We introduce and study the notion of entropy of affine permutations and prove that it coincides with the atomic length associated with the sum of the fundamental weights for a type $A$ affine root system, as defined by the first two authors. We then establish an analogue of the Granville-Ono theorem by showing that any nonnegative integer can be realised as the entropy of an affine permutation or alternatively, as the size of a core multipartition as introduced by the last two authors. Our proof uses an additive combinatorics theorem due to Hall on difference sets of permutations modulo $n$. More generally, we give a polynomial expression of the atomic length associated with any dominant weight in affine type $A$ and investigate the problem of its universality. Beyond type $A$, we are able to prove that the entropy of affine type $C_n$ permutations is universal when $2n+1$ is prime. This is achieved by establishing an analogue of Hall's theorem for the hyperoctahedral group based on Alon's combinatorial Nullstellensatz. We also propose conjectures generalising the results presented in the paper, each supported by computational evidence. Finally, we show that in any affine classical type, the problem of the universality of the atomic length simplifies in large rank when the weight considered is conveniently adjusted.

In this paper, we explicitly classify the corank 4 unitary representations of symplectic or split odd special orthogonal groups over non-Archimedean local fields of characteristic zero, by classifying Arthur representations of corank 4 and verifying the corresponding unitary dual conjecture recently proposed by Hazeltine-Jiang-Liu-Lo-Zhang in [HJLLZ24].

Plastic pollution has become a critical global challenge, with microplastics pervading ecosystems and entering human food chains. Effectively monitoring this widespread contamination demands rapid, reliable, and portable material identification techniques that often elude conventional Raman and FTIR spectroscopy. Undetected photon spectroscopy within a nonlinear interferometer (NLI) offers a solution, allowing the retrieval of mid-infrared absorption spectra by detecting only near-infrared signal photons using standard silicon-based technology. Here, we demonstrate a highly compact, micro-integrated, thermally-stabilised NLI with a Michelson-like geometry designed for the rapid spectroscopy of plastics. We benchmarked its room-temperature performance, demonstrating a signal-to-noise ratio of 34 with a measurement rate of 100 Hz and a spectral resolution of 6 cm$^{-1}$. We show that we can accurately and rapidly retrieve the characteristic vibrational absorption spectra of common polymers such as polypropylene, polyethene, and polystyrene, without using mid-infrared technology. These results establish our compact module as a promising field-deployable platform for robust, real-time environmental monitoring systems and other mid-infrared spectroscopy applications.

The increasing heterogeneity of high-performance computing (HPC) systems and the transition to exascale architectures require systematic and reproducible performance evaluation across diverse workloads. While continuous integration (CI) ensures functional correctness in software engineering, performance and energy efficiency in HPC are typically evaluated outside CI workflows, motivating continuous benchmarking (CB) as a complementary approach. Integrating benchmarking into CI workflows enables reproducible evaluation, early detection of regressions, and continuous validation throughout the software development lifecycle. We present exaCB, a framework for continuous benchmarking developed in the context of the JUPITER exascale system. exaCB enables application teams to integrate benchmarking into their workflows while supporting large-scale, system-wide studies through reusable CI/CD components, established harnesses, and a shared reporting protocol. The framework supports incremental adoption, allowing benchmarks to be onboarded easily and to evolve from basic runnability to more advanced instrumentation and reproducibility. The approach is demonstrated in JUREAP, the early-access program for JUPITER, where exaCB enabled continuous benchmarking of over 70 applications at varying maturity levels, supporting cross-application analysis, performance tracking, and energy-aware studies. These results illustrate the practicality using exaCB for continuous benchmarking for exascale HPC systems across large, diverse collections of scientific applications.

We present a new approach to hyperbolic plugs, via a construction of bicontact plugs on 3-manifolds with boundary that are surface bundles over the circle. The boundary components are quasi transverse tori, and we prove a gluing theorem that allows us to produce closed manifolds carrying new transitive Anosov flows. We show that a toroidal manifold produced by gluing two copies of the figure eight knot complement may carry many nonequivalent Anosov flows, and likewise a manifold composed of a figure eight complement and a trefoil complement. We further show that certain generalized Handel--Thurston surgeries can be realized as sequences of Goodman--Fried surgeries and produce new examples of different surgery sequences resulting in the same Anosov flow.

Dense neutrino media in core-collapse supernovae and neutron star mergers can experience collective flavor transformations in the form of neutrino oscillation waves. It was recently reported that the stable fast modes of collective oscillations can be damped through a mechanism similar to the Landau damping of plasma waves. In this work, we show that the actual damping rates of fast oscillation waves are usually very small and vanishes in the pure fast limit. This result does not affect the unstable modes that eventually drive collective neutrino flavor conversions in supernovae and neutron star mergers.

Global entanglement in quantum many-body systems is inherently nonlocal, raising the question of whether it can be inferred from local observations. We investigate this problem in monitored quantum circuits, where projective measurements generate classical records distributed across spacetime. Using graph neural networks (GNNs), we represent individual quantum trajectories as directed spacetime graphs and reconstruct the half-chain entanglement entropy from local measurement data alone. Because information propagates through the network via local message passing, the architecture directly controls the spacetime region over which correlations can be aggregated. By systematically varying this accessible scale -- through network depth and hierarchical spacetime coarse-graining -- we probe how much measurement information is required to reconstruct global entanglement. We find that prediction accuracy improves as the accessible spacetime region grows and that results from different architectures collapse when expressed in terms of an effective spacetime scale combining depth and coarse-graining. These results demonstrate that the information required to reconstruct global entanglement is organized in spacetime scales and show that graph-based learning architectures provide a controlled operational framework for probing how global quantum correlations emerge from local measurement data.

In this work we study a classic generalization of the Vertex Cover (VC) problem, called the Component Order Connectivity (COC) problem. In COC, given an undirected graph $G$, integers $d \geq 1$ and $k$, the goal is to determine if there is a set of at most $k$ vertices whose deletion results in a graph where each connected component has at most $d$ vertices. When $d=1$, this is exactly VC. This work is inspired by polynomial kernelization results with respect to structural parameters for VC. On one hand, Jansen & Bodlaender [TOCS 2013] show that VC admits a polynomial kernel when the parameter is the distance to treewidth-$1$ graphs, on the other hand Cygan, Lokshtanov, Pilipczuk, Pilipczuk & Saurabh [TOCS 2014] showed that VC does not admit a polynomial kernel when the parameter is distance to treewidth-$2$ graphs. Greilhuber & Sharma [IPEC 2024] showed that, for any $d \geq 2$, $d$-COC cannot admit a polynomial kernel when the parameter is distance to a forest of pathwidth $2$. Here, $d$-COC is the same as COC only that $d$ is a fixed constant not part of the input. We complement this result and show that like for the VC problem where distance to treewidth-$1$ graphs versus distance to treewidth-$2$ graphs is the dividing line between structural parameterizations that allow and respectively disallow polynomial kernelization, for COC this dividing line happens between distance to pathwidth-$1$ graphs and distance to pathwidth-$2$ graphs. The main technical result of this work is that COC admits a polynomial kernel parameterized by distance to pathwidth-$1$ graphs plus $d$.

We derive a dressed-state master equation in Lindblad form for two strongly coupled two-level atoms. The resulting decay dynamics are governed by Lindblad operators that couple different dressed states. We show that the eigenvalues and eigenvectors of the Liouvillian can be obtained in a compact form, since each off-diagonal element in the dressed-state basis constitutes an eigenvector. Depending on the interatomic distance and the atomic transition frequency, two distinct time scales emerge. On a short time scale, the system relaxes toward two states, one of which corresponds to a transient, maximally entangled configuration. On a longer time scale, this entangled state gradually decays to the steady state.

Many natural and social science systems are described using probability distributions over elements that are related to each other: for instance, occupations with shared skills or species with similar traits. Standard information theory quantities such as entropies and $f$-divergences treat elements interchangeably and are blind to the similarity structure. We introduce a family of divergences that are sensitive to the geometry of the underlying domain. By virtue of being the Bregman divergences of structure-aware entropies, they provide a framework that retains several advantages of Kullback-Leibler divergence and Shannon entropy. Structure-aware divergences recover planted patterns in a synthetic clustering task that conventional divergences miss and are orders of magnitude faster than optimal transport distances. We demonstrate their applicability in economic geography and ecology, where structure plays an important role. Modelling different notions of occupation relatedness yields qualitatively different regionalisations of their geographic distribution. Our methods also reproduce established insights into functional $β$-diversity in ecology obtained with optimal transport methods.

The Quick-Look Pipeline (QLP; Huang et al. 2020, Kunimoto et al. 2021 and references therein) generates light curves for up to 2 million stars every 27.4 days observed by TESS as part of its planet search. As machine learning methods enable deeper searches and scientific priorities shift toward fainter stars, there is a motivation for QLP to perform better at fainter magnitudes. We have adopted the photometry methods employed by the TESS-Gaia Light Curve package (Han & Brandt 2023), which has been shown to have better noise characteristics than the original QLP photometry from 10.5 $<$ $T$ $<$ 13.5. We still perform aperture photometry and deliver 3 apertures, and 3 levels of detrending for all stars brighter than $T$ = 13.5, so the changes should be seamless for external users. This method is implemented as of Sector 94 in QLP light curves and is providing users with higher precision light curves and allows detection of fainter signals in our planet searches.

A search for Higgs boson (\PH) production at high transverse momentum ($p_\mathrm{T}$) in the WW decay channel is presented. The analysis uses proton-proton collisions at $\sqrt{s}$ = 13 TeV recorded by the CMS experiment in 2016$-$2018, corresponding to an integrated luminosity of 138 fb$^{-1}$. The visible decay products of the Higgs boson are reconstructed as a single large-radius jet with one isolated lepton or none (1$\ell$ and 0$\ell$, respectively; $\ell$ = e, $μ$). The \PH-candidate jets are identified using an advanced transformer-based algorithm and are calibrated with the Lund jet plane reweighting technique. The 1$\ell$ channel is further split into gluon fusion, vector boson fusion, and associated production with hadronically decaying vector boson categories, while the 0$\ell$ channel considers all production processes inclusively. The measured cross section times the H $\to$ WW branching fraction relative to the standard model expectation is $μ$ = $-$0.19$^{+0.48}_{-0.46}$, indicating no evidence of a signal above the background. This measurement represents the first dedicated study of highly Lorentz-boosted H $\to$ WW decays, complementing earlier searches for high-$p_\mathrm{T}$ Higgs boson in other decay channels.

In this work, we theoretically propose a broadband asymmetric transmission (AT) device based on periodic Si nanoring arrays embedded in a SiO2 substrate. Results indicate that the device achieves a remarkable broadband AT effect in the near-infrared region (1750-2400 nm), with forward transmissivity exceeding 0.8 (maximum of 0.98), backward transmissivity less than 0.15 (minimum of 0.015) and an isolation ratio (IR) reaching a maximum of 17.8 dB at 2280 nm. Furthermore, the transmissivity spectrum exhibits excellent scalability and tunability through uniform scaling of the structure, allowing the operational band to be tailored across a wide spectral range, from 890 to 3300 nm. This Si-based nanostructure offers a robust and flexible platform for applications in optical isolation, multi-channel sensing, and integrated photonic circuits.

We give a sufficient combinatorial condition for the non-negativity of the coefficients of polynomial quotients of products of $q$-integers, also known as cyclotomic generating functions (CGFs). This slightly extends work by Iano-Fletcher, Pizzato, Sano and Tasin, who studied this condition as a criterion for quasismoothness of complete intersections in weighted projective spaces. As a consequence, we solve a problem by Billey and Swanson, prove most cases of an unpublished conjecture by Stanton and most cases of two conjectures by Gatzweiler and Krattenthaler. We also study sufficient conditions given by structural properties of the division lattice.

We consider random polytopes in the $d$-dimensional Euclidean space that are the convex hulls i.i.d. random points selected according to beta-prime distributions. These distributions are rotationally symmetric, heavy-tailed, and their support is the entire space, making them distinct from other commonly studied distributions, for instance, the uniform and Gaussian distributions. We prove lower bounds for the variances of the intrinsic volumes and the $f$-vector of such random polytopes. Beta-prime random polytopes are the push-forwards of spherical random polytopes, which are the convex hulls of random points chosen in the upper open hemisphere according to some rotationally symmetric distribution, including the uniform distribution in the open half-sphere. Our variance lower bounds also transfer to the spherical settings.

The vision of electrolytic hydrogen as a clean energy vector prompts the emergence of hydrogen-centric companies that must simultaneously engage in electricity, hydrogen, and green certificate markets while operating complex, geographically distributed asset portfolios. This paper proposes a portfolio-level optimization framework tailored for the integrated operational scheduling and market participation of such companies. The model co-optimizes asset scheduling and market decisions across multiple sites, incorporating spatial distribution, technical constraints, and company-level policy requirements. It supports participation in the electricity market, physical and virtual Power Purchase Agreements (PPAs), bundled and unbundled hydrogen markets, and green certificate transactions. The model is applied to three operational scenarios to evaluate the economic and operational impacts of different compliance strategies. Results show that centralized, portfolio-level control unlocks the full flexibility of geographically distributed assets, enabling a 2.42-fold increase in hydrogen production and a 9.4% reduction in daily operational costs, while satisfying all company policy constraints.

The Takagi function $T:[0,1]\to \mathbb{R}$ is a classical example of a continuous nowhere differentiable function. In this paper, we study the discrete dynamical system generated by the Takagi function. First, we prove that for almost every point $x\in [0,1]$, the orbit $(T^n(x))_n$ converges to $2/3$. We introduce the family of Takagi maps, given by $\textbf{T}_γ=γ\cdot T$, where $γ>0$ is a parameter. We also study the shadowing property for this family of maps. We show that the Takagi function has the shadowing property. Additionally, we provide two distinct techniques that allow us to find values of the parameter $γ$ for which $\textbf{T}_γ$ fails to have the shadowing property. Finally, we pose some open questions.

Recently, we have seen an increasing need for fresh data exploration, where data analysts seek to explore the main characteristics or detect anomalies of data being actively collected. In addition to the common challenges in classic data exploration, such as a lack of prior knowledge about the data or the analysis goal, fresh data exploration also demands an ingestion system with sufficient throughput to keep up with rapid data accumulation. However, leveraging traditional Extract-Transform-Load (ETL) pipelines to achieve low query latency can still be extremely resource-intensive as they must conduct an excessive amount of data preprocessing routines (DPRs) (e.g., parsing and indexing) to cover unpredictable data characteristics and analysis goals. To overcome this challenge, we seek to approach it from a different angle: leveraging occasional idle system capacity or cheap preemptive resources (e.g., Amazon Spot Instance) during ingestion. In particular, we introduce a new type of data ingestion system called fluid ETL pipelines, which allow users to start/stop arbitrary DPRs on demand without blocking data ingestion. With fluid ETL pipelines, users can start potentially useful DPRs to accelerate future exploration queries whenever idle/cheap resources are available. Moreover, users can dynamically change which DPRs to run with limited resources to adapt to users' evolving interests. We conducted experiments on a real-world dataset and verified that our vision is viable. The introduction of fluid ETL pipelines also raises new challenges in handling essential tasks, such as ad-hoc query processing, DPR generation, and DPR management. In this paper, we discuss open research challenges in detail and outline potential directions for addressing them.

We prove that P = NP implies #P = FP by exploiting the topological structure of 3SAT solution spaces. The argument proceeds via a dichotomy: any polynomial-time algorithm for 3SAT either operates without global knowledge of the solution-space topology, in which case it cannot certify unsatisfiability for instances with second Betti number b_2 = 2^{Omega(N)} (leading to contradiction), or it computes global topological invariants, which are #P-hard. As local information is provably insufficient and any useful global invariant is #P-hard, the dichotomy is exhaustive. The proof is non-relativizing, consistent with oracles separating P = NP from #P = FP, and therefore necessarily exploits non-oracle properties of computation. Combined with Toda's theorem, the result yields P = NP => #P = FP => PH = P, providing new structural evidence for P != NP via a topological mechanism. We complement the theoretical framework with empirical validation of solution-space shattering at scale (N up to 500), demonstrating that these topological barriers manifest as measurable hardness across five independent algorithm classes.

This article addresses multi-servicer on-orbit servicing mission planning in geosynchronous Earth orbit, where routing decisions are tightly coupled with time-dependent orbital phasing and strict propellant and mission-duration constraints. We propose a Route-Phasing-Split Genetic Algorithm (RPS-GA) that simultaneously optimizes target sequencing, discrete phasing rotation decisions (i.e., the number of phasing revolutions/waiting cycles), and route partitioning across multiple servicing spacecrafts (SSCs). An RPS triplet chromosome encodes route order, phasing rotations, and route splits in a unified structure, enabling split-aware recombination without disrupting feasible multi-servicer route blocks. Feasibility is enforced through a constraint-aware fitness function that ranks feasible solutions based on total $ΔV$, while penalizing propellant and mission duration violations, using aggregate and imbalance penalties. This formulation discourages the concentration of violations on a single servicing spacecraft (SSC). Once a feasible best solution is identified, it is preserved as feasible in subsequent generations, thereby enhancing convergence stability. The framework incorporates split-aware crossover, mutation and a regret-based Large Neighborhood Search for local intensification. Experiments on representative GEO servicing scenarios demonstrate that RPS-GA produces feasible multi-servicer plans with substantially improved fuel efficiency, reducing total $ΔV$ by $24.5\%$, (from $1956.36 \ m/s$ to $ 1476.32\ m/s $) compared with a state-of-the-art LNS-AGA baseline.

We use the theoretical framework introduced in the companion paper to provide simple formulas as regards the resonance conditions for CBET with smoothed laser beams.Our analytical CBET model with optical smoothing shows that these fusion-critical lasers produce a significantly broader resonance than conventional plane wave models predict. In particular, temporal smoothing, as used in many high energy laser facilities, and flow components normal to the CBET ion acoustic waves, significantly modify the power transfer between smoothed beams. Our model predicts that the energy transfer rate out of resonance is substantially higher with optical smoothing than without, a result that has profound implications for optimizing predicting and interpreting future fusion experiments. We provide a simple criterion which pinpoints the laser and plasma parameters for which laser smoothing impacts CBET. These findings pave the way for experimental investigations in high-energy-density physics and fusion energy.

We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced separator of size $O_d(m^{1/d}n^{1-2/d})$. This bound is best possible in terms of the parameters involved. The same result holds if the balls and spheres are replaced by fat convex bodies and their boundaries.

By providing a suitable generalization of Newman's bijective correspondence known for cocommutative Hopf algebras, we prove that the category of cocommutative Hopf monoids in any abelian symmetric monoidal category is semi-abelian, once faithful (co)flatness conditions are satisfied. This result unifies and generalizes the semi-abelianness of cocommutative Hopf algebras and of cocommutative color Hopf algebras known up to now. As a consequence of the semi-abelianness, the category of cocommutative Hopf monoids is also action representable. Finally, we prove that abelian objects in the category of cocommutative Hopf monoids coincide exactly with commutative and cocommutative Hopf monoids, which form so an abelian category.

We present a systematic analysis of high-cadence optical light curves of the non-blazar AGN PKS~0521$-$36 obtained with \textit{TESS} across three sectors: Sectors~5 and~6 (Cycle~1, 30~min cadence) and Sector~32 (Cycle~3, 10~min cadence). The source exhibits moderate variability with $F_\mathrm{var} \approx 0.69$--$1.19\%$, consistent with a mildly beamed jet. The power spectral density (PSD) in all sectors is better described by a bending power-law than a simple power law, with high-frequency slopes $α_1 \approx 2.1$--2.9, indicating red-noise dominated variability. Flux distributions require two-component models, with a double log-normal providing the best description, suggesting the presence of two distinct optical flux states associated with quiescent jet emission and episodic flaring activity. A significant QPO at $P = 2.838 \pm 0.078$~d is detected in Sector~5 at $>99.99\%$ confidence in the Lomb--Scargle periodogram, independently confirmed by WWZ ($2.839 \pm 0.110$~d) and supported at the $3σ$ level by DRW analysis. The signal spans $\sim$9 cycles within the 26.1-day baseline in Sector~5 and is absent in Sectors~6 and~32, indicating a transient feature. The PSD bending frequency ($ν_b \approx 0.308$~d$^{-1}$; $\sim$3.2~d) is consistent with the QPO period, suggesting a common origin. We interpret the oscillation as magnetohydrodynamic kink instabilities in the relativistic jet, consistent with the observed helicoidal structure. A moderate Doppler factor ($δ\approx 5$--10) naturally explains the day-scale periodicity. Together with previously reported $γ$-ray QPOs on longer timescales, this suggests a hierarchical variability structure, and, to the best of our knowledge, provides the first evidence for an optical QPO in a non-blazar AGN with a directly imaged helical jet.

This paper re-evaluates Jozsef Sutak (1865-1954), a Hungarian scholar-priest and professor, as a grey eminence rather than a genius, offering a counter-narrative to the history of Hungarian university mathematics. By examining his career - including his 1897 Bolyai translation and his defense of set theory during the 1911 Grundlagenkrise - the study illuminates the overlooked substructure of the academic system. Key institutional moments, such as his 1912 appointment over Frigyes Riesz and Alfred Haar and his administrative role during the Numerus Clausus era, reveal a system prioritizing rigorous pedagogy and stability over avant-garde research. Sutak's legacy is the foundation and ethical commitment that enabled the next generation of Hungarian mathematical giants to emerge.

We express the Kodaira-Iitaka dimension and the multiplicity of graded linear series in terms of the intersection theory of the plurisubharmonic envelope associated with the linear series, and obtain two refined versions of these formulas at the pointwise and at the metric levels. At the pointwise level, we focus on the weak convergence of the partial Bergman kernel associated with the linear series and a Bernstein-Markov measure. At the metric level, we compute the asymptotic ratio of the volumes of unit balls defined by the sup-norms on the linear series. Based on our findings, we introduce a non-pluripolar version of the numerical Kodaira-Iitaka dimension for a line bundle, show that this invariant dominates the classical Kodaira-Iitaka dimension and is, in turn, bounded above by the numerical versions proposed so far.

In this work, we show that for all statistical estimation problems, a natural MMSE instability (discontinuity) condition implies the failure of stable algorithms, serving as a version of OGP for estimation tasks. Using this criterion, we establish separations between stable and polynomial-time algorithms for the following MMSE-unstable tasks (i) Planted Shortest Path, where Dijkstra's algorithm succeeds, (ii) random Parity Codes, where Gaussian elimination succeeds, and (iii) Gaussian Subset Sum, where lattice-based methods succeed. For all three, we further show that all low-degree polynomials are stable, yielding separations against low-degree methods and a new method to bound the low-degree MMSE. In particular, our technique highlights that MMSE instability is a common feature for Shortest Path and the noiseless Parity Codes and Gaussian subset sum. Last, we highlight that our work places rigorous algorithmic footing on the long-standing physics belief that first-order phase transitions--which in this setting translates to MMSE-instability impose fundamental limits on classes of efficient algorithms.

The use of Internet of Things (IoT) devices is growing at a rapid rate. While much of this growth is consumer devices, IoT devices are also commonly found in corporate and industrial environments, as well. These devices can be organization-owned and managed by an information technology unit, deployed organizationally without the knowledge and involvement of technology staff or brought in to the corporate environment by user-owners. In each case, these devices may have access to corporate networks and data and are, thus, important to consider as part of organizational cybersecurity risk assessment. Despite the prevalence of these devices, there is little literature about how to audit their security. This paper presents a risk-based auditing framework which can be used by both internal and external auditors, of any experience level and in any industry, to assess IoT devices.

Simulation methods have become important tools for quantifying partisan and racial bias in redistricting plans. We generalize the Sequential Monte Carlo (SMC) algorithm of McCartan and Imai (2023), one of the commonly used approaches. First, our generalized SMC (gSMC) algorithm can split off regions of arbitrary size, rather than a single district as in the original SMC framework, enabling the sampling of multi-member districts. Second, the gSMC algorithm can operate over various sampling spaces, providing additional computational flexibility. Third, we derive optimal-variance incremental weights and show how to compute them efficiently for each sampling space. Finally, we incorporate Markov chain Monte Carlo (MCMC) steps, creating a hybrid gSMC-MCMC algorithm that can be used for large-scale redistricting applications. We demonstrate the effectiveness of the proposed methodology through analyses of the Irish Parliament, which uses multi-member districts, and the Pennsylvania House of Representatives, which has more than 200 single-member districts.

Let $\mathbb{F}_\ell$ be a finite field with $\ell$ elements and let $G = C_p \rtimes C_m$ be a faithful split metacyclic group. In this paper, we develop a complete theory for the twisted group algebra $\mathbb{F}_\ell^αG$. Using the Lyndon--Hochschild--Serre spectral sequence, we prove that the second cohomology group of $G$ is isomorphic to $\mathbb{F}_\ell^\times/(\mathbb{F}_\ell^\times)^m$, and we show that all twisting occurs only on the $C_m$ factor. We determine the primitive central idempotents by analyzing the combined action of the Frobenius automorphism and the group action on the character group of $C_p$. Using crossed product theory and the structure of finite fields, we obtain the complete Wedderburn decomposition of $\mathbb{F}_\ell^αG$ into matrix algebras over explicitly determined fields $\mathbb{F}_{\ell^{d_j}}$. Finally, the irreducible projective representations of $G$ over $\mathbb{F}_\ell$ are also determined.