arXiv subjects

Yumeng Wu

Publications and source records attributed to Yumeng Wu.

8 recordsLinked to original sources

arXiv2026

Construction of PBW and canonical bases in the folding extension algebras for symmetrizable types

We study extension algebras arising from Lusztig's construction of perverse sheaves on quiver representation spaces equipped with an admissible automorphism $a$, leading to folding Khovanov--Lauda--Rouquier (KLR) algebras. Extending the method of M. Varagnolo and E. Vasserot to Lusztig's symmetrizable setting, we construct the skew group algebra $R(\nu)\langle\mathbf{a}\rangle$, where $\mathbf{a}$ is induced by the admissible automorphism $a$, and show that $\Ext^{\bullet}(\ ,L)$ preserves the signed basis and the crystal structure. In this framework, for finite types, we define standard and dual standard modules, analyze their behavior under dualities, and study the resulting Grothendieck groups. We prove that the transition matrix from the classes of indecomposable projective modules, which give the canonical basis, to the dual standard modules, which give the PBW basis, is upper triangular with diagonal entries equal to one. This provides a geometric categorification of these bases and of their relation in the symmetrizable setting.

math.QA↗

arXiv2026

Geometric realization of affine bases: the Kronecker quiver case

In this paper, we study the transition matrix between the PBW basis and the canonical basis for the negative part of the quantized enveloping algebra of the Kronecker quiver from a geometric viewpoint. Building on Lusztig's geometric construction of the canonical basis, we construct sheaf-complex realizations of PBW basis elements by means of flag sheaf complexes over the strata $X(\alpha,m)$ of representation varieties. Our first goal is to give a geometric description of the simple constituents appearing in the restrictions of these flag sheaf complexes to the strata $X(\alpha,m)$. This allows us to compare the PBW-type sheaf complexes with the simple perverse sheaves $IC(X(\alpha),L_\chi)$ arising in Lusztig's construction. Using this description together with a purity result for the relevant $\mathbb{F}_q$-structures, we obtain another proof that the elements defined by Lusztig's perverse sheaves indeed form a basis of the composition algebra.Our second goal is to make the transition coefficients between the PBW basis and the canonical basis geometrically explicit. More precisely, we show that these coefficients are governed by the multiplicities of local systems in the restrictions of intersection cohomology complexes to smaller strata. As a consequence, the transition matrix from the canonical basis to the PBW basis is upper triangular with diagonal entries equal to $1$, and its coefficients admit a direct geometric interpretation. In particular, in the Kronecker quiver case we recover the triangularity of the transition matrix and obtain positivity properties of the corresponding coefficient polynomials.

math.QA↗

arXiv2025

Structure coefficients for quantum groups

According to the Hall algebras of quivers with automorphisms under Lusztig's construction, the polynominal forms of several structure coefficients for quantum groups of all finite types are presented in this note. We first provide a geometric realization of the coefficients between PBW basis and the canonical basis via standard sheaves on quiver moduli spaces with admissible automorphisms. This realization is constructed through Lusztig sheaves equipped with periodic functors and their modified Grothendieck groups. Second, within this geometric framework, we present an alternative proof for the existence of Hall polynomials originally due to Ringel. Finally, we give a slight generalization of the Reineke-Caldero expression for the bar involution of PBW basis elements in symmetrizable cases. When the periodic functor $\mathbf{a}^*$ is taken $\operatorname{id}$, our results are the same as Lusztig's and Caldero-Reineke's.

math.RT↗

arXiv2024

Lusztig sheaves and integrable highest weight modules in the symmetrizable case

This paper continues the work of \cite{fang2023lusztigsheavesintegrablehighest} and \cite{fang2023lusztigsheavestensorproducts}. For a symmetrizable generalized Cartan matrix $C$ and the corresponding quantum group $\mathbf{U}$, we consider an associated quiver $Q$ equipped with an admissible automorphism $a$. We construct a category $\widetilde{\mathcal{Q}/\mathcal{N}}$ obtained from localizations of Lusztig sheaves for the corresponding framed and $2$-framed quivers with automorphism. The Grothendieck groups of these categories realize the integrable highest weight module $L(\lambda)$ and the tensor product $L(\lambda_1)\otimes L(\lambda_2)$ of integrable highest weight $\mathbf{U}$-modules. After quotienting by traceless objects, Lusztig sheaves yield the signed canonical bases of $L(\lambda)$ and $L(\lambda_1)\otimes L(\lambda_2)$. As applications, we recover symmetrizable crystal structures on Nakajima quiver varieties, Nakajima tensor product varieties, and Lusztig nilpotent varieties of preprojective algebras.

math.RT↗

arXiv2024

An Improved Height Difference Based Model of Height Profile for Drop-on-Demand 3D Printing With UV Curable Ink

This paper proposes an improved height profile model for drop-on-demand 3D printing with UV curable ink. It is extended from a previously validated model and computes height profile indirectly from volume and area propagation to ensure volume conservation. To accommodate 2D patterns using multiple passes, volume change and area change within region of interest are modeled as a piecewise function of height difference before drop deposition. Model coefficients are experimentally obtained and validated with bootstrapping of experimental samples. Six different drop patterns are experimentally validated. The RMS height profile errors for 2D patterns from the proposed model are consistently smaller than existing models from literature and are on the same level as 1D patterns reported in our previous publication.

physics.flu-dyn↗

arXiv2024

An improved model of height profile for Drop-on-demand print of UV curable ink

This paper proposes an improved model of height profile for drop-on-demand printing of UV curable ink. Unlike previous model, the proposed model propagates volume and covered area based on height difference between adjacent drops. Height profile is then calculated from the propagated volume and area. Measurements of 2-drop and 3-drop patterns are used to experimentally compute model parameters. The parameters are used to predict and validate height profiles of 4 and more drops in a straight line. Using the same root mean square (RMS) error as benchmark, this model achieves 5.9% RMS height profile error on 4-drop lines. This represents more than 60% reduction from graph-based model and an improvement from our previous effort.

physics.flu-dyn↗

arXiv2024

Modeling height profile for Drop-on-demand print of UV curable ink

This paper proposes a height profile model for drop-on-demand printing of UV curable ink. Existing models includesuperposition of single drops, numerical models, and graphic-based model. They are either too complicated or over simplified.Graphic model intends to find a sweet spot in between, however, accuracy is marginally improved from superposition modelwhile it demands more computation. The proposed model aimsto achieve the same as graphic model by introducing volumeand area propagation matrices to reflect the localized ink flowfrom higher location to the lower, while avoiding the detailedphysics behind it. This model assumes a constant volume andarea propagation of subsequent drop due to height profile difference. It is validated with experiments on single drop, 2-drop and 3-drop line printing. Stability of this model is analyzed.. Usingroot mean square (RMS) error as benchmark, proposed modelachieves 6.6% along the center row and 7.4% overall, better thanexisting models.

physics.flu-dyn↗

arXiv2024

The parity of Lusztig's restriction functor and Green's formula for a quiver with automorphism

In [8], Fang-Lan-Xiao proved a formula about Lusztig's induction and restriction functors which can induce Green's formula for the path algebra of a quiver over a finite field via the trace map. In this paper, we generalize their formula to that for the mixed semisimple perverse sheaves for a quiver with an automorphism. By applying the trace map, we obtain Green's formula for any finite-dimensional hereditary algebra over a finite field.

math.RT↗