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Nicholas Proudfoot

Publications and source records attributed to Nicholas Proudfoot.

At least 37 records · Page 2Linked to original sources
arXiv2022

Equivariant Kazhdan-Lusztig theory of paving matroids

We study the way in which equivariant Kazhdan-Lusztig polynomials, equivariant inverse Kazhdan-Lusztig polynomials, and equivariant Z-polynomials of matroids change under the operation of relaxation of a collection of stressed hyperplanes. This allows us to compute these polynomials for arbitrary paving matroids, which we do in a number of examples, including various matroids associated with Steiner systems that admit actions of Mathieu groups.

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arXiv2021

Equivariant log concavity and representation stability

We expand upon the notion of equivariant log concavity, and make equivariant log concavity conjectures for Orlik--Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik--Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{sl}_n$, we exploit the theory of representation stability to give computer assisted proofs of these conjectures in low degree.

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arXiv2021

Intersection cohomology of hypertoric varieties

A hypertoric variety is a quaternionic analogue of a toric variety. Just as the topology of toric varieties is closely related to the combinatorics of polytopes, the topology of hypertoric varieties interacts richly with the combinatorics of hyperplane arrangements and matroids. Using finite field methods, we obtain combinatorial descriptions of the Betti numbers of hypertoric varieties, both for ordinary cohomology in the smooth case and intersection cohomology in the singular case. We also introduce a conjectural ring structure on the intersection cohomology of a hypertoric variety.

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arXiv2020

Stability phenomena for resonance arrangements

We prove that the ith graded pieces of the Orlik-Solomon algebras or Cordovil algebras of resonance arrangements form a finitely generated FS^op-module, thus obtaining information about the growth of their dimensions and restrictions on the irreducible representations of symmetric groups that they contain.

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arXiv2020

A type B analogue of the category of finite sets with surjections

We define a type B analogue of the category of finite sets with surjections, and we study the representation theory of this category. We show that the opposite category is quasi-Grobner, which implies that submodules of finitely generated modules are again finitely generated. We prove that the generating functions of finitely generated modules have certain prescribed poles, and we obtain restrictions on the representations of type B Coxeter groups that can appear in such modules. Our main example is a module that categorifies the degree i Kazhdan-Lusztig coefficients of type B Coxeter arrangements.

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arXiv2020

A semi-small decomposition of the Chow ring of a matroid

We give a semi-small orthogonal decomposition of the Chow ring of a matroid M. The decomposition is used to give simple proofs of Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations for the Chow ring, recovering the main result of [AHK18]. We also show that a similar semi-small orthogonal decomposition holds for the augmented Chow ring of M.

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arXiv2020

The contraction category of graphs

We study the category whose objects are graphs of fixed genus and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian and we study two families of modules over these categories. The first takes a graph to a graded piece of the homology of its unordered configuration space and the second takes a graph to an intersection homology group whose dimension is given by a Kazhdan-Lusztig coefficient; in both cases we prove that the module is finitely generated. This allows us to draw conclusions about torsion in the homology groups of graph configuration spaces, and about the growth of Betti numbers of graph configuration spaces and Kazhdan-Lusztig coefficients of graphical matroids. We also explore the relationship between our category and outer space, which is used in the study of outer automorphisms of free groups.

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arXiv2020

The quantum Hikita conjecture

The Hikita conjecture relates the coordinate ring of a conical symplectic singularity to the cohomology ring of a symplectic resolution of the dual conical symplectic singularity. We formulate a quantum version of this conjecture, which relates the quantized coordinate ring of the first variety to the quantum cohomology of a symplectic resolution of the dual variety. We prove this conjecture for hypertoric varieties and for the Springer resolution. Our paper includes an appendix, written by Ben Webster, which studies highest weights for quantizations of symplectic resolutions with isolated torus actions.

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arXiv2019

Configuration spaces, $\operatorname{FS^{op}}$-modules, and Kazhdan-Lusztig polynomials of braid matroids

The equivariant Kazhdan-Lusztig polynomial of a braid matroid may be interpreted as the intersection cohomology of a certain partial compactification of the configuration space of n distinct labeled points in the plane, regarded as a graded representation of the symmetric group. We show that, in fixed cohomological degree, this sequence of representations of symmetric groups naturally admits the structure of an FS-module, and that the dual FS^op-module is finitely generated. Using the work of Sam and Snowden, we give an asymptotic formula for the dimensions of these representations and obtain restrictions on which irreducible representations can appear in their decomposition.

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arXiv2019

Functorial invariants of trees and their cones

We study the category whose objects are trees (with or without roots) and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian, and we study two natural families of modules over these categories. The first takes a tree to a graded piece of the homology of its unordered configuration space, or to the homology of the unordered configuration space of its cone. The second takes a tree to a graded piece of the intersection homology of the reciprocal plane of its cone, which is a vector space whose dimension is given by a Kazhdan-Lusztig coefficient. We prove finite generation results for each of these modules, which allow us to obtain results about the growth of Betti numbers of configuration spaces and of Kazhdan-Lusztig coefficients of graphical matroids.

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arXiv2018

Equivariant Kazhdan-Lusztig polynomials of $q$-niform matroids

We introduce $q$-analogues of uniform matroids, which we call $q$-niform matroids. While uniform matroids admit actions of symmetric groups, $q$-niform matroids admit actions of finite general linear groups. We show that the equivariant Kazhdan-Lusztig polynomial of a $q$-niform matroid is the unipotent $q$-analogue of the equivariant Kazhdan-Lusztig polynomial of the corresponding uniform matroid, thus providing evidence for the positivity conjecture for equivariant Kazhdan-Lusztig polynomials.

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arXiv2018

The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials

Kazhdan-Lusztig-Stanley polynomials are a combinatorial generalization of Kazhdan-Lusztig polynomials of for Coxeter groups that include g-polynomials of polytopes and Kazhdan-Lusztig polynomials of matroids. In the cases of Weyl groups, rational polytopes, and realizable matroids, one can count points over finite fields on flag varieties, toric varieties, or reciprocal planes to obtain cohomological interpretations of these polynomials. We survey these results and unite them under a single geometric framework.

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arXiv2017

The Z-polynomial of a matroid

We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion, obtaining a closed formula for Kazhdan-Lusztig coefficients as alternating sums of multi-indexed Whitney numbers. For realizable matroids, we give a cohomological interpretation of the Z-polynomial in which the symmetry is a manifestation of Poincare duality.

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arXiv2016

The equivariant Kazhdan-Lusztig polynomial of a matroid

We define the equivariant Kazhdan-Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.

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arXiv2016

The Kazhdan-Lusztig polynomial of a matroid

We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M, in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always non-negative, and we prove this conjecture for representable matroids by interpreting our polynomials as intersection cohomology Poincare polynomials. We also introduce a q-deformation of the Mobius algebra of M, and use our polynomials to define a special basis for this deformation, analogous to the canonical basis of the Hecke algebra. We conjecture that the structure coefficients for multiplication in this special basis are non-negative, and we verify this conjecture in numerous examples.

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arXiv2016

The Orlik-Terao algebra and the cohomology of configuration space

We give a recursive algorithm for computing the Orlik-Terao algebra of the Coxeter arrangement of type A_{n-1} as a graded representation of S_n, and we give a conjectural description of this representation in terms of the cohomology of the configuration space of n points in SU(2) modulo translation. We also give a version of this conjecture for more general graphical arrangements.

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