Related papers: Random Balanced Cayley Complexes

Let $G$ be a finite additive abelian group with exponent $n$ and $S=g_{1}\cdots g_{t}$ be a sequence of elements in $G$. For any element $g$ of $G$ and $A\subseteq\{1,2,\ldots,n-1\}$, let $N_{A,g}(S)$ denote the number of subsequences…

Let $k$ be an algebraically closed field of characteristic $p \geq 0$, let $G$ be a simple simply connected classical linear algebraic group of rank $\ell$ and let $T$ be a maximal torus in $G$ with rational character group $X(T)$. For a…

Let $k$ be an algebraically closed field of characteristic 0, $Y=k^{r}\times {(k^{\times})}^{s}$ and let $G$ be an algebraic torus acting diagonally on the ring of differential operators $\cD (Y)^G$. We give necessary and sufficient…

Let $G$ be a linear connected non-compact real simple Lie group and let $K\subset G$ be a maximal compact subgroup of $G$. Suppose that the centre of $K$ isomorphic to $\mathbb{S}^1$ so that $G/K$ is a global Hermitian symmetric space. Let…

For an irreducible complex reflection group $W$ of rank $n$ containing $N$ reflections, we put $g=2N/n$ and construct a $(g+1)^n$-dimensional irreducible representation of the Cherednik algebra which is (as a vector space) a quotient of the…

Let $G = \mathrm{SCl}_n(q)$ be a quasisimple classical group with $n$ large, and let $x_1, \dots, x_k \in G$ random, where $k \geq q^C$. We show that the diameter of the resulting Cayley graph is bounded by $q^2 n^{O(1)}$ with probability…

Let $\mathcal{C}(n,k)$ be the set of $k$-dimensional simplicial complexes $C$ over a fixed set of $n$ vertices such that: (1) $C$ has a complete $k-1$-skeleton; (2) $C$ has precisely ${{n-1}\choose {k}}$ $k$-faces; (3) the homology group…

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an…

We construct an explicit sequence $V_{k_n,a_n}$ of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semi-stable representation $V_{k,{\mathcal{L}}}$ of…

We show an analogue of a theorem of An, Ghosh, Guan, and Ly on weighted badly approximable vectors for totally imaginary number fields. We show that for $G=\mathrm{SL}_2(\mathbb{C})\times\dots\times\mathrm{SL}_2(\mathbb{C})$ and $\Gamma<G$…

The $k$-representation number of a graph $G$ is the minimum cardinality of the system of vertex subsets with the property that every edge of $G$ is covered at least $k$ times while every non-edge is covered at most $(k-1)$ times. In…

Let G be a connected, complex reductive Lie group with maximal compact subgroup K, and let X denote the moduli space of G- or K-valued representations of a rank r free group. In this article, we develop methods for studying the…

We explore a relationship between the classical representation theory of a complex, semisimple Lie algebra \g and the resonance varieties R(V,K)\subset V^* attached to irreducible \g-modules V and submodules K\subset V\wedge V. In the…

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup of $G$. We obtain both asymptotic and explicit bounds for the number of odd-degree irreducible complex representations of $G$ in terms of the size of the abelianization of $P$. To do…

Let $G$ be a simple real Lie group with maximal parabolic subgroup $P$ whose nilradical is abelian. Then $X=G/P$ is called a symmetric $R$-space. We study the degenerate principal series representations of $G$ on $C^\infty(X)$ in the case…

We prove a quantitative refinement of the statement that groups of polynomial growth are finitely presented. Let $G$ be a group with finite generating set $S$ and let $\operatorname{Gr}(r)$ be the volume of the ball of radius $r$ in the…

We prove that any minimal valid function for the k-dimensional infinite group relaxation that is piecewise linear with at most k+1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a…

We prove a sharp bound for the minimal doubling of a small measurable subset of a compact connected Lie group. Namely, let $G$ be a compact connected Lie group of dimension $d_G$, we show that for for all measurable subsets $A$, we have…

Let G=(V,E) be an undirected graph, lambda_k be the k-th smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that lambda_k > 0 if and only if G has at most k-1 connected components.…

Let $\Gamma =(V,E)$ be a reflexive relation with a transitive automorphisms group. Let $v\in V$ and let $F$ be a finite subset of $V$ with $v\in F.$ We prove that the size of $\Gamma (F)$ (the image of $F$) is at least $$ |F|+ |\Gamma…