arXiv subjects
Publications and source records attributed to GyeongHyeon Nam.
In this paper, we answer the question posed by Goodwin and Röhrle for reductive groups and their parabolic subgroups. In addition, we consider an additive analogue of this problem. By studying this additive analogue, we identify similar properties between the Deligne-Lusztig character of a finite reductive group and the Harish-Chandra induction over the corresponding finite Lie algebra.
Absolutely indecomposable vector bundle and parabolic vector bundles are well-studied via quiver representations. In this paper, we study absolutely indecomposable quasi-parabolic $G$-bundles over $\mathbb{P}^1$ with generic additive character varieties. Furthermore, we give a geometric interpretation of the multiplicity of the tensor product of irreducible characters of finite reductive groups using generic additive character varieties.
The goal of this paper is to consider symplectic singularities of the moduli space of $G$-Higgs bundles over a compact Riemann surface with genus at least $2$. Furthermore, when $G$ is semisimple, we also study its symplectic resolution.
In this paper, we compute the multiplicities of tensor products of almost unipotent characters and Deligne Lusztig characters of a finite reductive group $G^F$, and these multiplicities are related to the ring structure of the complex irreducible characters of $G^F$. In addition, we consider Frobenius Schur indicators of modules over the Drinfeld doubles of finite reductive groups. In the final section, we study the multiplicities of tensor products of almost unipotent characters and pose the question of whether their non vanishing can be detected through the multiplicities of tensor products of irreducible characters of the Weyl group.
We count points on character varieties associated with punctured surfaces and regular semisimple generic conjugacy classes in reductive groups. We find that the number of points are palindromic polynomials. This suggests a $P=W$ conjecture for these varieties. We also count points on the corresponding additive character varieties and find that the number of points are also polynomials, which we conjecture have non-negative coefficients. These polynomials can be considered as the reductive analogues of the Kac polynomials of comet-shaped quivers.
We consider the proportion of zero entries in the character table of a sequence of reductive groups over a finite field. We prove an asymptotic lower bound when the reductive group is fixed and the size of the finite field increases. Furthermore, we prove that when considering a sequence of reductive groups with increasing semisimple rank, the proportion is asymptotically one. We also establish an additive analogue of this phenomenon in the context of a fixed reductive Lie algebra.
For a field $L$ of characteristic $p$, a polynomial $f \in \overline{\mathbb{F}}_p[x]$ and $α, β\in L$, let $\mathrm{Prep}(f;α,β)$ be the set of all $λ\in \overline{L}$ such that both $α$ and $β$ are preperiodic under the action of $f_λ(x) := f(x) + λ$. Ghioca and Hsia proved that for certain families of polynomials, this set is infinite if and only if $f(α)=f(β)$ or $α, β\in \overline{\mathbb{F}}_p$. Building on their work, we determine when $\mathrm{Prep}(f;α,β)$ is infinite for most of the remaining binomial cases that were left open. Specifically, let $f(x)=c_1 x^{d_1} + c_2 x^{d_2} \in \overline{\mathbb{F}}_p[x]$, where $c_i \in \overline{\mathbb{F}}_p^*$, $1 \le d_1 < d_2$ and $d_i=p^{\ell_i}s_i$ with $\ell_i \ge 0$ and $p \nmid s_i$. We prove that if $p^{\ell_2}(s_2-1) < p^{\ell_1}(s_1-1)$, then $\mathrm{Prep}(f;α,β)$ is infinite if and only if $f(α)=f(β)$ or $α, β\in \overline{\mathbb{F}}_p$. The key idea of the proof is to use the parameters $λ_{\overlineα} := \overlineα - f(\overlineα)$ associated to suitable elements $\overlineα \in \overline{L}$ satisfying $f(\overlineα)=f(α)$. As an application, we extend the work of Asgarli and Ghioca on the colliding orbits problem to binomials satisfying $s_2>1$ and $p^{\ell_2}(s_2-1) < p^{\ell_1}(s_1-1)$.
We know from Letellier that if for some triple of partitions the corresponding Kronecker coefficient is non-zero then the corresponding multiplicities for unipotent characters of GL(n,q) is also non-zero. A conjecture of Saxl says that the tensor square of an irreducible character of the symmetric group corresponding to a staircase partition contains all the irreducible characters. Therefore Saxl conjecture implies its analogue for unipotent characters. In this paper we prove the analogue of Saxl conjecture for unipotent characters and we describe conjecturally the set of all partitions for which the tensor square of the associated unipotent character contains all the unipotent characters.
We study character varieties arising as moduli of representations of an orientable surface group into a reductive group $G$. We first show that if $G/Z$ acts freely on the representation variety, then both the representation variety and the character variety are smooth and equidimensional. Next, we count points on a family of smooth character varieties; namely, those involving both regular semisimple and regular unipotent monodromy. In particular, we show that these varieties are polynomial count and obtain an explicit expression for their $E$-polynomials. Finally, by analysing the $E$-polynomial, we determine certain topological invariants of these varieties such as the Euler characteristic and the number of connected components. As an application, we give an example of a cohomologically rigid representation which is not physically rigid.