Related papers: Block theory and Brauer's first main theorem for p…

Let $\pi_1$ and $\pi_2$ be absolutely irreducible smooth representations of $G=GL_2(Q_p)$ with a central character, defined over a finite field of characteristic $p$. We show that if there exists a non-split extension between $\pi_1$ and…

This paper contains a proof of a conjecture of Breuil and Schneider, on the existence of an invariant norm on any locally algebraic representation of $\GL(n)$, with integral central character, whose smooth part is given by a generalized…

We study the periplectic Brauer algebra introduced by Moon in the study of invariant theory for periplectic Lie superalgebras. We determine when the algebra is quasi-hereditary and, for fields of characteristic zero, describe the block…

This paper proposes a conjectural framework for Alperin's Main Problem of Block Theory from 1976. The character sets considered here are defined by nonvanishing at given elements, not only by degree conditions. From this point of view,…

We classify the Morita equivalence classes of blocks with elementary abelian defect groups of order $16$ with respect to a complete discrete valuation ring with algebraically closed residue field of characteristic two. As a consequence,…

We give formulas for calculating the unramified Brauer group of a homogeneous space $X$ of a semisimple simply connected group $G$ with finite geometric stabilizer $\bar F$ over a wide family of fields of characteristic 0. When $k$ is a…

This paper introduces the notion of Brauer-friendly modules, a generalisation of endo-p-permutation modules. A module over a block algebra OGe is said to be Brauer-friendly if it is a direct sum of indecomposable modules with compatible…

Let $k$ be a number field, $\mathbf{G}$ an algebraic group defined over $k$, and $\mathbf{G}(k)$ the group of $k$-rational points in $\mathbf{G}.$ We determine the set of functions on $\mathbf{G}(k)$ which are of positive type and…

We introduce two families of diagrammatic monoidal supercategories. The first family, depending on an associative superalgebra, generalizes the oriented Brauer category. The second, depending on an involutive superalgebra, generalizes the…

The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of $\ell$-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that…

The aim of this short research note is to present some results about a conjecture of Barker and Gelvin claiming that any source algebra of a block of a finite group has the unit group containing a basis stabilised by the left and right…

For a positive integer r we prove that if G is a profinite group in which the centralizer of every nontrivial element has rank at most r, then G is either a pro-p group or a group of finite rank. Further, if G is not virtually a pro-p…

Let $k$ be an algebraically closed field of prime characteristic $p$ and $P$ a finite $p$-group. We compute the Scott $kG$-module with vertex $P$ when $\mathcal{F}$ is a constrained fusion system on $P$ and $G$ is Park's group for…

In 1941, Brauer-Nesbitt established a characterization of a block with trivial defect group as a block $B$ with $k(B) = 1$ where $k(B)$ is the number of irreducible ordinary characters of $B$. In 1982, Brandt established a characterization…

Let $G$ be a finitely generated abelian-by-finite group and $k$ a field of characteristic $p\ge 0$. The Euler class $[k_G]$ of $G$ over $k$ is the class of the trivial $kG$-module in the Grothendieck group $G_0(kG)$. We show that $[k_G]$…

Let $G$ be a finite solvable or symmetric group and let $B$ be a $2$-block of $G$. We construct a canonical correspondence between the irreducible characters of height zero in $B$ and those in its Brauer first main correspondent. For…

Let $H$ be a Krull monoid with infinite cyclic class group $G$ and let $G_P \subset G$ denote the set of classes containing prime divisors. We study under which conditions on $G_P$ some of the main finiteness properties of factorization…

We show that the decomposition matrix of a given group $G$ is unitriangular, whenever $G$ has a normal subgroup $N$ such that the decomposition matrix of $N$ is unitriangular, $G/N$ is abelian and certain characters of $N$ extend to their…

Let $G$ be an inner form of a general linear group over a non-archimedean locally compact field of residue characteristic $p$, let $R$ be an algebraically closed field of characteristic different from $p$ and let $\mathscr{R}_R(G)$ be the…

We study the cohomology with modular coefficients of Deligne-Lusztig varieties associated to Coxeter elements. Under some torsion-free assumption on the cohomology we derive several results on the principal l-block of a finite reductive…