Related papers: Homomorphisms between Weyl modules for SL_3(k)

We show that for any set of primes $\mathcal{P}$ there exists a space $M_{\mathcal{P}}$ which is a homology and cohomology 3-manifold with coefficients in $\mathbb{Z}_{p}$ for $p\in \mathcal{ P}$ and is not a homology or cohomology…

We prove that if $Y$ is a closed, oriented 3-manifold with first homology $H_1(Y;\mathbb{Z})$ of order less than $5$, then there is an irreducible representation $\pi_1(Y) \to \mathrm{SL}(2,\mathbb{C})$ unless $Y$ is homeomorphic to $S^3$,…

Given a Hilbert modular form for a totally real field $F$, and a prime $p$ split completely in $F$, the $f$-eigenspace in $p$-adic de Rham cohomology of the Hilbert modular variety has a family of partial filtrations and partial Frobenius…

We study the category of modules of minimal dimension over completed Weyl algebras in equal characteristic zero. In particular we prove finiteness of de Rham cohomology of such modules.

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $k$. The aim of this paper is to present a method to find triples $(G,M,H)$ with the following three properties. Property 1: $G$ is simple and $k$…

We study three related homological properties of modules in the BGG category O for basic classical Lie superalgebras, with specific focus on the general linear superalgebra. These are the projective dimension, associated variety and…

Let $g$ be a finite-dimensional simple Lie algebra over the complex number field. We classify the homomorphisms between $g$-modules induced from one-dimensional modules of maximal parabolic subalgebras.

Let $K$ be any field, and let $E$ be any graph. We explicitly construct the projective resolution of simple left modules over the Leavitt path algebra $L_K(E)$ associated to cycles and irreducible polynomials. Then we study the dimension of…

Let $p$ be a prime. Given a split semisimple group scheme $G$ over a normal integral domain $R$ which is a faithfully flat $\mathbb Z_{(p)}$-algebra, we classify all finite dimensional representations $V$ of the fiber $G_K$ of $G$ over…

Given an algebraically closed field $\Bbbk$ of characteristic zero, a Lie superalgebra $\mathfrak{g}$ over $\Bbbk$ and an associative, commutative $\Bbbk$-algebra $A$ with unit, a Lie superalgebra of the form $\mathfrak{g} \otimes_\Bbbk A$…

Let F be an algebraically closed field of characteristic p>0. Suppose that SL_{n-1}(F) is naturally embedded into SL_n(F) (either in the top left corner or in the bottom right corner). We prove that certain Weyl modules over SL_{n-1}(F) can…

Given a field $\mathbb K$, for any $n\geq 3$ the first cohomology group $H^1(G_n,A^*_n)$ of the special linear group $G_n = \mathrm{SL}(n,{\mathbb K})$ over the dual $A^*_n$ of its adjoint module $A_n$ is isomorphic to the space…

Let G be a finite group of Lie type, defined over a field k of characteristic p > 0. We find explicit bounds for the dimension of the first cohomology group for G with coefficients in a simple kG-module. We proceed by bounding the number of…

The submodule structure of general Specht modules in prime characteristic is a difficult open problem. Kleshchev and Sheth [Journal of Algebra, 221(2), pp.705-722] gave a combinatorial description of the submodule structure of Specht…

The period morphism of polarized hyper-K\"ahler manifolds of K3$^{[m]}$-type gives an embedding of each connected component of the moduli space of polarized hyper-K\"ahler manifolds of K3$^{[m]}$-type into their period space, which is the…

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some…

Given a field $F$ of characteristic $3$ and division symbol $p$-algebras $[\alpha,\beta)_{3,F}$ and $[\alpha,\gamma)_{3,F}$ of degree $3$ over $F$, we prove that if $\alpha \text{dlog}(\beta)\wedge \text{dlog}(\gamma)$ is trivial in the…

We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as…

We study Hom-quantum groups, their representations, and module Hom-algebras. Two Twisting Principles for Hom-type algebras are formulated, and construction results are proved following these Twisting Principles. Examples include Hom-quantum…

We describe polar homology groups for complex manifolds. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincare residue…