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Li Luo

Publications and source records attributed to Li Luo.

At least 55 records · Page 3Linked to original sources
arXiv2022

Cells in affine q-Schur algebras

We develop algebraic and geometrical approaches toward canonical bases for affine q-Schur algebras of arbitrary type introduced in this paper. A duality between an affine q-Schur algebra and a corresponding affine Hecke algebra is established. We introduce an inner product on the affine q-Schur algebra, with respect to which the canonical basis is shown to be positive and almost orthonormal. We then formulate the cells and asymptotic forms for affine q-Schur algebras, and develop their basic properties analogous to the cells and asymptotic forms for affine Hecke algebras established by Lusztig. The results on cells and asymptotic algebras are also valid for q-Schur algebras of arbitrary finite type.

math.RT↗
arXiv2022

Multiplication formulas and isomorphism theorem of $\imath$Schur superalgebras

We introduce the notion of $\imath$Schur superalgebra, which can be regarded as a type B/C counterpart of the $q$-Schur superalgebra (of type A) formulated as centralizer algebras of certain signed $q$-permutation modules over Hecke algebras. Some multiplication formulas for $\imath$Schur superalgebra are obtained to construct their canonical bases. Furthermore, we established an isomorphism theorem between the $\imath$Scuhr superalgebras and the $q$-Schur superalgebras of type A, which helps us derive a semisimplicity criteria of the $\imath$Schur superalgebras.

math.RT↗
arXiv2022

Distance Guided Channel Weighting for Semantic Segmentation

Recent works have achieved great success in improving the performance of multiple computer vision tasks by capturing features with a high channel number utilizing deep neural networks. However, many channels of extracted features are not discriminative and contain a lot of redundant information. In this paper, we address above issue by introducing the Distance Guided Channel Weighting (DGCW) Module. The DGCW module is constructed in a pixel-wise context extraction manner, which enhances the discriminativeness of features by weighting different channels of each pixel's feature vector when modeling its relationship with other pixels. It can make full use of the high-discriminative information while ignore the low-discriminative information containing in feature maps, as well as capture the long-range dependencies. Furthermore, by incorporating the DGCW module with a baseline segmentation network, we propose the Distance Guided Channel Weighting Network (DGCWNet). We conduct extensive experiments to demonstrate the effectiveness of DGCWNet. In particular, it achieves 81.6% mIoU on Cityscapes with only fine annotated data for training, and also gains satisfactory performance on another two semantic segmentation datasets, i.e. Pascal Context and ADE20K. Code will be available soon at https://github.com/LanyunZhu/DGCWNet.

cs.CV↗
arXiv2022

Observation of quadruple Weyl point in hybrid-Weyl phononic crystals

The discovery of Weyl semimetals opens the door for searching topological semimetals in physical science. The Weyl points are generally recognized as conventional, quadratic, spin-1, and those of high topological charges. Here we report the observation of the quadruple Weyl point of charge 4, the highest topological charge a twofold degenerate node can carry. Besides the quadruple Weyl point, the phononic semimetal also hosts conventional, quadratic, and spin-1 Weyl points, which stands as a system with yet the richest types of Weyl points. The quadruple-helicoid surface states, specific to the quadruple Weyl point, are demonstrated. The finding of the high-charge Weyl point enriches the knowledge of Weyl semimetals and may stimulate related researches in other systems, such as photonic, mechanical and cold atom systems.

cond-mat.mes-hall↗
arXiv2022

Lectures on dualities ABC in representation theory

In these lecture notes for a summer mini-course, we provide an exposition on quantum groups and Hecke algebras, including (quasi) R-matrix, canonical basis, and $q$-Schur duality. Then we formulate their counterparts in the setting of $\imath$quantum groups arising from quantum symmetric pairs, including (quasi) K-matrix, $\imath$-canonical basis, and $\imath$Schur duality. As an application, the ($\imath$-)canonical bases are used to formulate Kazhdan-Lusztig theories and character formulas in the BGG categories for Lie (super)algebras of type A-D. Finally, geometric constructions for $q$-Schur and $\imath$Schur dualities are provided.

math.RT↗
arXiv2021

Topological materials for full-vector elastic waves

Elastic wave manipulation is important in a wide variety of scales in applications including information processing in tiny elastic devices and noise control in big solid structures. The recent emergence of topological materials opens a new avenue toward modulating elastic waves in solids. However, because of the full-vector feature, and the complicated couplings of the longitudinal and transverse components of elastic waves, manipulating elastic waves is generally difficult, compared with manipulating acoustic waves (scalar waves) and electromagnetic waves (vectorial waves but transverse only). Up to date, topological materials, including insulators and semimetals, have been realized for acoustic and electromagnetic waves. Although topological materials of elastic waves have also been reported, the topological edge modes observed all lie on the domain wall. A natural question can be asked: whether there exists an elastic metamaterial with the topological edge modes on its own boundary only? Here, we report a 3D metal-printed bilayer metamaterial, insulating topologically the elastic waves. By introducing the chiral interlayer couplings, the spin-orbit couplings for elastic waves are induced, which give rise to nontrivial topological properties. The helical edge states with the vortex feature are demonstrated on the boundary of the single topological phase. We further show a heterostructure of the metamaterial, which exhibits tunable edge transport. Our work may have potential in devices based on elastic waves in solids.

cond-mat.mtrl-sci↗
arXiv2020

Blocks and characters of $G(3)$-modules of non-integral weights

We classify blocks in the BGG category $\mathcal O$ of modules of non-integral weights for the exceptional Lie superalgebra $G(3)$. We compute the characters for tilting modules of non-integral weights in $\mathcal O$. Reduction methods are established to connect non-integral blocks of $G(3)$ with blocks of the special linear Lie algebra $\mathfrak{sl}(2)$, the exceptional Lie algebra $G_2$, the general linear Lie superalgebras $\mathfrak{gl}(1|1)$, $\mathfrak{gl}(2|1)$ and the ortho-symplectic Lie superalgebra $\mathfrak{osp}(3|2)$.

math.RT↗
arXiv2020

Observation of a phononic higher-order Weyl semimetal

Weyl semimetals are extraordinary systems where exotic phenomena such as Fermi arcs, pseudo-gauge fields and quantum anomalies arise from topological band degeneracy in crystalline solids for electrons and metamaterials for photons and phonons. On the other hand, higher-order topological insulators unveil intriguing multidimensional topological physics beyond the conventional bulk-edge correspondences. However, it is unclear whether higher-order topology can emerge in Weyl semimetals. Here, we report the experimental discovery of higher-order Weyl semimetals in its phononic analog which exhibit topologically-protected boundary states in multiple dimensions. We create the physical realization of the higher-order Weyl semimetal in a chiral phononic crystal with uniaxial screw symmetry. Using near-field spectroscopies, we observe the chiral Fermi arcs on the surfaces and a new type of hinge arc states on the hinge boundaries. These topological boundary arc states link the projections of Weyl points in different dimensions and directions, and hence demonstrate higher-order multidimensional topological physics in Weyl semimetals. Our study establishes the fundamental connection between higher-order topology and Weyl physics in crystalline materials and unveils a new horizon of higher-order topological semimetals where unprecedented materials such as higher-order topological nodal-lines may emerge.

cond-mat.mes-hall↗
arXiv2020

Acoustic square-root topological insulators

Square-root topological states are new topological phases, whose topological property is inherited from the square of the Hamiltonian. We realize the first-order and second-order square-root topological insulators in phononic crystals, by putting additional cavities on connecting tubes in the acoustic Su-Schrieffer-Heeger model and the honeycomb lattice, respectively. Because of the square-root procedure, the bulk gap of the squared Hamiltonian is doubled. In both two bulk gaps, the square-root topological insulators possess multiple localized modes, i.e., the end and corner states, which are evidently confirmed by our calculations and experimental observations. We further propose a second-order square-root topological semimetal by stacking the decorated honeycomb lattice to three dimensions.

cond-mat.mes-hall↗
arXiv2020

Decomposition of the Total Effect for Two Mediators: A Natural Counterfactual Interaction Effect Framework

Mediation analysis has been used in many disciplines to explain the mechanism or process that underlies an observed relationship between an exposure variable and an outcome variable via the inclusion of mediators. Decompositions of the total causal effect of an exposure variable into effects characterizing mediation pathways and interactions have gained an increasing amount of interest in the last decade. In this work, we develop decompositions for scenarios where the two mediators are causally sequential or non-sequential. Current developments in this area have primarily focused on either decompositions without interaction components or with interactions but assuming no causally sequential order between the mediators. We propose a new concept called natural counterfactual interaction effect that captures the two-way and three-way interactions for both scenarios that extend the two-way mediated interactions in literature. We develop a unified approach for decomposing the total effect into the effects that are due to mediation only, interaction only, both mediation and interaction, neither mediation nor interaction within the counterfactual framework. Finally, we illustrate the proposed decomposition method using a real data analysis where the two mediators are causally sequential.

stat.ME↗
arXiv2020

Decomposition of Total Effect with the Notion of Natural Counterfactual Interaction Effect

Mediation analysis serves as a crucial tool to obtain causal inference based on directed acyclic graphs, which has been widely employed in the areas of biomedical science, social science, epidemiology and psychology. Decomposition of total effect provides a deep insight to fully understand the casual contribution from each path and interaction term. Since the four-way decomposition method was proposed to identify the mediated interaction effect in counterfactual framework, the idea had been extended to a more sophisticated scenario with non-sequential multiple mediators. However, the method exhibits limitations as the causal structure contains direct causal edges between mediators, such as inappropriate modeling of dependence and non-identifiability. We develop the notion of natural counterfactual interaction effect and find that the decomposition of total effect can be consistently realized with our proposed notion. Furthermore, natural counterfactual interaction effect overcomes the drawbacks and possesses a clear and significant interpretation, which may largely improve the capacity of researchers to analyze highly complex causal structures.

stat.ME↗
arXiv2020

Blocks and characters of $D(2|1;ζ)$-modules of non-integral weights

We classify blocks in the BGG category $\mathcal O$ of modules of non-integral weights for the exceptional Lie superalgebra $D(2|1;ζ)$. We establish various reduction methods, which connect some types of non-integral blocks of $D(2|1;ζ)$ with integral blocks of general linear Lie superalgebras $\mathfrak{gl}{(1|1)}$ and $\mathfrak{gl}{(2|1)}$. We compute the characters for irreducible $D(2|1;ζ)$-modules of non-integral weights in $\mathcal O$.

math.RT↗
arXiv2020

The $q$-Schur algebras and $q$-Schur dualities of finite type

We formulate a $q$-Schur algebra associated to an arbitrary $W$-invariant finite set $X_{\texttt f}$ of integral weights for a complex simple Lie algebra with Weyl group $W$. We establish a $q$-Schur duality between the $q$-Schur algebra and Hecke algebra associated to $W$. We then realize geometrically the $q$-Schur algebra and duality, and construct a canonical basis for the $q$-Schur algebra with positivity. With suitable choices of $X_{\texttt f}$ in classical types, we recover the $q$-Schur algebras in the literature. Our $q$-Schur algebras are closely related to the category $\mathcal O$, where the type $G_2$ is studied in detail.

math.RT↗
arXiv2019

Affine Hecke algebras and quantum symmetric pairs

We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra ${\mathbf K}^{\mathfrak c}_n$. We show that ${\mathbf K}^{\mathfrak c}_n$ is a coideal subalgebra of quantum affine algebra ${\bf U}(\hat{\mathfrak{gl}}_n)$, and $\big({\mathbf U}(\hat{ \mathfrak{gl}}_n), {\mathbf K}^{\mathfrak c}_n)$ forms a quantum symmetric pair. The modified coideal subalgebra is shown to admit monomial and stably canonical bases. We also formulate several variants of the affine Schur algebra and the (modified) coideal subalgebra above, as well as their monomial and canonical bases. This work provides a new and algebraic approach which complements and sheds new light on our previous geometric approach on the subject. In the appendix by four of the authors, new length formulas for the Weyl groups of affine classical types are obtained in a symmetrized fashion.

math.QA↗
arXiv2019

Schur algebras and quantum symmetric pairs with unequal parameters

We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson-Lusztig-MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type A III/IV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated to any weight function. These bases are the counterparts of Lusztig's bar-invariant basis for Hecke algebras with unequal parameters. In the appendix we provide an algebraic version of a type D Beilinson-Lusztig-MacPherson construction which is first introduced by Fan-Li from a geometric viewpoint.

math.RT↗
arXiv2019

Skewness correction in tail probability approximations for sums of local statistics

Correcting for skewness can result in more accurate tail probability approximations in the central limit theorem for sums of independent random variables. In this paper, we extend the theory to sums of local statistics of independent random variables and apply the result to $k$-runs, U-statistics, and subgraph counts in the Erdös-Rényi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order Cramér-type moderate deviations via Stein's method.

math.PR↗
arXiv2019

Odd singular vector formula for general linear Lie superalgebras

We establish a closed formula for a singular vector of weight $λ-β$ in the Verma module of highest weight $λ$ for Lie superalgebra $\mathfrak{gl}(m|n)$ when $λ$ is atypical with respect to an odd positive root $β$. It is further shown that this vector is unique up to a scalar multiple, and it descends to a singular vector, again unique up to a scalar multiple, in the corresponding Kac module when both $λ$ and $λ-β$ are dominant integral.

math.RT↗