15 research outputs found
Kazhdan-Lusztig polynomials of matroids under deletion
We present a formula which relates the Kazhdan-Lusztig polynomial of a
matroid , as defined by Elias, Proudfoot and Wakefield, to the
Kazhdan--Lusztig polynomials of the matroid obtained by deleting an element,
and various contractions and localizations of . We give a number of
applications of our formula to Kazhdan--Lusztig polynomials of graphic
matroids, including a simple formula for the Kazhdan--Lusztig polynomial of a
parallel connection graph.Comment: 21 pages, two figures. Minor updates and correction
Topology of Arrangements and Representation Stability
The workshop “Topology of arrangements and representation stability” brought together two directions of research: the topology and geometry of hyperplane, toric and elliptic arrangements, and the homological and representation stability of configuration spaces and related families of spaces and discrete groups. The participants were mathematicians working at the interface between several very active areas of research in topology, geometry, algebra, representation theory, and combinatorics. The workshop provided a thorough overview of current developments, highlighted significant progress in the field, and fostered an increasing amount of interaction between specialists in areas of research
Geometric, Algebraic, and Topological Combinatorics
The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics"
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference included (1) Karim Adiprasito presented his
very recent proof of the -conjecture for spheres (as a talk and as a "Q\&A"
evening session) (2) Federico Ardila gave an overview on "The geometry of matroids",
including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz
Singular Hodge theory for combinatorial geometries
We introduce the intersection cohomology module of a matroid and prove that
it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the
Hodge-Riemann relations. As applications, we obtain proofs of Dowling and
Wilson's Top-Heavy conjecture and the nonnegativity of the coefficients of
Kazhdan-Lusztig polynomials for all matroids.Comment: 101 pages; v3: major improvements to the exposition, particularly in
Sections 8, 9, 10. Fixed a gap in the proof of the main result of Section 10.
Added one new result (Theorem 1.4), giving the monotonicity of coefficients
of Kazhdan-Lusztig polynomials of matroids under contraction of flat