The Eaton--Moret\'o Conjecture Holds True for \(p\)-Solvable Groups
In this paper we prove that the Eaton--Moret\'{o} conjecture holds true for all $p$-solvable groups.
math.RT↗
arXiv subjects
Jiping Zhang
Publications and source records attributed to Jiping Zhang.
A Proof of the Isaacs_Navarro_Wolf Conjecture
An element of a finite group is non-vanishing if no irreducible complex character vanishes on it. We prove that every non-vanishing element of a finite solvable group belongs to the Fitting subgroup. The proof combines the reduction of Isaacs, Navarro, and Wolf with an argument using finite symplectic spaces and Clifford theory.
math.GR↗
Thompson's Conjecture for Finite Simple Groups of Lie Type over Small Fields
We prove that every finite simple group of Lie type over a field of order at most 8 contains a conjugacy class whose square is the whole group. Together with the previously established cases, our results complete the proof of Thompson's conjecture. We explicitly construct the required class, address all low-rank identifications, and provide the precise computational certificates used in the exceptional character calculations.
math.GR↗
The Blockwise Navarro Alperin Weight Conjecture for Double Covers of Symmetric and Alternating Groups
We prove the blockwise Navarro Alperin weight conjecture for double covers of symmetric and alternating groups.
math.GR↗
Generic weights for finite reductive groups
This paper is motivated by the study of Alperin's weight conjecture in the representation theory of finite groups. We generalize the notion of $e$-cuspidality in the $e$-Harish-Chandra theory of finite reductive groups, and define generic weights in non-defining characteristic. We show that the generic weights play an analogous role as the weights defined by Alperin in the investigation of the inductive Alperin weight condition for simple groups of Lie type at most good primes. We hope that our approach will constitute a step towards an eventual proof of Alperin's weight conjecture.
math.RT↗
WaveHiTS: Wavelet-Enhanced Hierarchical Time Series Modeling for Wind Direction Nowcasting in Eastern Inner Mongolia
Wind direction forecasting plays a crucial role in optimizing wind energy production, but faces significant challenges due to the circular nature of directional data, error accumulation in multi-step forecasting, and complex meteorological interactions. This paper presents a novel model, WaveHiTS, which integrates wavelet transform with Neural Hierarchical Interpolation for Time Series to address these challenges. Our approach decomposes wind direction into U-V components, applies wavelet transform to capture multi-scale frequency patterns, and utilizes a hierarchical structure to model temporal dependencies at multiple scales, effectively mitigating error propagation. Experiments conducted on real-world meteorological data from Inner Mongolia, China demonstrate that WaveHiTS significantly outperforms deep learning models (RNN, LSTM, GRU), transformer-based approaches (TFT, Informer, iTransformer), and hybrid models (EMD-LSTM). The proposed model achieves RMSE values of approximately 19.2{\deg}-19.4{\deg} compared to 56{\deg}-64{\deg} for deep learning recurrent models, maintaining consistent accuracy across all forecasting steps up to 60 minutes ahead. Moreover, WaveHiTS demonstrates superior robustness with vector correlation coefficients (VCC) of 0.985-0.987 and hit rates of 88.5%-90.1%, substantially outperforming baseline models. Ablation studies confirm that each component-wavelet transform, hierarchical structure, and U-V decomposition-contributes meaningfully to overall performance. These improvements in wind direction nowcasting have significant implications for enhancing wind turbine yaw control efficiency and grid integration of wind energy.
cs.LG↗
Categorical action for finite classical groups and its applications: characteristic 0
In this paper, we construct a categorical double quantum Heisenberg action on the representation category of finite classical groups $\mathrm{O}_{2n+1}(q)$, $\mathrm{Sp}_{2n}(q)$ and $\mathrm{O}^{\pm}_{2n}(q)$ with $q$ odd. Over a field of characteristic zero or characteristic $\ell$ with $\ell\nmid q(q-1)$, we deduce a categorical action of a Kac-Moody algebra $\mathfrak{s}\mathfrak{l}'_{I_+}\oplus\mathfrak{s}\mathfrak{l}'_{I_-}$ on the representation category of finite classical groups. We show that the colored weight functions $\mathbb{O}^+(u)(\bullet)$, $\mathbb{O}^-(v)(\bullet)$ and uniform projection can distinguish all irreducible characters of finite classical groups. In particular, the colored weight functions are complete invariants of quadratic unipotent characters. We also show that using the theta correspondence and extra symmetries of categorical double quantum Heisenberg action, the Kac-Moody action on the Grothendieck group of the whole category can be determined explicitly.
math.RT↗
Control of fusion by elementary abelian subgroups of rank at least 2
In this paper, we focus on the subgroups control $p$-fusion, and we improve the Theorem B of [4] for odd prime. For odd prime, we prove that elementary abelian subgroups of rank at least 2 can control $p$-fusion(see our Theorem B).
math.GR↗
Radical subgroups of finite reductive groups
Radical subgroups play an important role in both finite group theory and representation theory. This is the first of a series of papers of ours in classifying radical $p$-subgroups of finite reductive groups and in verifying the inductive blockwise Alperin weight condition for them, contributing to the program of proving the Alperin weight conjecture by verifying its inductive condition for finite simple groups. In this paper we present a uniform method for classifying radical subgroups of finite reductive groups. As applications, we investigate the inductive blockwise Alperin weight condition for classical groups, as well as groups of type $F_4$.
math.RT↗
Are Your Models Still Fair? Fairness Attacks on Graph Neural Networks via Node Injections
Despite the remarkable capabilities demonstrated by Graph Neural Networks (GNNs) in graph-related tasks, recent research has revealed the fairness vulnerabilities in GNNs when facing malicious adversarial attacks. However, all existing fairness attacks require manipulating the connectivity between existing nodes, which may be prohibited in reality. To this end, we introduce a Node Injection-based Fairness Attack (NIFA), exploring the vulnerabilities of GNN fairness in such a more realistic setting. In detail, NIFA first designs two insightful principles for node injection operations, namely the uncertainty-maximization principle and homophily-increase principle, and then optimizes injected nodes' feature matrix to further ensure the effectiveness of fairness attacks. Comprehensive experiments on three real-world datasets consistently demonstrate that NIFA can significantly undermine the fairness of mainstream GNNs, even including fairness-aware GNNs, by injecting merely 1% of nodes. We sincerely hope that our work can stimulate increasing attention from researchers on the vulnerability of GNN fairness, and encourage the development of corresponding defense mechanisms. Our code and data are released at: https://github.com/CGCL-codes/NIFA.
cs.LG↗
Descent of splendid Rickard equivalences in ${\rm GL}_n(q)$
Let $n$ be a positive integer and $q$ a prime power. We prove that a refined version of Broué's abelian defect group conjecture holds for unipotent $\ell$-blocks of ${\rm GL}_n(q)$, where $\ell\nmid q$. We also give a sufficient condition on general $\ell$-blocks of ${\rm GL}_n(q)$ to satisfy the refined abelian defect group conjecture. We explain by an example that this sufficient condition does not hold in general.
math.RT↗
Categorical actions and derived equivalences for finite odd-dimensional orthogonal groups
In this paper we prove that Broué's abelian defect group conjecture is true for the finite odd-dimensional orthogonal groups $\SO_{2n+1}(q)$ at linear primes with $q$ odd. We first make use of the reduction theorem of Bonnafé-Dat-Rouquier to reduce the problem to isolated blocks. Then we construct a categorical action of a Kac-Moody algebra on the category of quadratic unipotent representations of the various groups $\SO_{2n+1}(q)$ in non-defining characteristic, by generalizing the corresponding work of Dudas-Varagnolo-Vasserot for unipotent representations. This is one of the main ingredients of our work which may be of independent interest. To obtain derived equivalences of blocks and their Brauer correspondents, we define and investigate isolated RoCK blocks. Finally, we establish the desired derived equivalence based on the work of Chuang-Rouquier that categorical actions provide derived equivalences between certain weight spaces.
math.RT↗
The strengthened Broué abelian defect group conjecture for ${\rm SL}(2,p^n)$ and ${\rm GL}(2,p^n)$
We show that each $p$-block of ${\rm SL}(2,p^n)$ and ${\rm GL}(2,p^n)$ over an arbitrary complete discrete valuation ring is splendidly Rickard equivalent to its Brauer correspondent, hence give new evidence for a refined version of Broué's abelian defect group conjecture proposed by Kessar and Linckelmann.
math.RT↗
On polynomial representations of finite groups
By generalizing Frobenius' polynomial method to good partition algebra, we will develop new character theories for a finite group $G$. A uniform defining equations are derived for these kinds of character theories. The new character theories leads to various factorizations of the group determinant. We will show that these new character theories are equivalent to the Frobenius polynomials of the correspondent good partition algebras. In particular, the character table of a finite group $G$ can be replaced by the Frobenius polynomial of $G$ which is a kind of degenerate of the group determinant. As applications, we find a new series of invariants $p_{ijl}$ for a finite group. In particular, a finite simple group is determined by these invariants. As further applications, the degrees of all irreducible characters can also be realized as the solutions of a polynomial of $G$ and we can combine the refined McKay conjecture and part of Galois conjecture of Navarro on degrees of irreducible characters into a new general conjecture.
math.RT↗
Morita equivalences and the inductive blockwise Alperin weight condition for type $\mathsf A$
As a step to establish the blockwise Alperin weight conjecture for all finite groups, we verify the inductive blockwise Alperin weight condition introduced by Navarro--Tiep and Späth for simple groups of Lie type $\mathsf A$, split or twisted. Key to the proofs is to reduce the verification of the inductive condition to the isolated (that means unipotent) blocks, using the Jordan decomposition for blocks of finite reductive groups given by Bonnafé, Dat and Rouquier.
math.RT↗
Jordan decomposition for weights and the blockwise Alperin weight conjecture
The Alperin weight conjecture was reduced to simple groups by the work of Navarro, Tiep and Späth. To prove Alperin weight conjecture, it suffices to show that all finite non-abelian simple groups are BAW-good. We reduce the verification of the inductive conditons for groups of Lie type in non-defining characteristic to quasi-isolated blocks.
math.RT↗
On the inductive blockwise Alperin weight condition for type $\mathsf A$
In this paper we prove the blockwise Alperin weight conjecture for finite special linear and unitary groups, for finite groups with abelian Sylow $3$-subgroups, and verify the inductive blockwise Alperin weight condition for certain cases of groups of type $\mathsf A$. We also give a classification for the 2-blocks of special linear and unitary groups.
math.RT↗
On the local rank of fusion systems
In this paper we define the notion of local rank for fusion systems so as to reformulate the Alperin's weight conjecture in the framework of block fusion systems following the work by Knörr and Robinson.
math.GR↗