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Chao-Ping Dong

Publications and source records attributed to Chao-Ping Dong.

29 records · Page 2Linked to original sources
arXiv2020

Unitary Representations with Dirac cohomology: a finiteness result for complex Lie groups

Let $G$ be a connected complex simple Lie group, and let $\widehat{G}^{\mathrm{d}}$ be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that $\widehat{G}^{\mathrm{d}}$ consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of $\widehat{G}^{\mathrm{d}}$ come from $\widehat{L}^{\mathrm{d}}$ via cohomological induction and they are all in the good range. Here $L$ runs over the Levi factors of proper $θ$-stable parabolic subgroups of $G$. It follows that figuring out $\widehat{G}^{\mathrm{d}}$ requires a finite calculation in total. As an application, we report a complete description of $\widehat{F}_4^{\mathrm{d}}$.

math.RT↗
arXiv2019

Fast Diffusion Inhibits Disease Outbreaks

We show that the basic reproduction number of an SIS patch model with standard incidence is either strictly decreasing and strictly convex with respect to the diffusion coefficient of infected subpopulation if the patch reproduction numbers of at least two patches in isolation are distinct or constant otherwise. Biologically, it means that fast diffusion of the infected people reduces the risk of infection. This completely solves and generalizes a conjecture by Allen et al. ({\it SIAM J Appl Math}, 67: 1283-1309, 2007). Furthermore, a substantially improved lower bound on the multipatch reproduction number, a generalized monotone result on the spectral bound the Jacobian matrix of the model system at the disease-free equilibrium, and the limiting endemic equilibrium are obtained. The approach and results can be applied to a class of epidemic patch models where only one class of infected compartments migrate between patches and one transmission route is involved.

q-bio.PE↗
arXiv2019

Antichain generating polynomials of posets

This paper gives a formula for the antichain generating polynomial $\mathcal{N}_{[k]\times Q}$ of the poset $[k]\times Q$, where $[k]$ is an arbitrary chain and $Q$ is any finite graded poset. When $Q$ specializes to be a connected minuscule poet, which was classified by Proctor in 1984, we find that the polynomial $\mathcal{N}_{[k]\times Q}$ bears nice properties. For instance, we will recover the $B_n$-Narayana polynomial and the $D_{2n+2}$-Narayana polynomial. We collect evidence for the conjecture that whenever $\mathcal{N}_{[k]\times P}(x)$ is palindromic, it must be $γ$-positive. Moreover, the family $\mathcal{N}_{[2]\times [n]\times [m]}$ should be real-rooted and $\mathcal{N}_{[2]\times [n]\times [n+1]}$ should be $γ$-positive. We also conjecture that $\mathcal{N}_{Q}(x)$ is log-concave (thus unimodal) for any connected Peck poset $Q$.

math.CO↗
arXiv2018

Unitary representations with Dirac cohomology: finiteness in the real case

Let $G$ be a complex connected simple algebraic group with a fixed real form $σ$. Let $G(\mathbb{R})=G^σ$ be the corresponding group of real points. This paper reports a finiteness theorem for the classification of irreducible unitary Harish-Chandra modules of $G(\mathbb{R})$ (up to equivalence) having non-vanishing Dirac cohomology. Moreover, we study the distribution of the spin norm along Vogan pencils for certain $G(\mathbb{R})$, with particular attention paid to the unitarily small convex hull introduced by Salamanca-Riba and Vogan.

math.RT↗
arXiv2017

Orbits of antichains in certain root posets

This paper gives another proof of Propp and Roby's theorem saying that the average antichain size in any reverse operator orbit of the poset $[m]\times [n]$ is $\frac{mn}{m+n}$. It is conceivable that our method should work for other situations. As a demonstration, we show that the average size of antichains in any reverse operator orbit of $[m]\times K_{n-1}$ equals $\frac{2mn}{m+2n-1}$. Here $K_{n-1}$ is the minuscule poset $[n-1]\oplus ([1] \sqcup [1]) \oplus [n-1]$. Note that $[m]\times [n]$ and $[m]\times K_{n-1}$ can be interpreted as sub-families of certain root posets. We guess these root posets should provide a unified setting to exhibit the homomesy phenomenon defined by Propp and Roby.

math.CO↗
arXiv2017

Minuscule representations and Panyushev conjectures

Recently, Panyushev raised five conjectures concerning the structure of certain root posets arising from $\mathbb{Z}$-gradings of simple Lie algebras. This paper aims to provide proofs for four of them. Our study also links these posets with Kostant-Macdonald identity, minuscule representations, Stembridge's "$t=-1$ phenomenon", and the cyclic sieving phenomenon due to Reiner, Stanton and White.

math.RT↗
arXiv2016

On the nonvanishing hypothesis for Rankin-Selberg convolutions for $\mathrm{GL}_n(\mathbb C)\times \mathrm{GL}_n(\mathbb C)$

Inspired by Sun's breakthrough in establishing the nonvanishing hypothesis for Rankin-Selberg convolutions for the groups $\mathrm{GL}_n (\mathbb R)\times \mathrm{GL}_{n-1} (\mathbb R)$ and $\mathrm{GL}_n (\mathbb C)\times \mathrm{GL}_{n-1} (\mathbb C)$, we confirm it for $\mathrm{GL}_{n} (\mathbb C)\times \mathrm{GL}_n (\mathbb C)$ at the central critical point.

math.RT↗
arXiv2013

Remarks on the abelian ideals of a Borel subalgebra

Let $\frb$ be a fixed Borel subalgebra of a finite-dimensional complex simple Lie algebra $\frg$. The Shi bijection associates to every ad-nilpotent ideal $\fri$ of $\frb$ a region $V_{\fri}$. In this paper, we show that $\fri$ is abelian if and only if $V_{\fri}\cap 2A$ is empty, if and only if the volume of $V_{\fri}\cap 2A$ equals to that of $A$, where $A$ is the fundamental alcove of the affine Weyl group. For certain flag of abelian ideals, we record an ascending property of their associated regions. We also determine the maximal eigenvalue $m_{r-1}$ of the Casimir operator on $\wedge^{r-1} \frg$ and the corresponding eigenspace $M_{r-1}$, where $r$ is the number of positive roots.

math.RT↗
arXiv2013

Ad-nilpotent ideals and The Shi arrangement

We extend the Shi bijection from the Borel subalgebra case to parabolic subalgebras. In the process, the $I$-deleted Shi arrangement $\texttt{Shi}(I)$ naturally emerges. This arrangement interpolates between the Coxeter arrangement $\texttt{Cox}$ and the Shi arrangement $\texttt{Shi}$, and breaks the symmetry of $\texttt{Shi}$ in a certain symmetrical way. Among other things, we determine the characteristic polynomial $χ(\texttt{Shi}(I), t)$ of $\texttt{Shi}(I)$ explicitly for $A_{n-1}$ and $C_n$. More generally, let $\texttt{Shi}(G)$ be an arbitrary arrangement between $\texttt{Cox}$ and $\texttt{Shi}$. Armstrong and Rhoades recently gave a formula for $χ(\texttt{Shi}(G), t)$ for $A_{n-1}$. Inspired by their result, we obtain formulae for $χ(\texttt{Shi}(G), t)$ for $B_n$, $C_n$ and $D_n$.

math.CO↗