Related papers: Lower defect groups and vertices of simple modules

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on its set of conjugacy class sizes: this is the (simple undirected) graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, and two…

In the theory of finite groups, the irreducible representations of G over a field F are classified into blocks based on a direct decompositions of the group algebra FG. This gives a natural decomposition of FG-modules into direct summands,…

A finite abelian $p$-group having an automorphism $x$ such that $1+\ldots+x^{p-1}=0$, can be viewed as a module over an appropriate discrete valuation ring $\mathcal{O}$ containing $\mathbb{Z}_p$ (the ring of $p$-adic integer). This yields…

We perform a general study of the structure of locally compact modules over compactly generated abelian groups. We obtain a devissage result for such modules of the form "compact-by-sheer-by-discrete", and then study more specifically the…

In this paper, we study moduli spaces of finite-dimensional Lie algebras with flat center, proving that the forgetful map from Lie p-algebras to Lie algebras is an affine fibration, and we point out a new case of existence of a p-mapping.…

Slattery has generalized Brauer's theory of p-blocks of finite groups to pi-blocks of pi-separable groups where pi is a set of primes. In this setting we show that the order of a defect group of a pi-block B is bounded in terms of the…

It is well known that the internal suplattices in the topos of sheaves on a locale are precisely the modules on that locale. Using enriched category theory and a lemma on KZ doctrines we prove (the generalization of) this fact in the case…

Vertex $F$-algebras are a deformation of the concept of an ordinary vertex algebra in which the additive formal group law is replaced by an arbitrary formal group law $F$. The main theorem of this paper constructs a Lie algebra from a…

Let ${\mathscr M}(p)$ $(p=2,3,\ldots)$ be the singlet vertex operator algebra and $\omega$ its conformal vector. We classify the simple weak ${\mathscr M}(p)$-modules with a non-zero element $u$ such that for some integer $s\geq 2$,…

In this short note, we prove a formula for the group inverse of a block matrix and consider the pseudo principal pivot transform expressed in terms of group inverses. Extensions of the usual principal pivot transform, where the usual…

We show that if a graded submodule of a Noetherian module cannot be written as a proper intersection of graded submodules, then it cannot be written as a proper intersection of submodules at all. More generally, we show that a natural…

In representation theory of finite groups an important role is played by irreducible characters of p-defect 0, for a prime p dividing the group order. These are exactly those vanishing at the p-singular elements. In this paper we generalize…

We explain some simple methods to establish the property of Rapid Decay for a number of groups arising geometrically. We also give new examples of groups with the property of Rapid Decay. In particular we establish the property of Rapid…

We revisit the modular flavor symmetry from a more general perspective. The scalar modular forms of principal congruence subgroups are extended to the vector-valued modular forms, then we have more possible finite modular groups including…

It is proved that if any Z-graded weak module for vertex operator algebra V is completely reducible, then V is rational and C_2-cofinite. That is, V is regular. This gives a natural characterization of regular vertex operator algebras.

Building on previous work by Caicedo and the second author, we develop a method that decides the existence of units of finite order in blocks of $\mathbb{Z}_p G$ of defect 1. This allows us to prove that if $p$ is a prime and $G$ is a…

In this paper, we first classify all irreducible modules of the vertex algebra $V_L^+$ when $L$ is a negative definite even lattice of arbitrary rank. In particular, we show that any irreducible $V_L^+$-module is isomorphic to a submodule…

Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen $\epsilon>0$, describe the structure of $A$ and…

We investigate PBW deformations H of k[x,y]#G where G is the cyclic group of order p and k also has characteristic p; in these deformations, [x,y] takes a value in kG. These algebras are versions of symplectic reflection algebras that only…

The problem of classifying all short multiplets of superconformal algebras still seems to be an open question. A generic short multiplet is non-unitary, which nevertheless is of interest in various contexts. Even if one is interested in…