CyberSec Research

Let $G$ be a reductive algebraic group defined over a non-Archimedean local field $F$ of residue characteristic $p$. Let $\sigma$ be an automorphism of $G$ of order $\ell$ -- a prime number -- with $\ell\neq p$. Let $\Pi$ be a finite length $\overline{\mathbb{F}}_\ell$-representation of $G(F)\rtimes \langle\sigma\rangle$. We show that the Tate cohomology $\widehat{H}^i(\langle\sigma\rangle, \Pi)$ is a finite length representation of $G^\sigma(F)$. We give an application to genericity of these Tate cohomology spaces.

The study of modular representation theory of the double covering groups of the symmetric and alternating groups reveals rich and subtle combinatorial and algebraic phenomena involving their irreducible characters and the structure of their p-blocks, where p is an odd prime number. In this paper, we investigate the action of certain Galois automorphisms, those that act on p'-roots of unity by a power of p, on spin characters, with an emphasis on their interaction with perfect isometries and block theory. In particular, we prove that perfect isometries constructed by the first author and J.\,B. Gramain in \cite{BrGr3}, which were used to establish a weaker form of the Kessar--Schaps conjecture, remain preserved under this Galois action whenever certain natural compatibility conditions occur.

Let $F$ be a locally compact non-Archimedean field of characteristic $0$, and let $G$ be either the split special orthogonal group $\mathrm{SO}_{2n+1}(F)$ or the symplectic group $\mathrm{Sp}_{2n}(F)$. The goal of this paper is to give an explicit description of the Aubert-Zelevinsky duality for $G$ in terms of Langlands parameters. We present a new algorithm, inspired by the Moeglin-Waldspurger algorithm for $\mathrm{GL}_n(F)$, which computes the dual Langlands data in a recursive and combinatorial way. Our method is simple enough to be carried out by hand and provides a practical tool for explicit computations. Interestingly, the algorithm was discovered with the help of machine learning tools, guiding us toward patterns that led to its formulation.

A recent result of Cohen and Zemel provides an elegant expansion of the Rasala polynomials for symmetric group character degrees. In this note we present an alternative short algebraic proof. Extensions to polynomials of character values follow.

The purpose of this note is to demonstrate the advantages of Y.-Z.\ Huang's definition of the Zhu algebra (Comm.\ Contemp.\ Math., 7 (2005), no.\ 5, 649--706) for an arbitrary vertex algebra, not necessarily equipped with Hamiltonian operators or Virasoro elements, by achieving the following two goals: (1) determining the Zhu algebras of $N=1,2,3,4$ and big $N=4$ superconformal vertex algebras, and (2) introducing the Zhu algebras of $N_K=N$ supersymmetric vertex algebras.

Rooted trees are essential for describing numerical schemes via the so-called B-series. They have also been used extensively in rough analysis for expanding solutions of singular Stochastic Partial Differential Equations (SPDEs). When one considers scalar-valued equations, the most efficient combinatorial set is multi-indices. In this paper, we investigate the existence of intermediate combinatorial sets that will lie between multi-indices and rooted trees. We provide a negative result stating that there is no combinatorial set encoding elementary differentials in dimension $d\neq 1$, and compatible with the rooted trees and the multi-indices aside from the rooted trees. This does not close the debate of the existence of such combinatorial sets, but it shows that it cannot be obtained via a naive and natural approach.

For a non-Archimedean local field $F$ of residue cardinality $q=p^r$, we give an explicit classical generator $V$ for the bounded derived category $D_{fg}^b(\mathsf{H}_1(G))$ of finitely generated unipotent representations of $G=\mathrm{GL}_n(F)$ over an algebraically closed field of characteristic $l\neq p$. The generator $V$ has an explicit description that is much simpler than any known progenerator in the underived setting. This generalises a previous result of the author in the case where $n=2$ and $l$ is odd dividing $q+1$, and provides a triangulated equivalence between $D_{fg}^b(\mathsf{H}_1(G))$ and the category of perfect complexes over the dg algebra of dg endomorphisms of a projective resolution of $V$. This dg algebra can be thought of as a dg-enhanced Schur algebra. As an intermediate step, we also prove the analogous result for the case where $F$ is a finite field.

In 2018, Chen, Han and Zhou introduced a criterion to determine whether the HRS-tilt at a given torsion pair induces derived equivalence. We showcase four applications of this criterion: to stable torsion pairs in arbitrary abelian categories (which we prove to always induce derived equivalence), to abelian categories of global dimension at most two, to (co)hereditary torsion pairs over artin algebras (for which we give a purely combinatorial criterion for derived equivalence), and to study whether irreducible silting mutation acts transitively on two-term tilting complexes over a finite dimensional algebra.

Higher Lie characters form a distinguished family of symmetric group characters, which appear in many areas of algebra and combinatorics. An old open problem of Thrall is to decompose them into irreducibles. We propose a novel asymptotic approach to this problem, showing that many families of higher Lie characters tend to be proportional to the regular character. In particular, a random higher Lie character tends, in probability, to be proportional to the regular character.

Let $A$ be an artin algebra. The aim of this work is to describe the enlargements of an indecomposable complex in $\mathbf{C}_{n}(\mbox{proj} \,A)$, and to study the irreducible morphisms between them. Precisely, we prove that any indecomposable complex in $\mathbf{C}_{[0,n]}(\mbox{proj} \,A)$ or in $\mathbf{C}_{n+1}(\mbox{proj} \,A)$ for $n$ a positive integer is a shift or an enlargement of an indecomposable complex in $\mathbf{C}_{n}(\mbox{proj} \,A)$. We also describe the entrances of the irreducible morphisms in $\mathbf{C}_{[0,n]}(\mbox{proj} \,A)$ between enlargements of an indecomposable complex $X$ in $\mathbf{C}_{n}(\mbox{proj} \,A)$.

We produce graded monoidal categorifications of the quantum boson algebras in any symmetrizable Kac-Moody type. Our categories are defined in terms of diagrammatic generators and relations and have a faithful 2-representation on Khovanov-Lauda and Rouquier's categorification of the corresponding positive part quantum group. We use our construction to produce interesting bases of the quantum boson algebras and quantum bosonic extensions.

The objective of this article is to prove the necessity statement in Crawley-Boevey's conjectural solution to the (tame) Deligne-Simpson problem. We use the nonabelian Hodge correspondence, variation of parabolic weights and results of Schedler-Tirelli to reduce to simpler situations, where every conjugacy class is semi-simple and the underlying quiver is (1) an affine Dynkin diagram or (2) an affine Dynkin diagram with an extra vertex. In case (1), a nonexistence result of Kostov applies. In case (2), the key step is to show that simple representations, if exist, lie in the same connected component as direct sums of lower dimensional ones.

These notes give a short introduction to finite Coxeter groups, their classification, and some parts of their representation theory, with a focus on the infinite families. They are based on lectures delivered by the author at the conferences Recent Trends in Group Theory at IIT Bhubaneswar and the Asian - European School in Mathematics at NEHU, Shillong. The first draft was prepared by Archita Gupta (IIT Kanpur) and Sahanawaz Sabnam (NISER Bhubaneswar), to whom the author is deeply grateful. We hope these notes will be useful both for beginners and for readers who wish to study the subject further. The author also thanks the organizers and participants of the above conferences for their support and encouragement.

The periodic Temperley-Lieb category consists of connectivity diagrams drawn on a ring with $N$ and $N'$ nodes on the outer and inner boundary, respectively. We consider families of modules, namely sequences of modules $\mathsf{M}(N)$ over the enlarged periodic Temperley-Lieb algebra for varying values of $N$, endowed with an action $\mathsf{M}(N') \to \mathsf{M}(N)$ of the diagrams. Examples of modules that can be organised into families are those arising in the RSOS model and in the XXZ spin-$\frac12$ chain, as well as several others constructed from link states. We construct a fusion product which outputs a family of modules from any pair of families. Its definition is inspired from connectivity diagrams drawn on a disc with two holes. It is thus defined in a way to describe intermediate states in lattice correlation functions. We prove that this fusion product is a bifunctor, and that it is distributive, commutative, and associative.

Let $\mathcal{A}_{q}$ be an arbitrary quantum cluster algebra with principal coefficients. We give the fundamental relations between the quantum cluster variables arising from one-step mutations from the initial cluster in $\mathcal{A}_{q}$. Immediately and directly, we obtain an algebra homomorphism from the corresponding (untwisted) quantum group to $\mathcal{A}_{q}$.

In this paper, we investigate the structural and characterizing properties of the so-called {\it 2-UQ rings}, that are rings such that the square of every unit is the sum of an idempotent and a quasi-nilpotent element that commute with each other. We establish some fundamental connections between 2-UQ rings and relevant widely classes of rings including 2-UJ, 2-UU and tripotent rings. Our novel results include: (1) complete characterizations of 2-UQ group rings, showing that they force underlying groups to be either 2-groups or 3-groups when $3 \in J(R)$; (2) Morita context extensions preserving the 2-UQ property when trace ideals are nilpotent; and (3) the discovery that potent 2-UQ rings are precisely the semi-tripotent rings. Furthermore, we determine how the 2-UQ property interacts with the regularity, cleanness and potent conditions. Likewise, certain examples and counter-examples illuminate the boundaries between 2-UQ rings and their special relatives. These achievements of ours somewhat substantially expand those obtained by Cui-Yin in Commun. Algebra (2020) and by Danchev {\it et al.} in J. Algebra \& Appl. (2025).

It is natural to consider extending the typical construction of relative Poisson algebras from commutative differential algebras to the context of bialgebras. The known bialgebra structures for relative Poisson algebras, namely relative Poisson bialgebras, are equivalent to Manin triples of relative Poisson algebras with respect to the symmetric bilinear forms which are invariant on both the commutative associative and Lie algebras. However, they are not consistent with commutative and cocommutative differential antisymmetric infinitesimal (ASI) bialgebras as the bialgebra structures for commutative differential algebras. Alternatively, with the invariance replaced by the commutative $2$-cocycles on the Lie algebras, the corresponding Manin triples of relative Poisson algebras are proposed, which are shown to be equivalent to certain bialgebra structures, namely relative PCA bialgebras. They serve as another approach to the bialgebra theory for relative Poisson algebras, which can be naturally constructed from commutative and cocommutative differential ASI bialgebras. The notion of the relative PCA Yang-Baxter equation (RPCA-YBE) in a relative PCA algebra is introduced, whose antisymmetric solutions give coboundary relative PCA bialgebras. The notions of $\mathcal{O}$-operators of relative PCA algebras and relative pre-PCA algebras are also introduced to give antisymmetric solutions of the RPCA-YBE.

We study Chabauty limits of the fixed-point group of $k$-points $H_k$ associated with an involutive $k$-automorphism $\theta$ of a connected linear reductive group $G$ defined over a non-Archimedean local field $k$ of characteristic zero. Leveraging the geometry of the Bruhat--Tits building, the structure of $(\theta,k)$-split tori, and the $K\mathcal{B}_kH_k$ decomposition of $G_k$, we establish that any nontrivial Chabauty limit $L$ of $H_k$ is $G_k$-conjugate to a subgroup of $$U_{\sigma_+}(k) \rtimes (Ker(\alpha)^0 \cdot (H_k \cap M_{\sigma_{\pm}})) \leq P_{\sigma_+}(k),$$ where $\alpha$ is a projection map arising from a Levi factor $M_{\sigma_{\pm}}$ of a parabolic subgroup $P_{\sigma_+} \subset G$, and $Ker(\alpha)^0$ denotes the subgroup of elliptic elements in the kernel of $\alpha$. Our analysis distinguishes between elliptic and hyperbolic elements and constructs explicit unipotent elements in the limit group $L$ using the Moufang property of $G_k$. Furthermore, we show that $L$ acts transitively on the set of ideal simplices opposite to $\sigma_+$. These results yield a detailed description of the Chabauty compactification of $H_k$, and provide new insights into its interaction with the non-Archimedean geometry of $G_k$.