We address two interrelated problems concerning permutation of roots of univariate polynomials whose coefficients depend on parameters. First, we compute the Galois group of polynomials \varphi(x)\in\mathbb{C}[t_{1},\ldots,t_{k}][x] over \mathbb{C}(t_{1},\ldots,t_{k}) . Provided that the corresponding multivariate polynomial \varphi(x,t_{1},\ldots,t_{k}) is generic with respect to its support set A\subset \mathbb{Z}^{k+1} , we determine the latter Galois group for any A . Second, we determine the Galois group of systems of polynomial equations of the form p(x,t)=q(t)=0 where p and q have prescribed support sets A_{1}\subset \mathbb{Z}^{2} and A_{2}\subset \{0\}\times \mathbb{Z} respectively. For each problem, we determine the image of an appropriate braid monodromy map in order to compute the sought Galois group. As applications, we compute the Galois group of any rational function that is generic with respect to its support. We also provide general obstructions on the Galois group of enumerative problems on algebraic groups. Eventually, the techniques we develop allow us to compute the kernel of the braid monodromy map associated to \varphi .

Abstract Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We define what it means for a finite dimensional representation of $\mathfrak{g}$ to be rectangular and completely classify faithful rectangular representations. As an application, we obtain new $\lambda $-independence results on the algebraic monodromy groups of compatible systems of $\lambda $-adic Galois representations of number fields.

Abstract We show that unitary representations of simply connected, semisimple algebraic groups over local fields of characteristic zero obey a spectral gap absorption principle: that is, that spectral gap is preserved under tensor products. We do this by proving that the unitary dual of simple algebraic groups is filtered by the integrability parameter of matrix coefficients. This is a filtration of closed ideals that captures every closed subset of the dual that does not contain the trivial representation. In other words, we show that a representation has a spectral gap if and only if there exists some $$p \in [2,\infty )$$ p ∈ [ 2 , ∞ ) such that its matrix coefficients are in $$L^{p+\varepsilon }(G)$$ L p + ε ( G ) for every $$\varepsilon >0$$ ε > 0 . Doing this, we continue the work of Bader and Sauer in this area and prove a conjecture they phrased. We also use this principle to give an affirmative solution to a conjecture raised by Bekka and Valette: the image of the restriction map from a semisimple group to a lattice is never dense in Fell topology.

Abstract Paley-type partial difference sets and skew–Hadamard difference sets are classical objects in algebraic combinatorics, known for their rich connections with graph theory, coding theory, and group theory. In this paper, we explore new links between these combinatorial structures and group codes arising as ideals in finite group algebras. We construct such codes from difference sets and determine their dimensions in several cases. As an application of our links, we explicitly compute the full set of primitive central idempotents in certain abelian $$ p $$ p -group algebras, by employing the classical sets of quadratic residues and non-residues modulo $$ p $$ p , which are well-studied examples of difference and partial difference sets—we also obtain their dimensions and estimate their minimum weights.

Tangent Lie algebras of automorphism groups of free algebras

Let G be an affine algebraic group scheme over a field k . We show there exists a unipotent normal subgroup of G which contains all other such subgroups; we call it the restricted unipotent radical Rad u ( G ) of G . We investigate some properties of Rad u ( G ) , and study those G for which Rad u ( G ) is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine k -groups.

The Rouquier blocks, also known as the RoCK blocks, are important blocks of the symmetric groups algebras and the Hecke algebras of type A , with the partitions labelling the Specht modules that belong to these blocks having a particular abacus configuration. We generalise the definition of Rouquier blocks to the Ariki–Koike algebras, where the Specht modules are indexed by multipartitions, and explore the properties of these blocks.

A new structure, based on joining copies of a group by means of a twist, has recently been introduced to describe the brackets of the two exceptional real Lie algebras of type G 2 in a highly symmetric way. In this work, we show that these are not isolated examples by providing a broad family of Lie algebras that can be realized as generalized group algebras over the group Z 2 3 . On the one hand, certain orthogonal Lie algebras arise quite naturally as generalized group algebras over this group. On the other hand, previous classifications of graded contractions can be applied in this context, yielding many additional examples involving solvable and nilpotent Lie algebras of dimensions 32,28,24,21,16, and 14.

Units in the Group Algebra FS3

Unitary units & Cayley unitary elements in group algebras

Isotypic blocks of finite group algebras that are not 𝑝-permutation equivalent

We show that the coordinate ring of a simply-connected simple algebraic group G G over the complex number field coincides with the Berenstein–Fomin–Zelevinsky cluster algebra and its upper cluster algebra, at least when G G is not of type F 4 F_4 .

In this paper, we study the cyclicity of binary group codes, identifying them as ideals in a group algebra. We focus on the construction of codes, proving that they are self-dual group codes over the abelian group . We demonstrate that for even integers , if the polynomial splits into self-reciprocal irreducible factors, these codes are not permutationally equivalent to any cyclic code. Additionally, we present computational results for binary group codes of length using the MAGMA software (V2.29-4). These results confirm that while all cyclic codes in this range are equivalent to abelian group codes, there exist non-cyclic group codes that cannot be realized as ideals in a cyclic group algebra, highlighting the strictly larger scope of the class of group codes.

The workshop focused on various directions of arithmetic statistics in algebra and number theory. These include statistical problems for random polynomials and varieties, probabilistic Galois theory, and counting and distribution problems for algebraic functions, algebraic number fields, elliptic curves, L -functions, as well as arithmetic problems in non-abelian settings (eg, arithmetic statistics for algebraic groups).

UDC 512.552 We explicitly describe each unit of a group algebra $Z_{p} S_{3}$ for each positive prime $p \geq 5$ by using a characterization of the group algebra of the metacyclic group $G= \langle x,c\colon x^{3}=1,\ c^{n}=1,\ cxc^{-1 } = x^{-1} \rangle$ over the finite field $F$ of characteristic $p,$ where $p$ is a positive prime such that $p \nmid 3n.$ Based on our findings, we pose a conjecture on the number of roots of some explicit polynomials over the prime field $\mathbb{Z}_{p}$ for further academic explorations.

A homomorphism of linear algebraic groups ϕ:K→G is called an epimorphism if it admits right cancellation. A subgroup H≤G is epimorphic if the inclusion map is an epimorphism. For G a simple algebraic group over an algebraically closed field of arbitrary characteristic we construct epimorphic subgroups of bounded dimension (at most five).

In this paper, we describe explicitly the structure of the derivation algebra and automorphism group of the symplectic oscillator Lie algebra [Formula: see text] ([Formula: see text]), where [Formula: see text] is the symplectic Lie algebra and [Formula: see text] is the [Formula: see text]-dimensional Heisenberg algebra.

The mathematical β-tilting theory idea (the mathematical β-t theory) originated from the conceptual paintings of Adachi and his colleagues in 2014 [1], and it rapidly emerged as a primary focus of investigation within representation theory(R-theory) of finite-dimensional algebras (F-D algebras ). Integrating the idea of tilt, this framework presents an efficient combinatorial and isomorphic tool for reading rotation training, silt complexes, and cluster- tilting (CT) objects within modular classes .This study provides a systematic and self-contained introduction to β-t theory , especially designed for readers with a standard background in representation theory. It aims to enable researchers to grasp the fundamental concepts without the need for extensive reference to external sources, while maintaining full commitment to high mathematical rigor. In addition, the study unifies silting theory and cluster-tilting theory (CT theory) within a common perspective, gathering in one place the basic definitions, important bijections, and mutation techniques that form the core of this field [1]. In addition to the classical foundations, the research criticizes major developments published between 2023 and 2026, including new properties of β-tilting ( β-t) finiteness for Borel–Schur algebras and group algebras of generalized symmetric groups, and their applications to Frobenius–Perron dimensions and generalized preprojective algebras [2–8]. Particular attention is given to recent developments in higher torsion classes, βd-tilting theory ( βd-t-theory) and duplicated algebras. The study concludes by highlighting several key open problems such as the classification of minimal β-tilting infinite algebras ( β-t-i algebras ) and the explicit description of the vital bijections for important families of algebras which continue to motivate current research [9].This work aims to be an accessible entry point for beginners and a comprehensive reference for active researchers in representation theory ( R- theory ) and related fields [1,2].

Abstract We consider quantum group representations Rep ⁡ ( G q ) \operatorname{Rep}(G_{q}) for a semisimple algebraic group 𝐺 at a complex root of unity 𝑞. Here we allow 𝑞 to be of any order. We first show that the Tannakian center in Rep ⁡ ( G q ) \operatorname{Rep}(G_{q}) is calculated via a twisting of Lusztig’s quantum Frobenius functor Rep ⁡ ( G ̌ ) → Rep ⁡ ( G q ) \operatorname{Rep}(\check{G})\to\operatorname{Rep}(G_{q}) , where G ̌ \check{G} is a dual group to 𝐺. We then consider the associated fiber category Vect ⊗ Rep ⁡ ( G ̌ ) Rep ⁡ ( G q ) \mathrm{Vect}\otimes_{\operatorname{Rep}{(\check{G})}}\operatorname{Rep}(G_{q}) over B ⁢ G ̌ B\check{G} , and show that this fiber is a finite, integral braided tensor category. Furthermore, when 𝐺 is simply connected and 𝑞 is of even order, the fiber in question is shown to be a modular tensor category. Finally, we exhibit a finite-dimensional quasitriangular quasi-Hopf algebra (also known as small quantum group) whose representations recover the tensor category Vect ⊗ Rep ⁡ ( G ̌ ) Rep ⁡ ( G q ) \mathrm{Vect}\otimes_{\operatorname{Rep}{(\check{G})}}\operatorname{Rep}(G_{q}) , and we describe the representation theory of this algebra in detail. At particular pairings of 𝐺 and 𝑞, our quasi-Hopf algebra is identified with Lusztig’s original finite-dimensional Hopf algebra from the ’90s. This work completes the author’s project from [C. Negron, Log-modular quantum groups at even roots of unity and the quantum Frobenius I, Comm. Math. Phys. 382 (2021), 2, 773–814].

This study proposes a new formalized approach to the stabilization of linear transformations in artificial neural networks, based on discrete algebraic properties. In contrast to existing stability methods that rely on spectral norms, regularization techniques, or empirical heuristics, this work introduces the concept of algebraic stabilization—stability that arises from the structural properties of the matrices defining linear operators. The central object of investigation is the class of integer-valued matrices for which exponentiation to a form of the type Wk=I+μD is possible, where D∈Zn×n,μ∈Z>1. A well-known problem in group algebra is considered that guarantees the existence of such an exponent under the condition that μ is coprime with the determinant of W. Within this framework, modular arithmetic, reduction modulo μ, and the group structure of GLnZμ are employed, thereby linking the proposed method to the theory of finite groups and linear automata. The advantages of the approach are discussed, including formal control over the iterative behavior of transformations, compatibility with quantized and finitely representable networks, the possibility of embedding stabilizing conditions directly into the network architecture, and the potential to improve model interpretability and reliability. At the same time, limitations are identified, particularly those related to constructive implementation, the selection of suitable hyperparameters, and generalization to broader classes of transformations.