Related papers: Stability of Branching Laws for Highest Weight Mod…

We consider two families X_n of varieties on which the symmetric group S_n acts: the configuration space of n points in C and the space of n linearly independent lines in C^n. Given an irreducible S_n-representation V, one can ask how the…

In the representation theory of real reductive Lie groups, many objects have finiteness properties. For example, the lengths of Verma modules and principal series representations are finite, and more precisely, they are bounded. In this…

Let $G$ denote the unramified quasi-split unitary group $\mathbb{U}(1,1)(F)$ over a $p$-adic field $F$ with residual characteristic $p \neq 2$. In this paper, we first construct a large family of irreducible representations of the maximal…

Let M_g^n be the moduli space of Riemann surfaces of genus g with n labeled marked points. We prove that, for g \geq 2, the cohomology groups {H^i(M_g^n;Q)}_{n=1}^{\infty} form a sequence of Sn representations which is representation stable…

Representation stability in the sense of Church-Farb is concerned with stable properties of representations of sequences of algebraic structures, in particular of groups. We study this notion on objects arising in toric topology. With a…

In a previous paper, arXiv:1206.5498, we introduced a new homological invariant $\e$ for the faithful action of a finite group G on an algebraic curve. We show here that the moduli space of curves admitting a faithful action of a finite…

Conjecture F from [VW12] states that the complements of closures of certain strata of the symmetric power of a smooth irreducible complex variety exhibit rational homological stability. We prove a generalization of this conjecture to the…

The virtual Betti number conjecture states that any hyperbolic three-manifold has a finite cover with positive first Betti number. We show that this would follow if it were known that the derived series of the fundamental group $G$ of a…

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X…

We consider in parallel pointed homotopy automorphisms of iterated wedge sums of topological spaces and boundary relative homotopy automorphisms of iterated connected sums of manifolds minus a disk. Under certain conditions on the spaces…

We study representation stability in the sense of Church, Ellenberg, and Farb \cite{FI-module} through the lens of symmetric function theory and the different symmetric function bases. We show that a sequence, $(F_n)_n$, where $F_n$ is a…

Let $G$ be a finite group. Let $U_1,U_2,\dots$ be a sequence of orthogonal representations in which any irreducible representation of $\oplus_{n \geq 1} U_n$ has infinite multiplicity. Let $V_n=\oplus_{i=1}^n U_n$ and $S(V_n)$ denote the…

The goal of this article is to study the branching problem for a holomorphic discrete series representation of the conformal group of a simple Euclidean Jordan algebra $V$ restricted to the subgroup…

For a fixed algebraic variety $X$, curve class $\alpha \in N_1(X)$, and genus $g \in \mathbb N$, we consider the sequence of $S_n$ representations obtained from the homology of the Kontsevich space of stable maps to $X$, $\bar…

Let $Q$ be an algebraic group and $V$ a $Q$-module. The index of $V$ is the minimal codimension of the $Q$-orbits in the dual space $V^*$. There is a general inequality, due to Vinberg, relating the index of $V$ and the index of…

We prove that the rational cohomology of the space of non-singular complex homogeneous polynomials of degree d in a fixed number of variables stabilizes to the cohomology of the general linear group for d sufficiently large.

We prove that the irreducible decomposition of the permutation representation of GL(n,q) on GL(n,q)/GL(n-m,q) stabilizes for large n. We deduce, as a consequence, a representation stability theorem for finitely generated VIC-modules.

Consider a family of modular forms of weight 2, all of whose residual $\pmod{p}$ Galois representations are isomorphic. It is well-known that their corresponding Iwasawa $\lambda$-invariants may vary. In this paper, we study this variation…

In this mostly expository article, we consider certain homological aspects of branching laws for representations of a group restricted to its subgroups in the context of $p$-adic groups. We follow our earlier paper, ICM 2018 proceedings,…

Generalizing results of Deroin-Dujardin, we introduce the notion of proximal stability for a holomorphic family $ (\rho_{\lambda}) $ of representations $ \Gamma \to \text{SL}(d, \mathbb{C}) $, where $ \Gamma $ is a finitely generated group,…