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David Plaza

Publications and source records attributed to David Plaza.

18 recordsLinked to original sources
arXiv2026

Positivity of Pre-Canonical Bases for Spherical Hecke Algebras

We introduce a new algorithm for computing Kostka-Foulkes polynomials. It arises from a combinatorial procedure that computes the transition coefficients between successive pre-canonical bases for spherical Hecke algebras, introduced by Libedinsky, Patimo, and the first author, in order to interpolate between the standard and canonical bases of the spherical Hecke algebras. Using this algorithm, we prove that the coefficients of the polynomials in the transition matrix from any pre-canonical basis to the next one has nonnegative coefficients. This establishes the positivity conjecture for pre-canonical bases and, moreover, yields a strictly stronger result that uncovers additional positivity phenomena beyond the original scope of the conjecture. As a by-product, this method yields a combinatorial algorithm for computing Lascoux's atomic decomposition of canonical basis elements. Finally, our results show that, after applying the Satake isomorphism and dualizing with respect to the Hall inner product, pre-canonical bases correspond to a new family of Schur-positive symmetric functions.

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arXiv2026

A differential characterization of volume polynomials of permutohedra

We study a graded vector space of polynomials associated to a square matrix, defined by a finite difference condition along the rows. We show this space coincides with one defined by directional derivatives, and prove it is finite-dimensional precisely when all principal minors are nonzero. In that case, its dimension in each degree equals a binomial coefficient, giving total dimension a power of two. For Cartan matrices of irreducible root systems, we construct an explicit basis of volume polynomials of faces of the associated permutohedra, yielding an elementary criterion, which we call geometricity, for expressing a polynomial as a linear combination of these volume polynomials.

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arXiv2026

Bruhat intervals that are large hypercubes

We study the question of finding big Bruhat intervals that are poset hypercubes in the symmetric group $S_n$. Using permutations suggested by AlphaEvolve (an evolutionary coding agent developed by Google DeepMind), we were led to an unusual situation in which the agent produced a pattern which performed well for the $n$ tested, and which we show works well for general $n$. When $n$ is a power of 2 we exhibit a hypercube of dimension $O(n\log n)$, matching the largest possible dimension up to a constant multiple. Furthermore, we give an exact characterization of the vertices of this hypercube: they are precisely the \emph{dyadically well-distributed} permutations -- a simple digitwise property that already appeared in connection with Monte Carlo integration and mathematical finance. The maximal dimension of a Bruhat interval that is an hypercube in $S_n$ gives a lower bound (and possibly is equal to) the maximal possible coefficient of the second-highest degree term in the Kazhdan--Lusztig $R$-polynomial in $S_n$. As a surprising consequence, we obtain a new lower bound of order $n\log n$ for the maximal number of frozen variables appearing in the cluster algebras attached to the open Richardson varieties in $S_n$, and a similar result for moduli spaces of embeddings of Bruhat graphs.

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arXiv2025

Atomic decomposition for an affine Weyl group of type $G_2$

We show that the elements of the Kazhdan--Lusztig basis of the spherical Hecke algebra of type $G_2$ have an atomic decomposition. As a by-product, we obtain a new algorithm to compute generalized Kostka--Foulkes polynomials in type $G_2$.

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arXiv2025

Paper BOAT

We derive a formula for computing the size of lower Bruhat intervals for elements in the dominant cone of an affine Weyl group of type $A$. This enumeration problem is reduced to counting lattice points in certain polyhedra. Our main tool is a decomposition -- or tiling -- of each interval into smaller, combinatorially tractable pieces, which we call paper boats. We also conjecture a generalization of this formula to all affine Weyl groups, restricted to elements in the lowest two-sided Kazhdan-Lusztig cell, which contains almost all of the elements.

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arXiv2023

On the size of Bruhat intervals

For affine Weyl groups and elements associated to dominant coweights, we present a convex geometry formula for the size of the corresponding lower Bruhat intervals. Extensive computer calculations for these groups have led us to believe that a similar formula exists for all lower Bruhat intervals.

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arXiv2022

Combinatorial invariance conjecture for $\widetilde{A}_2$

The combinatorial invariance conjecture (due independently to G. Lusztig and M. Dyer) predicts that if $[x,y]$ and $[x',y']$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan-Lusztig polynomials are equal, that is, $P_{x,y}(q)=P_{x',y'}(q)$. We prove this conjecture for the affine Weyl group of type $\widetilde{A}_2$. This is the first infinite group with non-trivial Kazhdan-Lusztig polynomials where the conjecture is proved.

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arXiv2022

Pre-canonical bases on affine Hecke algebras

For any affine Weyl group, we introduce the pre-canonical bases. They are a set of bases $\{\mathbf{N}^i\}_{1\leq i \leq m+1} $ (where $m$ is the height of the highest root) of the spherical Hecke algebra that interpolates between the standard basis $\mathbf{N}^1$ and the canonical basis $\mathbf{N}^{m+1}$. The expansion of $\mathbf{N}^{i+1}$ in terms of the $\mathbf{N}^i$ is in many cases very simple and we conjecture that in type $A$ it is positive.

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arXiv2021

Kazhdan-Lusztig polynomials for $\tilde{B}_2$

Kazhdan and Lusztig define, for an arbitrary Coxeter system $(W,S)$, a family of polynomials indexed by pairs of elements of $W$. Despite their relevance and elementary definition, the explicit computation of these polynomials is still one of the hardest open problems in algebraic combinatorics. In this paper we explicitly compute Kazhdan-Lusztig polynomials for a Coxeter system of type $\tilde{B}_2$.

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arXiv2020

The nil-blob algebra: An incarnation of type $\tilde{A}_1$ Soergel calculus and of the truncated blob algebra

We introduce a type $B$ analogue of the nil Temperley-Lieb algebra in terms of generators and relations, that we call the (extended) nil-blob algebra. We show that this algebra is isomorphic to the endomorphism algebra of a Bott-Samelson bimodule in type $\tilde{A}_1$. We also prove that it is isomorphic to an idempotent truncation of the classical blob algebra.Thus we provide strong evidence in favor of the recent Blob vs. Soergel conjecture.

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arXiv2020

Blob algebra approach to modular representation theory

Two decades ago P. Martin and D. Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix algebras) are Kazhdan-Lusztig polynomials in type $\tilde{A}_1$. In this paper we take that observation far beyond its original scope. We conjecture that for $\tilde{A}_n$ there is an equivalence of categories between the characteristic $p$ diagrammatic Hecke category and a "blob category" that we introduce (using certain quotients of KLR algebras called \emph{generalized blob algebras}). Using alcove geometry we prove the "graded degree" part of this equivalence for all $n$ and all prime numbers $p$. If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristic $p$ give the $p$-Kazhdan Lusztig polynomials in type $\tilde{A}_n$. We prove this for $\tilde{A}_1$, the only case where the $p$-Kazhdan Lusztig polynomials are known.

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arXiv2019

Type $\tilde{C}$ Temperley-Lieb algebra quotients and Catalan combinatorics

We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.

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arXiv2019

Blob algebra and two-color Soergel calculus

In 2003, Martin and Woodcock noticed a connection between the representation theory of the blob algebra and the Kazhdan--Lusztig polynomials associated with the infinite dihedral group. However, no conceptual explanation for this coincidence has yet been provided. In this study, a possible explanation of this phenomenon is suggested by enunciating a conjecture that relates the endomorphism algebra of Bott--Samelson bimodules to certain subalgebras of the blob algebra obtained by idempotent truncation. Evidence supporting this conjecture is provided.

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arXiv2018

Diagrammatics for Kazhdan-Lusztig R-polynomials

Let $(W,S)$ be an arbitrary Coxeter system. We introduce a family of polynomials, $\{ \tilde{\mathcal{R}}_{u,\underline{v}}(t)\}$, indexed by pairs $(u,\underline{v})$ formed by an element $u\in W$ and a (non-necessarily reduced) word $\underline{v}$ in the alphabet $S$. The polynomial $\tilde{\mathcal{R}}_{u,\underline{v}}(t)$ is obtained by considering a certain subset of Libedinsky's light leaves associated to the pair $(u,\underline{v})$. Given a reduced expression $\underline{v}$ of an element $v\in W$; we show that $\tilde{\mathcal{R}}_{u,\underline{v}}(t)$ coincides with the Kazhdan-- Lusztig $\tilde{R}$-polynomial $\tilde{R}_{u,v}(t)$. Using the diagrammatic approach, we obtain some closed formulas for $\tilde{R}$- polynomials.

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arXiv2014

Graded cellularity and the Monotonicity Conjecture

The graded cellularity of Libedinsky Double Leaves, which form a basis for the endomorphism ring of the Bott_Samelson_Soergel bimodules, allows us to view the Kazhdan_Lusztig polynomials as graded decomposition numbers. Using this point of view, I provide in this paper a new proof of the monotonicity conjecture for the Kazhdan_Lusztig polynomials of any Coxeter system.

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arXiv2013

Graded cellular bases for Temperley-Lieb algebras of type A and B

We show that the Temperley-Lieb algebra of type $A$ and the blob algebra (also known as the Temperley-Lieb algebra of type $ B$) at roots of unity are $ \mathbb Z$-graded algebras.We moreover show that they are graded cellular algebras, thus making their cell modules, or standard modules, graded modules for the algebras.

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