arXiv subjects

Daoji Huang

Publications and source records attributed to Daoji Huang.

At least 19 recordsLinked to original sources

arXiv2026

Dual Affine Robinson-Schensted Correspondence

We introduce the dual affine Robinson-Schensted correspondence that gives a bijection between the extended affine symmetric group and tuples $(\bar{P},\bar{Q},\lambda,N)$, where $\bar{P}$ and $\bar{Q}$ are tabloids, $\lambda$ is a partition, and $N$ is an integer, subject to compatibility conditions. The construction generalizes Fomin's growth diagrams and Viennot's shadow lines for the classical Robinson-Schensted correspondence on the symmetric group, and is dual to the affine matrix ball construction as well as Shi's correspondence, in the sense that the $P$-tabloids are the same, and the $Q$-tabloids are related by affine evacuation. As a consequence, our construction also parametrizes Kazhdan-Lusztig cells in affine type $A$. We conjecture that the growth diagrams we construct admit a natural geometric realization in terms of relative positions of affine flags, similar to the interpretation given by Steinberg and van Leeuwen in the classical case.

math.CO↗

arXiv2025

Bounded ratios for Lorentzian matrices

We study multiplicative inequalities among entries of Lorentzian matrices, referred to as bounded ratios. These inequalities can be viewed as generalizations of the classical Alexandrov--Fenchel inequalities for mixed volumes. Our main structural result identifies the cone of all bounded ratios on Lorentzian matrices with the dual of the cut cone, a finitely generated integral polyhedral cone extensively studied in metric geometry and graph theory. We examine in detail the pentagonal ratio, which first appears for Lorentzian matrices of size at least five. For Lorentzian matrices of size three, we determine the optimal bounding constants across the entire cone of bounded ratios, obtaining an explicit entropy-like formula. We conjecture that any normalized bounded ratio is, in fact, bounded by 2.

math.CO↗

arXiv2025

Realizations of homology classes and projection areas

The relationship between convex geometry and algebraic geometry has deep historical roots, tracing back to classical works in enumerative geometry. In this paper, we continue this theme by studying two interconnected problems regarding projections of geometric objects in four-dimensional spaces: (1) Let $A$ be a convex body in $\mathbb{R}^4$, and let $(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34})$ be the areas of the six coordinate projections of $A$ in $\mathbb{R}^2$. Which tuples of six nonnegative real numbers can arise in this way? (2) Let $S$ be an irreducible surface in $(\mathbb{P}^1)^4$, and let $(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34})$ be the degrees of the six coordinate projections from $S$ to $(\mathbb{P}^1)^2$. Which tuples of six nonnegative integers can arise in this way? We show that these questions are governed by the Pl\"ucker relations for the Grassmannian $\text{Gr}(2,4)$ over the triangular hyperfield $\mathbb{T}_2$. We extend our analysis by determining the homology classes in $(\mathbb{P}^m)^n$ proportional to the fundamental classes of irreducible algebraic surfaces, resolving the algebraic Steenrod problem in this setting. Our results lead to several conjectures on realizable homology classes in smooth projective varieties and on the projection volumes of convex bodies.

math.AG↗

arXiv2024

Fine multidegrees, universal Grobner bases, and matrix Schubert varieties

We give a criterion for a collection of polynomials to be a universal Gr\"{o}bner basis for an ideal in terms of the multidegree of the closure of the corresponding affine variety in $(\mathbb{P}^1)^N$. This criterion can be used to give simple proofs of several existing results on universal Gr\"{o}bner bases. We introduce fine Schubert polynomials, which record the multidegrees of the closures of matrix Schubert varieties in $(\mathbb{P}^1)^{n^2}$. We compute the fine Schubert polynomials of permutations $w$ where the coefficients of the Schubert polynomials of $w$ and $w^{-1}$ are all either 0 or 1, and we use this to give a universal Gr\"{o}bner basis for the ideal of the matrix Schubert variety of such a permutation.

math.AG↗

arXiv2024

Keys and Evacuation via Virtualization

In this paper, we study the relation between the key map and virtualization of crystals. Namely, we prove that virtualization between crystals in any two finite Cartan types commutes with the left and right key maps, thus embedding Demazure crystals and atoms correspondingly. In particular, this implies that the key map in any finite Cartan type can be reduced to the key map in a simply-laced type, provided an appropriate virtualization exists, generalizing the work of Azenhas--Santos. As an application, we study these maps in the context of orthogonal Kashiwara--Nakashima tableaux and show that the virtualizations from type B into C considered independently by Fujita and Pappe--Pfannerer--Schilling--Simone coincide with the splitting map of De Concini and Lecouvey. As a consequence, this enables us to give a new and purely combinatorial definition of orthogonal evacuation.

math.CO↗

arXiv2024

Marked Bumpless Pipedreams and Compatible Pairs

We construct a bijection between marked bumpless pipedreams with reverse compatible pairs, which are in bijection with not-necessarily-reduced pipedreams. This directly unifies various formulas for Grothendieck polynomials in the literature. Our bijection is a generalization of a variant of the bijection of Gao and Huang in the unmarked, reduced case.

math.CO↗

arXiv2024

A row analogue of Hecke column insertion

We introduce a new row insertion algorithm on decreasing tableaux and increasing tableaux, generalizing Edelman-Greene (EG) row insertion. Our row insertion algorithm is a nontrivial variation of Hecke column insertion which generalizes EG column insertion. Similar to Hecke column insertion, our row insertion is bijective and respects Hecke equivalence, and therefore recovers the expansions of stable Grothendieck functions into Grassmannian stable Grothendieck functions.

math.CO↗

arXiv2024

A Gröbner basis for Kazhdan-Lusztig ideals of the flag variety of affine type A

A Kazhdan-Lusztig variety is the intersection of a locally-closed Schubert cell with an opposite Schubert variety in a flag variety. We present a linear parametrization of the Schubert cells in the affine type A flag variety via Bott-Samelson maps, and give explicit equations that generate the Kazhdan-Lusztig ideals in these coordinates. Furthermore, our equations form a Gröbner basis for the Kazhdan-Lusztig ideals. Our result generalizes a result of Woo-Yong that gave a Gröbner basis for Kazhdan-Lusztig ideals in the type A flag variety.

math.AG↗

arXiv2024

Multimatroids and rational curves with cyclic action

We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids introduced by Bouchet, which naturally arise in topological graph theory. The vantage point of moduli of curves provides a tropical framework for studying multimatroids, generalizing the previous connection between type-A permutohedral varieties (Losev--Manin moduli spaces) and matroids, and the connection between type-B permutohedral varieties (Batyrev--Blume moduli spaces) and delta-matroids. Specifically, we equate a combinatorial nef cone of the moduli space with the space of $\mathbb{R}$-multimatroids, a slight generalization of multimatroids, and we introduce the independence polytopal complex of a multimatroid, whose volume is identified with an intersection number on the moduli space. As an application, for the generating set of the Chow ring of the moduli space consisting of all psi-classes and their pullbacks along certain forgetful maps, we give a combinatorial formula for their intersection numbers by relating to the volumes of independence polytopal complexes of multimatroids.

math.CO↗

arXiv2024

A pipe dream perspective on totally symmetric self-complementary plane partitions

We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams. Using this characterization and the recent bijection of [Gao-Huang] between reduced pipe dreams and reduced bumpless pipe dreams, we give a bijection between alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We also give a different bijection in the 1432- and 2143-avoiding case that preserves natural poset structures on the associated pipe dreams and bumpless pipe dreams.

math.CO↗

arXiv2023

The MatrixSchubert package for Macaulay2

We introduce the MatrixSchubert package for the computer algebra system Macaulay2. This package has tools to construct and study matrix Schubert varieties and alternating sign matrix (ASM) varieties. The package also introduces tools for quickly computing homological invariants of such varieties, finding the components of an ASM variety, and checking if a union of matrix Schubert varieties is an ASM variety.

math.AG↗

arXiv2023

Growth Diagrams for Schubert RSK

Motivated by classical combinatorial Schubert calculus on the Grassmannian, Huang--Pylyavskyy introduced a generalized theory of Robinson-Schensted-Knuth (RSK) correspondence for studying Schubert calculus on the complete flag variety via insertion algorithms. The inputs of the correspondence are certain biwords, the insertion objects are bumpless pipe dreams, and the recording objects are certain chains in Bruhat order. In particular, they defined plactic biwords and showed that classical Knuth relations can be generalized to plactic biwords. In this paper, we give an analogue of Fomin's growth diagrams for this generalized RSK correspondence on plactic biwords. We show that this growth diagram recovers the bijection between pipe dreams and bumpless pipe dreams of Gao--Huang.

math.CO↗

arXiv2023

PALM: Predicting Actions through Language Models

Understanding human activity is a crucial yet intricate task in egocentric vision, a field that focuses on capturing visual perspectives from the camera wearer's viewpoint. Traditional methods heavily rely on representation learning that is trained on a large amount of video data. However, a major challenge arises from the difficulty of obtaining effective video representation. This difficulty stems from the complex and variable nature of human activities, which contrasts with the limited availability of data. In this study, we introduce PALM, an approach that tackles the task of long-term action anticipation, which aims to forecast forthcoming sequences of actions over an extended period. Our method PALM incorporates an action recognition model to track previous action sequences and a vision-language model to articulate relevant environmental details. By leveraging the context provided by these past events, we devise a prompting strategy for action anticipation using large language models (LLMs). Moreover, we implement maximal marginal relevance for example selection to facilitate in-context learning of the LLMs. Our experimental results demonstrate that PALM surpasses the state-of-the-art methods in the task of long-term action anticipation on the Ego4D benchmark. We further validate PALM on two additional benchmarks, affirming its capacity for generalization across intricate activities with different sets of taxonomies.

cs.CV↗

arXiv2023

Standard Monomials for Positroid Varieties

We give an explicit characterization of the standard monomials for positroid varieties with respect to the Hodge degeneration and give a Gr\"obner basis. Furthermore, we show that promotion and evacuation biject standard monomials of a positroid variety with those of its cyclic shifts and $w_0$-reflection, respectively. The connection to promotion allows us to identify standard monomials of a positroid variety with Lam's cyclic Demazure crystal. Using a recurrence on the Hilbert series, we give an inductive formula for the character of cyclic Demazure modules.

math.AG↗

arXiv2023

Palm: Predicting Actions through Language Models @ Ego4D Long-Term Action Anticipation Challenge 2023

We present Palm, a solution to the Long-Term Action Anticipation (LTA) task utilizing vision-language and large language models. Given an input video with annotated action periods, the LTA task aims to predict possible future actions. We hypothesize that an optimal solution should capture the interdependency between past and future actions, and be able to infer future actions based on the structure and dependency encoded in the past actions. Large language models have demonstrated remarkable commonsense-based reasoning ability. Inspired by that, Palm chains an image captioning model and a large language model. It predicts future actions based on frame descriptions and action labels extracted from the input videos. Our method outperforms other participants in the EGO4D LTA challenge and achieves the best performance in terms of action prediction. Our code is available at https://github.com/DanDoge/Palm

cs.CV↗

arXiv2023

Knuth moves for Schubert polynomials

In our previous work we have introduced an analogue of Robinson-Schensted-Knuth correspondence for Schubert calculus of the complete flag varieties. The objects inserted are certain biwords, the outcomes of insertion are bumpless pipe dreams, and the recording objects are decorated chains in Bruhat order. In this paper we study a class of biwords that have a certain associativity property; we call them plactic biwords. We introduce analogues of Knuth moves on plactic biwords, and prove that any two plactic biwords with the same insertion bumpless pipe dream are connected by those moves.

math.CO↗

arXiv2023

Schubert Products for Permutations with Separated Descents

We say that two permutations $π$ and $ρ$ have separated descents at position $k$ if $π$ has no descents before position $k$ and $ρ$ has no descents after position $k$. We give a counting formula, in terms of reduced word tableaux, for computing the structure constants of products of Schubert polynomials indexed by permutations with separated descents, and recognize that these structure constants are certain Edelman-Greene coefficients. Our approach uses generalizations of Schützenberger's jeu de taquin algorithm and the Edelman-Greene correspondence via bumpless pipe dreams.

math.CO↗

arXiv2022

Bumpless pipe dream RSK, growth diagrams, and Schubert structure constants

We introduce analogs of left and right RSK insertion for Schubert calculus of complete flag varieties. The objects being inserted are certain biwords, the insertion objects are bumpless pipe dreams, and the recording objects are decorated chains in Bruhat order. As an application, we adopt Lenart's growth diagrams of permutations to give a combinatorial rule for Schubert structure constants in the separated descent case.

math.CO↗