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Dongkwan Kim

Publications and source records attributed to Dongkwan Kim.

30 records · Page 2Linked to original sources
arXiv2020

On total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case

Let $W$ be the Weyl group of type $BC_n$. We first provide restriction formulas of the total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case to the maximal parabolic subgroup of $W$ which is of type $BC_{n-1}$. Then we show that these two restriction formulas are equivalent, and discuss how the results can be used to examine the existence of affine pavings of Springer fibers corresponding to the symplectic Lie algebra in characteristic 2.

math.RT↗
arXiv2019

Two-row $W$-graphs in affine type $A$

For affine symmetric groups we construct finite $W$-graphs corresponding to two-row shapes, and prove their uniqueness. This gives the first non-trivial family of examples of finite $W$-graphs in an affine type. We compare our construction with quotients of periodic $W$-graphs defined by Lusztig. Under certain positivity assumption on the latter the two are shown to be isomorphic.

math.CO↗
arXiv2018

Homogeneity-Based Transmissive Process to Model True and False News in Social Networks

An overwhelming number of true and false news stories are posted and shared in social networks, and users diffuse the stories based on multiple factors. Diffusion of news stories from one user to another depends not only on the stories' content and the genuineness but also on the alignment of the topical interests between the users. In this paper, we propose a novel Bayesian nonparametric model that incorporates homogeneity of news stories as the key component that regulates the topical similarity between the posting and sharing users' topical interests. Our model extends hierarchical Dirichlet process to model the topics of the news stories and incorporates Bayesian Gaussian process latent variable model to discover the homogeneity values. We train our model on a real-world social network dataset and find homogeneity values of news stories that strongly relate to their labels of genuineness and their contents. Finally, we show that the supervised version of our model predicts the labels of news stories better than the state-of-the-art neural network and Bayesian models.

cs.CY↗
arXiv2018

On the affine Schützenberger involution

We consider an involution on the affine Weyl group of type $A$ induced from the nontrivial automorphism on the (finite) Dynkin diagram. We prove that the number of left cells fixed by this involution in each two-sided cell is given by a certain Green polynomial of type $A$ evaluated at -1.

math.RT↗
arXiv2018

On the Betti numbers of Springer fibers for classical types

For a Weyl group $W$ of classical type, we present a formula to calculate the restriction of (graded) Springer representations of $W$ to a maximal parabolic subgroup $W'$ where the types of $W$ and $W'$ are in the same series. As a result, we obtain recursive formulas for the Betti numbers of Springer fibers for classical types.

math.RT↗
arXiv2018

On total Springer representations for classical types

We give explicit formulas on total Springer representations for classical types. We also describe the characters of restrictions of such representations to a maximal parabolic subgroup isomorphic to a symmetric group. As a result, we give closed formulas for the Euler characteristic of Springer fibers.

math.RT↗
arXiv2016

Euler Characteristic of Springer fibers

For Weyl groups of classical types, we present formulas to calculate the restriction of Springer representations to a maximal parabolic subgroup of the same type. As a result, we give recursive formulas for Euler characteristics of Springer fibers for classical types. We also give tables of those for exceptional types.

math.RT↗
arXiv2016

Euler characteristic of analogues of a Deligne-Lusztig variety for $GL_n$

In this paper we give a combinatorial formula to calculate the Euler characteristic of an analogue of a Deligne-Lusztig variety if we replace Frobenius morphism with conjugation by an element for $GL_n$. The main theorem states that it only depends on the unipotent part of the Jordan decomposition of an element and the conjugacy class in the Weyl group. Also it generalizes the formula of the Euler characteristic of a Springer fiber for type A.

math.RT↗
arXiv2016

Homology Class of a Deligne-Lusztig variety and its analogues

In this paper we consider Deligne-Lusztig varieties and their analogues when the Frobenius endomorphism is replaced with conjugation by an element in a group, especially a regular semisimple or regular unipotent one. We calculate their classes in the Chow group of the flag variety in terms of Schubert classes. Also we give some sufficient criteria when different elements in the Weyl group result in the same class.

math.RT↗
arXiv2016

A Comparison of Two Complexes

In this paper we prove the conjecture of Lusztig in "Generic character sheaves on groups over $\mathbf{k}[ε]/(ε^r)$." Given a reductive group over $\mathbb{F}_q$ for some $r\geq 2$, there is a notion of a character sheaf defined in "Character sheaves and generalizations" by Lusztig. On the other hand, there is also a geometric analogue of the character constructed by Gérardin. The conjecture states that the two constructions are equivalent, which Lusztig also proved for $r=2, 3, 4$. Here we generalize his method to prove this conjecture for general $r$. As a corollary we prove that the characters derived from these two complexes are equal.

math.RT↗