Many previously studied path algebras or self-similar group algebras may be viewed as Steinberg algebras of self-similar groupoids. By way of inverse semigroup algebras, we characterize when the Steinberg algebra of a self-similar groupoid is simple. We show that the simplicity of the reduced $C^*$-algebra of a contracting self-similar groupoid coincides with the simplicity of the Steinberg algebra. As an aside, we show that simplicity of the two algebras sometimes depends only on the skeleton of the self-similar groupoid acting on a strongly connected graph. Finally, we apply our methods to examples including a self-similar groupoid akin to multispinal self-similar groups and a self-similar groupoid built from the well-known Basilica group.

Given a connected linear algebraic group G over an algebraically closed field of characteristic zero, we describe the subgroup of G generated by all semisimple elements.

Twisted group algebra of dihedral groups over finite fields

Abstract In this article, we study rational representations of groups of order p 5 p^{5} , where 𝑝 is an odd prime. For a 𝑝-group 𝐺 and an irreducible complex character 𝜒 of 𝐺, the construction of an irreducible rational matrix representation of 𝐺 affording the character Ω ⁢ ( χ ) \Omega(\chi) is equivalent to determining a pair ( H , ψ ) (H,\psi) , with 𝐻 a subgroup of 𝐺 and 𝜓 a linear character of 𝐻 such that ψ G = χ \psi^{G}=\chi and Q ⁢ ( ψ ) = Q ⁢ ( χ ) \mathbb{Q}(\psi)=\mathbb{Q}(\chi) , where Ω ⁢ ( χ ) = m Q ⁢ ( χ ) ⁢ ∑ σ ∈ Gal ⁢ ( Q ⁢ ( χ ) / Q ) χ σ \Omega(\chi)=m_{\mathbb{Q}}(\chi)\sum_{\sigma\in\mathrm{Gal}(\mathbb{Q}(\chi)/\mathbb{Q})}\chi^{\sigma} and m Q ⁢ ( χ ) m_{\mathbb{Q}}(\chi) denotes the Schur index of 𝜒 over ℚ. For each inequivalent irreducible rational representation of every group of order p 5 p^{5} , we determine such a pair, referred to as a required pair. We also derive combinatorial formulations for the Wedderburn decomposition of rational group algebras of these 𝑝-groups, using results from their rational representations.

Let G be an algebraic group over a field k, and M and N be G-modules. In 1961, Hochschild showed how one can define the cohomology groups Ext G i ( M , N ) . Kimura, in 1965, showed that one can generalize this to get relative cohomology for algebraic groups. The original cohomology groups play an important role in understanding the representation theory of G, but the role of relative cohomology is still not well understood. In this paper the author expands upon the work of Kimura to prove foundational results about the relative cohomology. The author starts by giving the definitions of relative exact sequences and relative injective modules and proves a variety of basic properties for each that will be essential to define relative cohomology and obtain a relative Grothendieck spectral sequence. In particular, the induction functor will play an important role when studying the relative injective modules. Once the necessary ground work is laid, the definition of relative cohomology is given. Finally, it is stated when there is a relative Grothendieck spectral sequence, and many consequences and examples are provided.

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.

Units in the Group Algebra FS3

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.

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.