3 citations · 8 across the 11 of their papers we have counts for
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Bounded Bruhat intervals and affine Coxeter groups
Grant T. Barkley, Christian Gaetz
Dyer (1991) proved that, for each fixed length , only finitely many isomorphism types of Bruhat intervals occur in finite Coxeter groups. In this short paper, we prove that this…
Totally nonnegative maximal tori and opposed Bruhat intervals
Grant T. Barkley, Steven N. Karp
Lusztig (2024) recently introduced the space of totally positive maximal tori of an algebraic group . Each such torus is the intersection of a totally positiv…
Combinatorial invariance for the coefficient of in Kazhdan-Lusztig polynomials
Grant T. Barkley, Christian Gaetz, Thomas Lam
We prove the combinatorial invariance of the coefficient of in Kazhdan--Lusztig polynomials for arbitrary Coxeter groups. As a result, we obtain the Combinatorial Invariance Co…
Extended weak order for the affine symmetric group
Grant T. Barkley
The extended weak order on a Coxeter group is the poset of biclosed sets in its root system. In (Barkley-Speyer 2024), it was shown that when is the affine…
The BBDVW Conjecture for Kazhdan-Lusztig polynomials of lower intervals
Grant T. Barkley, Christian Gaetz
Blundell, Buesing, Davies, Veličković, and Williamson (BBDVW) introduced the notion of a hypercube decomposition of an interval in Bruhat order. They conjectured a recursive formul…
A note on Combinatorial Invariance of Kazhdan--Lusztig polynomials
Francesco Esposito, Mario Marietti, Grant T. Barkley +1
We introduce the concepts of an amazing hypercube decomposition and a double shortcut for it, and use these new ideas to formulate a conjecture implying the Combinatorial Invarianc…
On combinatorial invariance of parabolic Kazhdan-Lusztig polynomials
Grant T. Barkley, Christian Gaetz
We show that the Combinatorial Invariance Conjecture for Kazhdan-Lusztig polynomials due to Lusztig and to Dyer, its parabolic analog due to Marietti, and a refined parabolic versi…
Affine extended weak order is a lattice
Grant T. Barkley, David E Speyer
Coxeter groups are equipped with a partial order known as the weak order, such that if the inversions of are a subset of the inversions of . In finite Coxeter gro…
Combinatorial invariance for Kazhdan-Lusztig -polynomials of elementary intervals
Grant T. Barkley, Christian Gaetz
We adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove the Combinatorial Invariance Conjecture for Kazhdan-Lusztig -polynomia…
Combinatorial descriptions of biclosed sets in affine type
Grant T. Barkley, David E Speyer
Let be a Coxeter group and let be its positive roots. A subset of is called biclosed if, whenever we have roots , and with $γ\in \mathbb{R}_{>0} α+ \…
Channels, Billiards, and Perfect Matching 2-Divisibility
Grant T. Barkley, Ricky Ini Liu
Let denote the number of perfect matchings of the graph . We introduce a number of combinatorial tools for determining the parity of and giving a lower bound on the…