Searcharxiv⌕ Search

arXiv subjects

Young Rock Kim

Publications and source records attributed to Young Rock Kim.

11 recordsLinked to original sources
arXiv2026

Quantum groups of Borcherds-Cartan type and Khovanov-Lauda-Rouquier algebras

We categorify a class of quantum groups associated with quivers, possibly with loops, by constructing the corresponding Khovanov-Lauda-Rouquier algebras (KLR) algebras $R$. We prove that the indecomposable projective $R$-modules realize the canonical basis of the negative part $U^-$ of the quantum group. Moreover, for $Λ\in P^+$, the cyclotomic KLR algebra $R^Λ$ provide a categorification of the irreducible highest weight $U$-module $V(Λ)$.

math.QA↗
arXiv2024

Perfect basis theory for quantum Borcherds-Bozec algebras

In this paper, we develop the perfect basis theory for quantum Borcherds-Bozec algebras $U_{q}(\mathfrak g)$ and their irreducible highest weight modules $V(λ)$. We show that the lower perfect graph (resp. upper perfect graph) of every lower perfect basis (resp. upper perfect basis) of $U_{q}^{-}(\mathfrak g)$ (resp. $V(λ)$) is isomorphic to the crystal $B(\infty)$ (resp. $B(λ)$).

math.QA↗
arXiv2024

Crystal bases and canonical bases for quantum Borcherds-Bozec algebras

Let $U_{q}^{-}(\mathfrak g)$ be the negative half of a quantum Borcherds-Bozec algebra $U_{q}(\mathfrak g)$ and $V(λ)$ be the irreducible highest weight module with $λ\in P^{+}$. In this paper, we investigate the structures, properties and their close connections between crystal bases and canonical bases of $U_{q}^{-}(\mathfrak g)$ and $V(λ)$. We first re-construct crystal basis theory with modified Kashiwara operators. While going through Kashiwara's grand-loop argument, we prove several important lemmas, which play crucial roles in the later developments of the paper. Next, based on the theory of canonical bases on quantum Bocherds-Bozec algebras, we introduce the notion of primitive canonical bases and prove that primitive canonical bases coincide with lower global bases.

math.RT↗
arXiv2024

A new Young wall realization of $B(λ)$ and $B(\infty)$

Using new combinatorics of Young walls, we give a new construction of the arbitrary level highest weight crystal $B(λ)$ for the quantum affine algebras of types $A^{(2)}_{2n}$, $D^{(2)}_{n+1}$, $A^{(2)}_{2n-1}$, $D^{(1)}_n$, $B^{(1)}_n$ and $C^{(1)}_n$. We show that the crystal consisting of reduced Young walls is isomorphic to the crystal $B(λ)$. Moreover, we provide a new realization of the crystal $B(\infty)$ in terms of reduced virtual Young walls and reduced extended Young walls.

math.QA↗
arXiv2020

Abstract crystals for quantum Borcherds-Bozec algebras

In this paper, we develop the theory of abstract crystals for quantum Borcherds-Bozec algebras. Our construction is different from the one given by Bozec. We further prove the crystal embedding theorem and provide a characterization of ${B}(\infty)$ and ${B}(λ)$ as its application, where ${B}(\infty)$ and ${B}(λ)$ are the crystals of the negative half part of the quantum Borcherds-Bozec algebra $U_q(\mathfrak g)$ and its irreducible highest weight module $V(λ)$, respectively.

math.RT↗
arXiv2016

Geometry and a natural symplectic structure of phase tropical hypersurfaces

First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are diffeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that phase tropical hypersurfaces with smooth tropicalization, possess naturally a smooth differentiable structure. Moreover, we prove that phase tropical hypersurfaces possess a natural symplectic structure.

math.AG↗
arXiv2006

Phylogenetic tree constructing algorithms fit for grid computing with SVD

Erikkson showed that singular value decomposition(SVD) of flattenings determined a partition of a phylogenetic tree to be a split. In this paper, based on his work, we develop new statistically consistent algorithms fit for grid computing to construct a phylogenetic tree by computing SVD of flattenings with the small fixed number of rows.

q-bio.QM↗
arXiv2006

Quartet consistency count method for reconstructing phylogenetic trees

Among the distance based algorithms in phylogenetic tree reconstruction, the neighbor-joining algorithm has been a widely used and effective method. We propose a new algorithm which counts the number of consistent quartets for cherry picking with tie breaking. We show that the success rate of the new algorithm is almost equal to that of neighbor-joining. This gives an explanation of the qualitative nature of neighbor-joining and that of dissimilarity maps from DNA sequence data. Moreover, the new algorithm always reconstructs correct trees from quartet consistent dissimilarity maps.

q-bio.PE↗
arXiv2006

Normal generation and Clifford index

Let $C$ be a smooth curve of genus $g\ge 4$ and Clifford index $c$. In this paper, we prove that if $C$ is neither hyperelliptic nor bielliptic with $g\ge 2c+5$ and $\mathcal M$ computes the Clifford index of $C$, then either $°\mathcal M\le \frac{3c}{2}+3$ or $|\mathcal M|=|g^1_{c+2}+h^1_{c+2}|$ and $g=2c+5$. This strengthens the Coppens and Martens' theorem (\cite{CM}, Corollary 3.2.5). Furthermore, for the latter case (1) $\mathcal M$ is half-canonical unless $C$ is a $\frac{c+2}{2}$-fold covering of an elliptic curve, (2) $\mathcal M(F)$ fails to be normally generated with $\cli(\mathcal M(F))=c$, $h^1(\mathcal M(F))=2$ for $F\in g^1_{c+2}$. Such pairs $(C,\mathcal M)$ can be found on a $K3$-surface whose Picard group is generated by a hyperplane section in $\mathbb P^r$. For such a $(C, \mathcal M)$ on a K3-surface, $\mathcal M$ is normally generated while $\mathcal M(F)$ fails to be normally generated with $\cli(\mathcal M)=\cli(\mathcal M(F))=c$.

math.AG↗