Related papers: Multiplicity upon restriction to the derived subgr…

Given a group $G$ we write $h(G)$ to denote the maximum number of times that a field extension of $\mathbb{Q}$ appears as the field of values of a conjugacy class of a group. In this work, we prove that $|G|$ is bounded in terms of $h(G)$.…

We examine various consequences of the existence of exceptional representations of an irreducible Weyl group. (These are notes from a talk in the MIT Lie groups seminar.)

Necessary and sufficient conditions for a group to possess a faithful irreducible representation are investigated

We prove one direction of a recently posed conjecture by Gan-Gross-Prasad, which predicts the branching laws that govern restriction from p-adic $GL_n$ to $GL_{n-1}$ of irreducible smooth representations within the Arthur-type class. We…

We investigate some situation in which automorphisms of a group G are uniquely determined by their restrictions to a proper subgroup H. Much of the paper is devoted to studying under which additional hypotheses this property forces G to be…

Let $X$ be a nonempty real variety that is invariant under the action of a reflection group $G$. We conjecture that if $X$ is defined in terms of the first $k$ basic invariants of $G$ (ordered by degree), then $X$ meets a $k$-dimensional…

We establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field, for codimension two algebras and for Gorenstein algebras of codimension three. In fact, we…

Zaremba's Conjecture concerns the formation of continued fractions with partial quotients restricted to a given alphabet. In order to answer the numerous questions that arrive from this conjecture, it is best to consider a semi-group, often…

The number of subgroups and the number of cyclic subgroups are natural combinatorial invariants of a finite group. We investigate how restrictions on these quantities, together with the number of distinct prime divisors of $|G|$, enforce…

We study the algebras generated by restriction and induction operations on complex modules over dihedral groups. In the case where the orders of all dihedral groups involved are not divisible by four, we describe the relations, a basis, the…

In this article, we study word equations in free semigroups and the conjecture that the existence of infinitely many solutions entails the existence of solutions with arbitrarily large exponent of periodicity. We examine this question in…

The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded $k$-algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several…

New cases of the multiplicity conjecture are considered.

The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A…

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…

This is a sequel to arXiv:1308.3604. We study applications to limit multiplicity generalizing the results of arXiv:1208.2257.

This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If $RG$ is a group ring, where $R$ is commutative and $S$ is a set of generators of $G$ then necessary and sufficient…

We determine the decomposition of the restriction of a length-one toral supercuspidal representation of a connected reductive group to the algebraic derived subgroup, in terms of parametrizing data, and show this restriction has…

We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive…

Let $G$ be a split reductive group over a local field $\bK$, and let $G((t))$ be the corresponding loop group. In \cite{GK} we have introduced the notion of a representation of (the group of $\bK$-points) of $G((t))$ on a pro-vector space.…