Related papers: Alvis-Curtis Duality for Finite General Linear Gro…

In this note, we prove that if every character of a finite group $G$ fixed by an order-2 Galois automorphism has odd degree, then $G$ has a normal Sylow $2$-subgroup. On the way, we study extensions of characters of $GL_n(q)$, $q$ odd, to…

As a sequel to our proof of the analog of Serre's conjecture for function fields in Part I of this work, we study in this paper the deformation rings of $n$-dimensional mod $\ell$ representations $\rho$ of the arithmetic fundamental group…

Let $\mathbb{F}_q$ be the finite field with $q = p^f$ elements. We study the restriction of two classes of mod $p$ representations of $G_q = \text{GL}_2({\mathbb{F}_q})$ to $G_p = \text{GL}_2(\mathbb{F}_p)$. We first study the restrictions…

A curve X over the field Q of rational numbers is modular if it is dominated by X_1(N) for some N; if in addition the image of its jacobian in J_1(N) is contained in the new subvariety of J_1(N), then X is called a new modular curve. We…

Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$…

A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We…

We use a unified elementary approach to prove the second part of classical, mixed, super, and mixed super Schur-Weyl dualities for general linear groups and supergroups over an infinite ground field of arbitrary characteristic. These…

We construct a special principal series representation for the modular double $U_{q\tilde{q}}(g_R)$ of type $A_r$ representing the generators by positive essentially self-adjoint operators satisfying the transcendental relations that also…

Given an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell$, the $\ell^n$-torsion points of $A$ give rise to a representation $\rho_{A, \ell^n} : \gal(\bar{K} / K) \to \gl_{2g}(\zz/\ell^n\zz)$. In particular,…

We propose a non-perturbative description of the moduli spaces encoding p-form generalized Maxwell theories in any dimension, using derived differential geometry. Our approach synthesizes the Batalin--Vilkovisky formalism with differential…

We prove that the complement of a very general pair of hypersurfaces of total degree $2n$ in $\mathbb{P}^n$ is algebraically hyperbolic modulo a proper closed subvariety. This provides evidence towards conjectures of Lang-Vojta and…

Let $O$ be a closed Poisson conjugacy class of the complex algebraic Poisson group GL(n) relative to the Drinfeld-Jimbo factorizable classical r-matrix. Denote by $T$ the maximal torus of diagonal matrices in GL(n). With every $a\in O\cap…

Let $G$ be a finite classical group of Lie type of rank $\ell$, defined over a field of characteristic $p>2$. In this work, we classify the irreducible representations of $G$ whose dimensions are bounded by a constant proportional to…

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness…

The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus $\L_g$ of such genus $g$ hyperelliptic curves is a $g$-dimensional subvariety of the moduli space of hyperelliptic curves $\H_g$. We discover a…

Let $X$ be a smooth projective connected curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ and $N_G(P)$ its normalizer in $G$. We show…

We adapt the commutator theory of universal algebra to the particular setting of racks and quandles, exploiting a Galois connection between congruences and certain normal subgroups of the displacement group. Congruence properties such as…

We prove that Alvis-Curtis duality preserves the Frobenius-Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius-Schur…

We define an involution on the space of compact tempered unipotent representations of inner twists of a split simple $p$-adic group $G$ and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact…

Given an abelian algebraic group $A$ over a global field $F$, $\alpha \in A(F)$, and a prime $\ell$, the set of all preimages of $\alpha$ under some iterate of $[\ell]$ generates an extension of $F$ that contains all $\ell$-power torsion…