace functions of mirabolic character sheaves
defined over a finite field. The answer is given in terms of the character
values of general linear groups over the finite field, and the structure
constants of multiplication in the mirabolic Hall-Littlewood basis of symmetric
functions, introduced by Shoji.Comment: 22 pages. The final version to appear in Selecta Mat
The Decomposition Theorem and the topology of algebraic maps
We give a motivated introduction to the theory of perverse sheaves,
culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and
Gabber. A goal of this survey is to show how the theory develops naturally from
classical constructions used in the study of topological properties of
algebraic varieties. While most proofs are omitted, we discuss several
approaches to the Decomposition Theorem, indicate some important applications
and examples.Comment: 117 pages. New title. Major structure changes. Final version of a
survey to appear in the Bulletin of the AM
On the top coefficients of Kazhdan-Lusztig polynomials
Kazhdan and Lusztig in 1979 defined, for any Coxeter group W, a family of polynomial that is know as Kazhdan-Lusztig polynomials. These polynomials are indexed by pairs of elements of W and they have fundamental
importance in several areas of mathematics as representation theory, geometry, combinatorics and topology of Schubert varieties. In this paper we want to show the combinatorial connection between special matchings and the top coefficient of Kazhdan-Lusztig polynomial. In particular we study a Conjecture, due to Brenti, in different Coxeter groups and pair of elements
Two-sided cells of Weyl groups and certain splitting Whittaker polynomials
Consider the subset of a Weyl group with a fixed descent set. For Weyl groups
of classical types, we determine the number of two-sided cells this subset
intersect. Moreover, we apply this result to prove that certain rational
Whittaker polynomials associated with covering groups split over the field of
rational numbers
Describing codimension two defects
Codimension two defects of the six dimensional theory
have played an important role in the understanding
of dualities for certain SCFTs in four dimensions. These
defects are typically understood by their behaviour under various dimensional
reduction schemes. In their various guises, the defects admit partial
descriptions in terms of singularities of Hitchin systems, Nahm boundary
conditions or Toda operators. Here, a uniform dictionary between these
descriptions is given for a large class of such defects in
.Comment: 74pp, lots of tables detailing order reversing duality; (v2)
Acknowledgement added. Notation simplified, refs added, minor fixes ; (v3)
Minor changes, version accepted in JHEP. I thank the referee for helpful
comments towards improving presentatio