Math Papers: Number Theory & Representation Theory (Oct 7, 2025)

Hey math enthusiasts! 👋 Get ready for the freshest batch of research dropping in the realms of Number Theory (math.NT) and Representation Theory (math.RT). This week's roundup, dated October 7, 2025, brings some seriously interesting reads. A big shoutout to the Github page for keeping us all in the loop with even more papers and details. Make sure to check them out for a deeper dive!

Fresh Papers from math.NT, math.RT

[2510.03208] Tilting Objects via Recollements and $p$-Cycles on Weighted Projective Lines

Tilting objects are crucial in understanding the structure of triangulated categories, and this paper introduces a novel approach for constructing them using recollements. Guys, this is where things get interesting! The authors, Qiang Dong and Hongxia Zhang, leverage the $p$-cycle method within exceptional curve processes, which provides significant advantages in building both recollements and ladders, thanks to the presence of reduction/insertion functors. To construct tilting objects within the stable category of vector bundles over a weighted projective line, the paper dives into explicit expressions for line bundles and extension bundles, all stemming from the $p$-cycles constructions. Furthermore, they provide a crucial proof for tilting cuboic objects and tilting objects composed of Auslander bundles. To top it off, they present some brand-new tilting objects in the stable category of vector bundles over a weighted projective line. This research could potentially reshape how we approach constructions in triangulated categories, offering new tools and insights for algebraic geometry and representation theory. The method promises to simplify complex structures, making it more accessible for further investigation and application. I'm personally stoked to see how this unfolds in future research. This is good stuff, guys.

[2510.03179] Feit's conjecture, the canonical Brauer induction formula, and Adams operations

Feit's conjecture gets a fresh look in this paper, connecting it to the canonical Brauer induction formula and Adams operations. Robert Boltje and Gabriel Navarro delve into a stronger version of Feit's conjecture, initially formulated in collaboration with A. Kleshchev and P. H. Tiep back in 2025. This conjecture revolves around the conductor $c(\chi)$ of an irreducible character $\chi$ of a finite group $G$. The authors introduce an integer-valued invariant $S(G,\chi,n)$ for any positive integer $n$ that divides the exponent of $G$, and for any character $\chi$ of $G$. This invariant can be defined as the sum of specific coefficients within the canonical Brauer induction formula of $\chi$, or as the multiplicity of the trivial character in a particular integral linear combination of Adams operations of $\chi$. They reveal that $S(G,\chi,n)$ is always non-negative, and it's positive if and only if a representation affording $\chi$ involves an eigenvalue of order $n$. Moreover, the strong version of Feit's conjecture holds for an irreducible character $\chi$ if and only if $S(G,\chi, c(\chi))>0$. This connection opens new avenues for understanding character theory, potentially leading to breakthroughs in the classification of finite groups. Keep an eye on this, it could be huge!

[2507.06130] Unit lattices of $D_4$-quartic number fields with signature $(2,1)$

Unit lattices in number fields are drawing increased attention, and this paper homes in on $D_4$-quartic fields with signature $(2,1)$. The team—Sergio Ricardo Zapata Ceballos, Sara Chari, Erik Holmes, Fatemeh Jalalvand, Rahinatou Yuh Njah Nchiwo, Kelly O'Connor, Fabian Ramirez, and Sameera Vemulapalli—explore the distributions of shapes of unit lattices, driven by both their number theory applications and the current lack of concrete results. They treat the unit lattice as a point in $GL_2(\mathbb{Z})\backslash \mathfrak{h}$ and demonstrate that every lattice arising this way must correspond to a transcendental point on the boundary of a fundamental domain of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$. Furthermore, they identify three explicit algebraic points of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$ that serve as limit points for the set of unit lattices of $D_4$-quartic fields with signature $(2,1)$. This work provides a more detailed understanding of the structure of unit lattices and their distribution, offering valuable insights into algebraic number theory and related areas. If you're into algebraic number theory, this is a must-read.

[2509.23521] Quasi-flag manifolds and moment graphs

Quasi-flag manifolds take center stage in this paper, generalizing the classic flag manifolds of compact connected Lie groups. Yuri Berest, Yun Liu, and Ajay C. Ramadoss introduce a new class of topological $G$-spaces, termed $m$-quasi-flag manifolds $F_m = F_m(G,T)$, which are topological realizations of the algebras $Q_k(W)$ of $k$-quasi-invariant polynomials of the Weyl group $W$. The $G$-equivariant cohomology $H_G(F_m, {\mathbb C})$ is naturally isomorphic to $Q_k(W)$, where $m$ is a $W$-invariant integer-valued multiplicity function on the system of roots of $W$ and $k = \frac{m}{2}$ or $\frac{m+1}{2}$ depending on whether $m$ is even or odd. Many topological properties and algebraic structures related to flag manifolds extend to quasi-flag manifolds. The authors compute the cohomology of quasi-flag manifolds by constructing rational algebraic models in terms of coaffine stacks, providing an algebro-geometric framework for rational homotopy theory. They also compute the equivariant K-theory of quasi-flag manifolds, extending cohomological results to the multiplicative setting. This approach is heavily influenced by classical work on homotopy decompositions of classifying spaces of compact Lie groups, but the diagrams used arise from moment graphs, combinatorial objects from GKM theory. This paper bridges topology and algebra, offering new perspectives on the structure and properties of flag manifolds. Guys, this paper is massive.

[2510.03084] A sparse canonical van der Waerden theorem

Van der Waerden's theorem gets a sparse twist in this paper, exploring canonical Ramsey properties in random subsets. José D. Alvarado, Yoshiharu Kohayakawa, Patrick Morris, Guilherme O. Mota, and Miquel Ortega determine the threshold at which the binomial random subset $[n]_p$ almost surely inherits the canonical Ramsey type property. Specifically, they address the question: For sufficiently large $n$, does every coloring of $[n]$ contain either a monochromatic or a rainbow arithmetic progression of length $k$ ($k$-AP)? As an application, they demonstrate the existence of sets $A \subseteq [n]$ such that the $k$-APs in $A$ define a $k$-uniform hypergraph of arbitrarily high girth, yet any coloring of $A$ induces a monochromatic or rainbow $k$-AP. This result sharpens our understanding of Ramsey properties in sparse environments and has implications for hypergraph theory. If you're into combinatorics, this one's for you.

[2510.03068] Local newforms for generic representations of $p$-adic ${\rm SO}_{2n+1}$: Uniqueness

Local newforms for $p$-adic groups are explored in this paper, focusing on the uniqueness of these forms for generic representations of ${\rm SO}_{2n+1}$. Yao Cheng investigates the conjectural theory of local newforms for the split $p$-adic group ${\rm SO}_{2n+1}$, proposed by Gross. This theory predicts that the space of local newforms in a generic representation is one-dimensional. The paper proves that this space is at most one-dimensional and verifies its expected arithmetic properties, conditional on existence. These results are crucial in proving the existence part of the newform conjecture. This paper provides a significant contribution to the Langlands program, deepening our understanding of representations of $p$-adic groups. This is good stuff, guys. Very formal math.

[2501.12504] Shapes of unit lattices in $D_p$-number fields

Unit lattices in $D_p$-number fields are the focus, examining their shapes via the logarithmic Minkowski embedding. Robert Harron, Erik Holmes, and Sameera Vemulapalli analyze the shape of the unit lattice within the family of prime degree $p$ number fields whose Galois closure has dihedral Galois group $D_p$ and a unique real embedding. For $p = 5$, they prove that the unit shapes lie on a single hypercycle on the modular surface (in this case, the modular surface is the space of shapes of rank $2$ lattices). For general $p$, they show that the unit shapes are contained in a finite union of translates of periodic torus orbits in the space of shapes. This work enriches our knowledge of the geometric properties of unit lattices and their distribution in number fields. Geometric Properties are the best!

[2510.02959] Abstract Cluster Structures

Cluster combinatorics gets a categorical treatment in this paper, introducing a framework for encoding this combinatorics using categorical methods. Jan E. Grabowski and Sira Gratz define an abstract cluster structure that captures the essence of cluster mutation at a tropical level. They demonstrate that cluster algebras, cluster varieties, cluster categories, and surface models all have associated abstract cluster structures. For cluster algebras and cluster varieties, they also show that these can be constructed from abstract cluster structures. By defining a suitable notion of morphism of abstract cluster structures, they introduce a category of these structures, showing that it has several desirable properties, such as initial and terminal objects and finite products and coproducts. They also prove that rooted cluster morphisms of cluster algebras give rise to morphisms of the associated abstract cluster structures, thus including a version of the extant category of cluster algebras. This framework allows for relating different types of representation of abstract cluster structures directly via morphisms, even when no direct map from, e.g., a cluster category to the associated cluster algebra is possible. The authors also extend this in the setting of abstract quantum cluster structures, analyzing the difference between the category of these and that of the unquantized version. This paper provides a unifying perspective on cluster theory, bridging different areas and offering new tools for investigation.

[2510.02953] New perspectives in Kac-Moody algebras associated to higher dimensional manifolds

Kac-Moody (KM) algebras associated to higher-dimensional manifolds are explored, presenting a general framework for their construction. Rutwig Campoamor-Stursberg, Alessio Marrani, and Michel Rausch de Traubenberg provide a review starting from loop algebras on the circle $\mathbb{S}^{1}$ and extending to compact and non-compact group manifolds, coset spaces, and soft deformations thereof. After recalling the geometric background on Riemannian manifolds, Hilbert bases, and Killing vectors, they present the construction of generalized current algebras $\mathfrak{g}(\mathcal{M})$, their semidirect extensions with isometry algebras, and their central extensions. The resulting algebras are controlled by the structure of the underlying manifold, illustrated through explicit realizations on $SU(2)$, $SU(2)/U(1)$, and higher-dimensional spheres, highlighting their relation to Virasoro-like algebras. The authors also discuss the compatibility conditions for cocycles, the role of harmonic analysis, and some applications in higher-dimensional field theory and supergravity compactifications. This paper unifies the understanding of KM algebras beyond one-dimensional settings, paving the way for further exploration of their mathematical and physical implications.

[2411.12661] Classicality of derived Emerton--Gee stack II: generalised reductive groups

Emerton--Gee stacks are examined using the Tannakian formalism, defining these stacks for general groups. Yu Min proves that for a flat algebraic group G over Z_p, the associated Emerton--Gee stack is a formal algebraic stack locally of finite presentation over Spf(Z_p). The author also defines a derived stack of Laurent F-crystals with G-structure on the absolute prismatic site, whose underlying classical stack is proved to be equivalent to the Emerton--Gee stack. In the case of connected reductive groups, the derived stack of Laurent F-crystals with G-structure is classical in the sense that when restricted to truncated animated rings, it is the étale sheafification of the left Kan extension of the Emerton--Gee stack along the inclusion from classical commutative rings to animated rings. Moreover, when G is a generalised reductive group, the classicality result still holds for a modified version of the Emerton--Gee stack. This completes the picture that the derived stack of local Langlands parameters for the Langlands dual group of a reductive group is classical. This paper provides a deeper understanding of the structure and properties of Emerton--Gee stacks and their connection to local Langlands parameters.

[2510.02924] Projective representations on operator algebras

Projective representations on operator algebras are investigated, establishing new restrictions on the values of the lifting obstruction. Sergio Girón Pacheco establishes new restrictions on the values of the lifting obstruction for projective unitary representations of second countable, locally compact Hausdorff groups on operator algebras. Using these, the author shows that every projective representation on the Jiang--Su algebra lifts to a genuine group representation. The possible values of lifting obstructions for finite group projective representations on UHF-algebras of infinite type and Cuntz algebras are also characterized. Finally, the author shows that certain 2-cocycles of Property (T) groups cannot arise as lifting invariants of projective representations on the Hyperfinite II$_{1}$ factor. This paper advances our understanding of the structure and properties of projective representations on operator algebras, with implications for representation theory and operator algebra theory.

[2510.02908] Notes on Cohomological Finite Generation for Finite Group Schemes

Cohomological Finite Generation for Finite Group Schemes are explored, providing an overview of recent results. Juan Omar Goméz and Chris J. Parker present extended notes based on lectures given by Vincent Franjou, Paul Sobaje, Peter Symonds, and Antoine Touzé at the Master Class on New Developments in Finite Generation of Cohomology at Bielefeld University in September 2023. These notes aim to give a panoramic overview of van der Kallen's recent result on the finite generation of cohomology for finite group schemes over an arbitrary Noetherian base. The notes bring together the many papers that this theorem relies on, supplying the necessary background and exposition. This work provides a comprehensive resource for researchers interested in the cohomology of finite group schemes. Guys, this is going to be useful.

[2509.25163] Totally positive Toeplitz matrices: classical and modern

Totally positive Toeplitz matrices are examined, linking classical results with modern interpretations. Konstanze Rietsch considers infinite totally positive Toeplitz matrices as limits of finite ones, obtaining two further asymptotic descriptions of the Schoenberg parameters that are related to quantum cohomology of the flag variety as n goes to infinity. One is related to asymptotics of normalised quantum parameters, and the other to asymptotics of the Chern classes of the tautological line bundles. The author also describes the asymptotics of (quantum) Schubert classes in terms of the Schoenberg parameters. These limit formulas relate to and were motivated by a tropical analogue of this theory that is surveyed. In the tropical setting, one finds an asymptotic relationship between the `tropical Schoenberg parameters' and the weight map from Lusztig's parametrisation of the canonical basis. This paper bridges classical matrix theory with modern algebraic geometry and representation theory, offering new insights into the structure and properties of totally positive Toeplitz matrices.

[2509.07883] Oriented matroids and type $\mathbb{A}$ cluster categories

Oriented matroids and type $\mathbb{A}$ cluster categories are connected, constructing a rank-four oriented matroid for any cluster-tilting object. Nicholas J. Williams constructs a rank-four oriented matroid $\mathcal{M}_{\mathsf{T}}$ for any cluster-tilting object $\mathsf{T}$ in the cluster category $\mathscr{C}_{n}$ of type $\mathbb{A}_{n}$. Stackable triangulations of $\mathcal{M}_{\mathsf{T}}$ are in bijection with equivalence classes of maximal green sequences with initial cluster $\mathsf{T}$. This generalizes the result that equivalence classes of maximal green sequences of linearly oriented $\mathbb{A}_{n}$ are in bijection with triangulations of a three-dimensional cyclic polytope. The definition of the oriented matroid $\mathcal{M}_{\mathsf{T}}$ arises from the extriangulated structure on $\mathscr{C}_{n}$ which makes $\mathsf{T}$ projective. This work provides a new geometric perspective on cluster categories, linking them to the theory of oriented matroids.

[2509.10120] Resolving subcategories for gentle algebras II: Resolving subcategories for gentle trees

Resolving subcategories for gentle algebras are studied, specifically for gentle trees. Benjamin Dequêne and Michaël Schoonheere focus on gentle quivers $(Q,R)$, where $Q$ is a directed tree, known as gentle trees. They introduce an upper join-decomposition, replacing the canonical one, and describe the resolving subcategories of any gentle tree. These techniques allow for explicitly constructing the resolving subcategory generated by any collection of indecomposable $\mathbb{K}Q/\langle R\rangle$-modules. This paper contributes to the classification and understanding of resolving subcategories for gentle algebras, furthering our knowledge of representation theory.

[2502.20994] Resolving subcategories for gentle algebras I: Monogeneous resolving subcategories for gentle trees

Resolving subcategories for gentle algebras are investigated, focusing on monogeneous resolving subcategories for gentle trees. Benjamin Dequêne and Michaël Schoonheere improve the precision of an algorithm from Takahashi for resolving closure calculations in well-behaved abelian categories. They modify the geometric model of Baur--Coelho-Simões and Opper--Plamondon--Schroll to compute such subcategories for gentle quivers that have a finite global dimension. They study the monogeneous resolving subcategories, which are the ones generated by a single non-projective indecomposable $\mathbb{K}Q/\langle R \rangle$-module, proving that these subcategories are the join-irreducible elements of the poset of all the resolving subcategories ordered by inclusion. This work provides foundational results for the study of resolving subcategories for gentle algebras.

[2509.11137] Gaussian periods and Shanks' cubic polynomials. II

A linear relation between cubic Gaussian period and a root of Shanks' cubic polynomial in wildly ramified cases is presented. Miho Aoki gives a linear relation between a cubic Gaussian period and a root of Shanks' cubic polynomial in wildly ramified cases. This paper establishes a connection between two important objects in number theory, providing new tools for their study.

[2510.02756] Modularity theorems for abelian surfaces

Modularity theorems for abelian surfaces are discussed, providing a brief account of results on the (potential) modularity of abelian surfaces. Toby Gee presents a brief account of results with George Boxer, Frank Calegari, and Vincent Pilloni on the (potential) modularity of abelian surfaces. This paper offers an overview of recent progress in the area of modularity of abelian surfaces, highlighting key results and future directions.

[2508.05966] Bounds on the Minkowski constants and a function involving $\varphi$

Minkowski constants and a function involving $\varphi$ are examined, finding explicit upper and lower bounds on $M(n)$ that improve on Katznelson's results. Giulia Pelizzari and James Punch use elementary techniques to find explicit upper and lower bounds on $M(n)$ that improve on Katznelson's results and recover his asymptotic result. Their results imply explicit bounds on functions closely related to $M(n)$, which appear in the study of abelian varieties. They also examine the function $\Phi(n)$, defined as the greatest positive integer $m$ for which $\varphi(m)$ divides $2n$, providing explicit upper bounds on $\Phi(n)$. This paper provides valuable estimates for important constants and functions in number theory.

[2504.09545] Note on a problem of Sárközy on multiplicative representation functions

A problem of Sárközy on multiplicative representation functions is addressed, classifying all situations satisfying a given condition. Yuchen Ding classifies all situations of the integers $b,c,e$ and $f$ satisfying a specified condition for any infinite $\mathcal{A}\subset \mathbb{N}$, where $ d(\mathcal{A},m)=#{a\in \mathcal{A}:a|m}$. This paper resolves a problem in the area of multiplicative representation functions, providing a complete classification of the relevant cases.

For more in-depth information on these papers, be sure to visit the Github page. Happy reading, and may your theorems be ever true!