arXiv subjects
Publications and source records attributed to Xun Xie.
In this paper, we give an efficient algorithm to compute the Gelfand-Kirillov dimensions of simple highest weight modules of exceptional Lie algebras. By using the Sommers-Achar duality, we also determine the annihilator varieties of these highest weight modules.
Filters typically play a crucial role in generating solitons by strictly confirming the working wavelength in a mode-locked laser. However, we have found that the broad smoothing filter with saturable gain also helps to regulate the pulses' operating wavelength. The effective gain, formed by the original gain spectrum and filters, shifts its center based on pulse energy, thereby affecting the pulse spectrum. A virtual mode-locked cavity is established to verify this approach in simulation. The results indicate that the spectrum changes with pulse energy, leading to a wider spectrum output. Our method paves the way for further investigations of the pulse spectrum and provides a practical approach to generating pulses at specific wavelengths.
Let $G$ be a Hermitian type Lie group with the complexified Lie algebra $\mathfrak{g}$. We use $L(λ)$ to denote a highest weight Harish-Chandra $G$-module with infinitesimal character $λ$. Let $w$ be an element in the Weyl group $W$. We use $L_w$ to denote a highest weight module with highest weight $-wρ-ρ$. In this paper we prove that there is only one Kazhdan--Lusztig right cell such that the corresponding highest weight Harish-Chandra modules $L_w$ have the same associated variety. Then we give a characterization for those $w$ such that $L_w$ is a highest weight Harish-Chandra module and the associated variety of $L(λ)$ will be characterized by the information of the Kazhdan--Lusztig right cell containing some special $w_λ$. We also count the number of those highest weight Harish-Chandra modules $L_w$ in a given Harish-Chandra cell.
Let $\mathfrak{g}$ be a complex simple Lie algebra. A simple $\mathfrak{g}$-module is called minimal if the associated variety of its annihilator ideal coincides with the closure of the minimal nilpotent coadjoint orbit. The main result of this paper is a classification of minimal highest weight modules for $\\mathfrak{g}$. This classification extends the work of Joseph, which focused on categorizing minimal highest weight modules annihilated by completely prime ideals. Furthermore, we have determined the associated varieties of these modules. In other words, we have identified all possible weak quantizations of minimal orbital varieties.
Let $\mathfrak{g}$ be a simple complex Lie algebra with a Cartan subalgebra $\mathfrak{h}$. We fix a standard parabolic subalgebra $\mathfrak{p}\supset \mathfrak{h}$. The socular simple modules play an important role in the parabolic versions of category $\mathcal{O}^{\mathfrak{p}}$. From Irving's work, we know that these modules are just those modules with largest possible Gelfand-Kirillov dimension in $\mathcal{O}^{\mathfrak{p}}$. In this article, we will give an explicit characterization for these modules of $\mathfrak{sl}(n,\mathbb{C})$. Our characterization is given in the information of the corresponding highest weight and Young tableau.
We prove a simple formula that calculates the associated variety of a highest weight Harish-Chandra module directly from its highest weight. We also give a formula for the Gelfand--Kirillov dimension of highest weight Harish-Chandra module which is uniform across Cartan types and is valid for arbitrary infinitesimal character.
In this paper, we present a uniform formula of Lusztig's $ \mathbf{a}$-functions on classical Weyl groups. Then we obtain an efficient algorithm for the Gelfand-Kirillov dimensions of simple highest weight modules of classical Lie algebras, whose highest weight is not necessarily regular or integral. To deal with type $ D $, we prove an interesting property about domino tableaux by introducing an invariant, called the hollow tableau. As an application, the associated varieties of all the simple highest weight Harish-Chandra modules are explicitly determined, including the exceptional cases.
Selective data protection is a promising technique to defend against the data leakage attack. In this paper, we revisit technical challenges that were neglected when applying this protection to real applications. These challenges include the secure input channel, granularity conflict, and sensitivity conflict. We summarize the causes of them and propose corresponding solutions. Then we design and implement a prototype system for selective data protection and evaluate the overhead using the RISC-V Spike simulator. The evaluation demonstrates the efficiency (less than 3% runtime overhead with optimizations) and the security guarantees provided by our system.
We prove Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3. Our proof enables us to give a description of the a-function and Kazhdan-Lusztig cells for these Coxeter groups.
We prove Lusztig's conjectures P1-P15 for Coxeter groups with complete graph, using deceasing induction on $ \mathbf{a} $-values and a kind of decomposition formula of Kazhdan-Lusztig basis elements. As a byproduct, we give a description of the left, right, and two-sided cells. In the appendix, we prove P1-P15 for right-angled Coxeter groups by the same methods.
Let $ (G,K) $ be an irreducible Hermitian symmetric pair of non-compact type with $G=SU(p,q)$, and let $ λ$ be an integral weight such that the simple highest weight module $ L(λ) $ is a Harish-Chandra $ (\mathfrak{g},K) $-module. We give a combinatoric algorithm for the Gelfand-Kirillov dimension of $ L(λ) $. This enables us to prove that the Gelfand-Kirillov dimension of $ L(λ) $ decreases as the integer $ \langleλ+ρ,β^\vee\rangle $ increases, where $ρ$ is the half sum of positive roots and $β$ is the maximal noncompact root. As a byproduct, we obtain a description on the associated variety of $ L(λ) $.
Let $\mathbf{c}_0$ be the lowest generalized two-sided cell of an extended affine Weyl group W. We determine the structure of the based ring of $\mathbf{c}_0$. For this we show that certain conjectures of Lusztig on generalized cells (called P1-P15) hold for $\mathbf{c}_0$. As an application, we use the structure of the based ring to study certain simple modules of Hecke algebras of $ W $ with unequal parameters, namely those attached to $\mathbf{c}_0$. Also we give a set of prime ideals $\mathfrak{p}$ of the center $\mathcal{Z}$ of the generic affine Hecke algebra $\mathcal{H}$ such that the reduced affine Hecke algebra $k_\mathfrak{p}\mathcal{H}$ is simple over $k_\mathfrak{p}$, where $k_\mathfrak{p}=\rm{Frac}(\mathcal{Z}/\mathfrak{p})$ is the residue field of $\mathcal{Z}$ at $\mathfrak{p}$. In particular, we show that the algebra $\mathcal{H}\otimes_\mathcal{Z}\rm{Frac}(\mathcal{Z})$ is a split simple algebra over the field $ \rm{Frac}(\mathcal{Z})$.
In this paper, we prove a decomposition formula for the Kazhdan-Lusztig basis of affine Hecke algebras of rank 2 with positive weight function. Then we discuss some applications of this kind of decomposition to Lusztig's conjectures P1-P15.