arXiv subjects

Jae-Ho Lee

Publications and source records attributed to Jae-Ho Lee.

17 recordsLinked to original sources

arXiv2026

On finite-dimensional multiplicity-free irreducible modules for a nil-DAHA of type $(C_1^\vee,C_1)$

Fix nonzero $r_0,r_1\in\mathbb{C}$. Let $\widetilde{\mathcal H}$ denote a nil-DAHA of type $(C_1^\vee,C_1)$ defined by generators $t_0,u_0,t_1,u_1$ and relations $(t_i-r_i)(t_i-r_i^{-1})=0$ for $i\in\{0,1\}$, $u_0^2=u_0$, $u_1^2=0$, and $u_0t_0t_1u_1=0=t_1u_1u_0t_0$. Set $A=u_0t_0$ and $B=t_1u_1$. A finite-dimensional $\widetilde{\mathcal H}$-module is called $(A,B)$-multiplicity-free, or simply multiplicity-free, if $A$ and $B$ are simultaneously diagonalizable and every nonzero common eigenspace is one-dimensional. We consider finite-dimensional irreducible multiplicity-free modules that have a certain ordered basis, which we call an adapted block basis. For $D\geq 1$, we construct a family of $2D$-dimensional $\widetilde{\mathcal H}$-modules $E_D$, and for $D\geq 0$, we construct a family of $(2D+1)$-dimensional $\widetilde{\mathcal H}$-modules $O_D$. We determine which members of these families are multiplicity-free and irreducible. We prove that every finite-dimensional irreducible $\widetilde{\mathcal H}$-module that is multiplicity-free and has an adapted block basis is isomorphic to a module of the form $E_D$ or $O_D$. We also determine when two members of the same family are isomorphic.

math.RT↗

arXiv2026

FacePlex: Full-Duplex Joint Speech-Facial Motion Generation for Conversational Avatars

Natural face-to-face conversation requires real-time speech generation together with synchronized facial motion. Existing systems only partially address this problem: speech-only full-duplex models can generate speech in real time but do not produce facial motion, while audio-driven facial motion models animate a face from already available audio rather than jointly generating speech and motion online. To bridge this gap, we first formalize full-duplex joint speech-facial motion generation, where speech tokens and facial motion tokens are produced together every step. Building on this formulation, we propose FacePlex, a unified streaming framework with two key components. First, Rolling Flow Matching adapts flow matching to online motion generation by committing new motion frames at each streaming step. Second, Rolling Cross-Attention couples the streaming audio queue with the motion queue, allowing speech and facial motion to condition each other as generation progresses. Through extensive experiments, ablation studies, and a user study, we show that FacePlex enables full-duplex joint speech-facial motion generation under online streaming constraints, while achieving stronger lip-sync quality and motion fidelity than audio-driven facial motion baselines.

cs.AI↗

arXiv2025

Every $Q$-polynomial distance-regular graph is sharp over $\mathbb{R}$

Let $\Gamma$ denote a distance-regular graph with vertex set $X$ and diameter $D \geq 3$. Fix a vertex $x \in X$. Let the field $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$. Let $\operatorname{Mat}_X(\mathbb{F})$ denote the $\mathbb{F}$-algebra of matrices whose rows and columns are indexed by $X$ and all entries in $\mathbb{F}$. The Terwilliger algebra $T^\mathbb{F} = T^\mathbb{F}(x)$ is the subalgebra of $\operatorname{Mat}_X(\mathbb{F})$ generated by the adjacency matrix $A$ of $\Gamma$ and the dual primitive idempotents $\{E_i^*\}_{i=0}^D$ of $\Gamma$ with respect to $x$. Let $\{E_i\}_{i=0}^D$ denote the primitive idempotents of $A$. Assume that the ordering $\{E_i\}_{i=0}^D$ is $Q$-polynomial. Let $W$ denote an irreducible $T^\mathbb{F}$-module. We say that $W$ is sharp over $\mathbb{F}$ whenever $\dim (E_r^* W) = 1$, where $r$ is the endpoint of $W$. It is known, by Nomura and Terwilliger (2008), that every irreducible $T^\mathbb{C}$-module is sharp. In this paper, we prove that every irreducible $T^\mathbb{R}$-module is sharp. Once this is established, we obtain four additional results: (i) if $W$ is an irreducible $T^\mathbb{R}$-module, then its complexification $W^\mathbb{C}= W \otimes_{\mathbb{R}} \mathbb{C}$ is an irreducible $T^\mathbb{C}$-module; (ii) two irreducible $T^\mathbb{R}$-modules $W_1$ and $W_2$ are isomorphic if and only if their complexifications $W_1^\mathbb{C}$ and $W_2^\mathbb{C}$ are isomorphic as $T^\mathbb{C}$-modules; (iii) if $\bigoplus_{i=1}^h \operatorname{Mat}_{n_i}(\mathbb{C})$ is the Wedderburn decomposition of $T^\mathbb{C}$, then $\bigoplus_{i=1}^h \operatorname{Mat}_{n_i}(\mathbb{R})$ is the Wedderburn decomposition of $T^\mathbb{R}$; (iv) each of the subalgebras $E_1^* T E_1^*$, $E_1 T E_1$, $E_D^* T E_D^*$, and $E_D T E_D$ is commutative and every element of these algebras is a symmetric matrix.

math.CO↗

arXiv2025

The nucleus of the Grassmann graph $J_q(N,D)$

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $N$ and $D$ denote integers with $N>D \ge 1$. Let $\mathcal{V}$ denote an $N$-dimensional vector space over $\mathbb{F}_q$. The Grassmann graph $J_q(N,D)$ is the graph with vertex set $X$ that consists of the $D$-dimensional subspaces of $\mathcal{V}$. Two vertices are adjacent whenever their intersection has dimension $D-1$. Fix a vertex $x$ in $X$. The Terwilliger algebra $T=T(x)$ of $J_q(N,D)$ with respect to $x$ is the subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^* = A^*(x)$. It is known that an irreducible $T$-module $W$ has certain parameters called the endpoint $r$, the dual endpoint $t$, and the diameter $d$. The displacement of $W$ is defined to be the integer $r+t-D+d$. Let $\mathcal{N}=\mathcal{N}(x)$ denote the span of all irreducible $T$-modules with displacement 0. We call $\mathcal{N}$ the nucleus of $J_q(N,D)$ with respect to $x$. In this paper, we study the structure of $\mathcal{N}$. Specifically, we present a formula for the dimension of $\mathcal{N}$, construct two explicit bases for $\mathcal{N}$, and describe the action of $A$ and $A^*$ on these bases. To obtain these results, we use the projective geometry $P_q(N)$, consisting of all subspaces of $\mathcal{V}$, as a key tool.

math.CO↗

arXiv2025

Four bases for the Onsager Lie algebra related by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ action

The Onsager Lie algebra $O$ is an infinite-dimensional Lie algebra defined by generators $A$, $B$ and relations $[A, [A, [A, B]]] = 4[A, B]$ and $[B, [B, [B, A]]] = 4[B, A]$. Using an embedding of $O$ into the tetrahedron Lie algebra $\boxtimes$, we obtain four direct sum decompositions of the vector space $O$, each consisting of three summands. As we will show, there is a natural action of $\mathbb{Z}_2 \times \mathbb{Z}_2$ on these decompositions. For each decomposition, we provide a basis for each summand. Moreover, we describe the Lie bracket action on these bases and show how they are recursively constructed from the generators $A$, $B$ of $O$. Finally, we discuss the action of $\mathbb{Z}_2 \times \mathbb{Z}_2$ on these bases and determine some transition matrices among the bases.

math.RA↗

arXiv2024

Versatile Incremental Learning: Towards Class and Domain-Agnostic Incremental Learning

Incremental Learning (IL) aims to accumulate knowledge from sequential input tasks while overcoming catastrophic forgetting. Existing IL methods typically assume that an incoming task has only increments of classes or domains, referred to as Class IL (CIL) or Domain IL (DIL), respectively. In this work, we consider a more challenging and realistic but under-explored IL scenario, named Versatile Incremental Learning (VIL), in which a model has no prior of which of the classes or domains will increase in the next task. In the proposed VIL scenario, the model faces intra-class domain confusion and inter-domain class confusion, which makes the model fail to accumulate new knowledge without interference with learned knowledge. To address these issues, we propose a simple yet effective IL framework, named Incremental Classifier with Adaptation Shift cONtrol (ICON). Based on shifts of learnable modules, we design a novel regularization method called Cluster-based Adaptation Shift conTrol (CAST) to control the model to avoid confusion with the previously learned knowledge and thereby accumulate the new knowledge more effectively. Moreover, we introduce an Incremental Classifier (IC) which expands its output nodes to address the overwriting issue from different domains corresponding to a single class while maintaining the previous knowledge. We conducted extensive experiments on three benchmarks, showcasing the effectiveness of our method across all the scenarios, particularly in cases where the next task can be randomly altered. Our implementation code is available at https://github.com/KHU-AGI/VIL.

cs.CV↗

arXiv2024

The standard generators of the tetrahedron algebra and their look-alikes

The tetrahedron algebra $\boxtimes$ is an infinite-dimensional Lie algebra defined by generators $\{x_{ij} \mid i, j \in \{0, 1, 2, 3\}, i \neq j\}$ and some relations, including the Dolan-Grady relations. These twelve generators are called standard. We introduce a type of element in $\boxtimes$ that "looks like" a standard generator. For mutually distinct $h, i, j, k \in \{0, 1, 2, 3\}$, consider the standard generator $x_{ij}$ of $\boxtimes$. An element $ξ\in \boxtimes$ is called $x_{ij}$-like whenever both (i) $ξ$ commutes with $x_{ij}$; (ii) $ξ$ and $x_{hk}$ satisfy a Dolan-Grady relation. Pick mutually distinct $i,j,k \in \{0,1,2,3\}$. In our main result, we find an attractive basis for $\boxtimes$ with the property that every basis element is either $x_{ij}$-like or $x_{jk}$-like or $x_{ki}$-like. We discuss this basis from multiple points of view.

math.RA↗

arXiv2024

On the (non-)existence of tight distance-regular graphs: a local approach

Let $Γ$ denote a distance-regular graph with diameter $D\geq 3$. Jurišić and Vidali conjectured that if $Γ$ is tight with classical parameters $(D,b,α,β)$, $b\geq 2$, then $Γ$ is not locally the block graph of an orthogonal array nor the block graph of a Steiner system. In the present paper, we prove this conjecture and, furthermore, extend it from the following aspect. Assume that for every triple of vertices $x, y, z$ of $Γ$, where $x$ and $y$ are adjacent, and $z$ is at distance $2$ from both $x$ and $y$, the number of common neighbors of $x$, $y$, $z$ is constant. We then show that if $Γ$ is locally the block graph of an orthogonal array (resp. a Steiner system) with smallest eigenvalue $-m$, $m\geq 3$, then the intersection number $c_2$ is not equal to $m^2$ (resp. $m(m+1)$). Using this result, we prove that if a tight distance-regular graph $Γ$ is not locally the block graph of an orthogonal array or a Steiner system, then the valency (and hence diameter) of $Γ$ is bounded by a function in the parameter $b=b_1/(1+θ_1)$, where $b_1$ is the intersection number of $Γ$ and $θ_1$ is the second largest eigenvalue of $Γ$.

math.CO↗

arXiv2024

Towards a classification of $1$-homogeneous distance-regular graphs with positive intersection number $a_1$

Let $\Gamma$ be a graph with diameter at least two. Then $\Gamma$ is said to be $1$-homogeneous (in the sense of Nomura) whenever for every pair of adjacent vertices $x$ and $y$ in $\Gamma$, the distance partition of the vertex set of $\Gamma$ with respect to both $x$ and $y$ is equitable, and the parameters corresponding to equitable partitions are independent of the choice of $x$ and $y$. Assume that $\Gamma$ is $1$-homogeneous distance-regular with intersection number $a_1>0$ and diameter $D\geqslant 5$. Define $b=b_1/(\theta_1+1)$, where $b_1$ is the intersection number and $\theta_1$ is the second largest eigenvalue of $\Gamma$. We show that if intersection number $c_2$ is at least $2$, then $b\geqslant 1$ and one of the following (i)--(vi) holds: (i) $\Gamma$ is a regular near $2D$-gon, (ii) $\Gamma$ is a Johnson graph $J(2D,D)$, (iii) $\Gamma$ is a halved $\ell$-cube with $\ell \in \{2D,2D+1\}$, (iv) $\Gamma$ is a folded Johnson graph $\bar{J}(4D,2D)$, (v) $\Gamma$ is a folded halved $4D$-cube, (vi) the valency of $\Gamma$ is bounded by a function of $b$. Using this result, we characterize $1$-homogeneous graphs with classical parameters and $a_1>0$, as well as tight distance-regular graphs.

math.CO↗

arXiv2023

Modernizing Old Photos Using Multiple References via Photorealistic Style Transfer

This paper firstly presents old photo modernization using multiple references by performing stylization and enhancement in a unified manner. In order to modernize old photos, we propose a novel multi-reference-based old photo modernization (MROPM) framework consisting of a network MROPM-Net and a novel synthetic data generation scheme. MROPM-Net stylizes old photos using multiple references via photorealistic style transfer (PST) and further enhances the results to produce modern-looking images. Meanwhile, the synthetic data generation scheme trains the network to effectively utilize multiple references to perform modernization. To evaluate the performance, we propose a new old photos benchmark dataset (CHD) consisting of diverse natural indoor and outdoor scenes. Extensive experiments show that the proposed method outperforms other baselines in performing modernization on real old photos, even though no old photos were used during training. Moreover, our method can appropriately select styles from multiple references for each semantic region in the old photo to further improve the modernization performance.

cs.CV↗

arXiv2023

A New Feasibility Condition for the AT4 Family

Let $Γ$ be an antipodal distance-regular graph with diameter $4$ and eigenvalues $θ_0>θ_1>θ_2>θ_3>θ_4$. Then $Γ$ is tight in the sense of Jurišić, Koolen, and Terwilliger [12] whenever $Γ$ is locally strongly regular with nontrivial eigenvalues $p:=θ_2$ and $-q:=θ_3$. Assume that $Γ$ is tight. Then the intersection numbers of $Γ$ are expressed in terms of $p$, $q$, and $r$, where $r$ is the size of the antipodal classes of $Γ$. We denote $Γ$ by $\mathrm{AT4}(p,q,r)$ and call this an antipodal tight graph of diameter $4$ with parameters $p,q,r$. In this paper, we give a new feasibility condition for the $\mathrm{AT4}(p,q,r)$ family. We determine a necessary and sufficient condition for the second subconstituent of $\mathrm{AT4}(p,q,2)$ to be an antipodal tight graph. Using this condition, we prove that there does not exist $\mathrm{AT4}(q^3-2q,q,2)$ for $q\equiv3$ $(\mathrm{mod}~4)$. We discuss the $\mathrm{AT4}(p,q,r)$ graphs with $r=(p+q^3)(p+q)^{-1}$.

math.CO↗

arXiv2022

Circular Hessenberg Pairs

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $M$ denote a Hessenberg matrix. Then $M$ is called circular whenever the upper-right corner entry of $M$ is nonzero and every other entry above the superdiagonal is zero. A circular Hessenberg pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a circular Hessenberg fashion. Let $A, A^*$ denote a circular Hessenberg pair. We investigate six bases for the underlying vector space that we find attractive. We display the transition matrices between certain pairs of bases among the six. We also display the matrices that represent $A$ and $A^*$ with respect to the six bases. We introduce a special type of circular Hessenberg pair, said to be recurrent. We show that a circular Hessenberg pair $A, A^*$ is recurrent if and only if $A, A^*$ satisfy the tridiagonal relations. For a circular Hessenberg pair, there is a related object called a circular Hessenberg system. We classify up to isomorphism the recurrent circular Hessenberg systems. To this end, we construct four families of recurrent circular Hessenberg systems. We show that every recurrent circular Hessenberg system is isomorphic to a member of one of the four families.

math.CO↗

arXiv2022

Remarks on pseudo-vertex-transitive graphs with small diameter

Let $Γ$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $Γ$. For a vertex $x\in X$ and for $0 \leq i \leq D$, let $E^*_i(x)$ denote the projection matrix to the $i$th subconstituent space of $Γ$ with respect to $x$. The Terwilliger algebra $T(x)$ of $Γ$ with respect to $x$ is the semisimple subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by $A, E^*_0(x), E^*_1(x), \ldots, E^*_D(x)$. Let $V$ denote a $\mathbb{C}$-vector space consisting of complex column vectors with rows indexed by $X$. We say $Γ$ is pseudo-vertex-transitive whenever for any vertices $x,y \in X$, there exists a $\mathbb{C}$-vector space isomorphism $ρ:V\to V$ such that $(ρA - A ρ)V=0$ and $(ρE^*_i(x) - E^*_i(y)ρ)V=0$ for all $0\leq i \leq D$. In this paper, we discuss pseudo-vertex transitivity for distance-regular graphs with diameter $D\in \{2,3,4\}$. For $D=2$, we show that a strongly regular graph is pseudo-vertex-transitive if and only if all its local graphs have the same spectrum. For $D = 3$, we consider the Taylor graphs and show that they are pseudo-vertex transitive. For $D=4$, we consider the antipodal tight graphs and show that they are pseudo-vertex transitive.

math.CO↗

arXiv2019

Grassmann graphs, degenerate DAHA, and non-symmetric dual $q$-Hahn polynomials

We discuss the Grassmann graph $J_q(N,D)$ with $N \geq 2D$, having as vertices the $D$-dimensional subspaces of an $N$-dimensional vector space over the finite field $\mathbb{F}_q$. This graph is distance-regular with diameter $D$; to avoid trivialities we assume $D\geq 3$. Fix a pair of a Delsarte clique $C$ of $J_q(N,D)$ and a vertex $x$ in $C$. We construct a $2D$-dimensional irreducible module $\mathbf{W}$ for the Terwilliger algebra $\mathbf{T}$ of $J_q(N,D)$ associated with the pair $x$, $C$. We show that $\mathbf{W}$ is an irreducible module for the confluent Cherednik algebra $\mathcal{H}_\mathrm{V}$ and describe how the $\mathbf{T}$-action on $\mathbf{W}$ is related to the $\mathcal{H}_\mathrm{V}$-action on $\mathbf{W}$. Using the $\mathcal{H}_\mathrm{V}$-module $\mathbf{W}$, we define non-symmetric dual $q$-Hahn polynomials and prove their recurrence and orthogonality relations from a combinatorial viewpoint.

math.CO↗

arXiv2018

Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials

Let $Γ$ be a dual polar graph with diameter $D \geqslant 3$, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field $\mathbb{F}_q$ equipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt index $D$. From a pair of a vertex $x$ of $Γ$ and a maximal clique $C$ containing $x$, we construct a $2D$-dimensional irreducible module for a nil-DAHA of type $(C^{\vee}_1, C_1)$, and establish its connection to the generalized Terwilliger algebra with respect to $x$, $C$. Using this module, we then define the non-symmetric dual $q$-Krawtchouk polynomials and derive their recurrence and orthogonality relations from the combinatorial points of view. We note that our results do not depend essentially on the particular choice of the pair $x$, $C$, and that all the formulas are described in terms of $q$, $D$, and one other scalar which we assign to $Γ$ based on the type of the form.

math.CO↗

arXiv2016

Nonsymmetric Askey-Wilson polynomials and $Q$-polynomial distance-regular graphs

In his famous theorem (1982), Douglas Leonard characterized the $q$-Racah polynomials and their relatives in the Askey scheme from the duality property of $Q$-polynomial distance-regular graphs. In this paper we consider a nonsymmetric (or Laurent) version of the $q$-Racah polynomials in the above situation. Let $Γ$ denote a $Q$-polynomial distance-regular graph that contains a Delsarte clique $C$. Assume that $Γ$ has $q$-Racah type. Fix a vertex $x \in C$. We partition the vertex set of $Γ$ according to the path-length distance to both $x$ and $C$. The linear span of the characteristic vectors corresponding to the cells in this partition has an irreducible module structure for the universal double affine Hecke algebra $\hat{H}_q$ of type $(C^{\vee}_1, C_1)$. From this module, we naturally obtain a finite sequence of orthogonal Laurent polynomials. We prove the orthogonality relations for these polynomials, using the $\hat{H}_q$-module and the theory of Leonard systems. Changing $\hat{H}_q$ by $\hat{H}_{q^{-1}}$ we show how our Laurent polynomials are related to the nonsymmetric Askey-Wilson polynomials, and therefore how our Laurent polynomials can be viewed as nonsymmetric $q$-Racah polynomials.

math.CO↗

arXiv2016

$Q$-polynomial distance-regular graphs and a double affine Hecke algebra of rank one

We study a relationship between $Q$-polynomial distance-regular graphs and the double affine Hecke algebra of type $(C^{\vee}_1,C_1)$. Let $Γ$ denote a $Q$-polynomial distance-regular graph with vertex set $X$. We assume that $Γ$ has $q$-Racah type and contains a Delsarte clique $C$. Fix a vertex $x \in C$. We partition $X$ according to the path-length distance to both $x$ and $C$. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a $\mathbb{C}$-vector space ${\bf W}$. The universal double affine Hecke algebra of type $(C^{\vee}_1,C_1)$ is the $\mathbb{C}$-algebra $\hat{H}_q$ defined by generators $\{t^{\pm1}_n\}^3_{n=0}$ and relations (i) $t_nt_n^{-1}=t_n^{-1}t_n=1$; (ii) $t_n+t_n^{-1}$ is central; (iii) $t_0t_1t_2t_3 = q^{-1/2}$. In this paper, we display an $\hat{H}_q$-module structure for ${\bf W}$. For this module and up to affine transformation, (i) $t_0t_1+(t_0t_1)^{-1}$ acts as the adjacency matrix of $Γ$; (ii) $t_3t_0+(t_3t_0)^{-1}$ acts as the dual adjacency matrix of $Γ$ with respect to $C$; (iii) $t_1t_2+(t_1t_2)^{-1}$ acts as the dual adjacency matrix of $Γ$ with respect to $x$. To obtain our results we use the theory of Leonard systems.

math.RT↗