Related papers: Modular Character Sheaves on Reductive Lie Algebra…

We consider a generalisation of the classical Lehmer problem about the distribution of modular inverses in arithmetic progression, introduced by E. Alkan, F. Stan and A. Zaharescu. Using bounds of sums of multiplicative characters instead…

Let $G$ be connected reductive algebraic group defined over an algebraically closed field of characteristic $p > 0$ and suppose that $p$ is a good prime for the root system of $G$, the derived subgroup of $G$ is simply connected and the Lie…

The theory of character sheaves on a reductive group is extended to a class of varieties which includes the strata of the De Concini-Procesi completion of an adjoint group.

Cohen and Taylor introduced Plesken Lie algebras of finite groups and studied their structural properties. As a further step, we will introduce Plesken Lie algebra representations, Plesken Lie algebra modules and discuss the irreducibility…

We study the moduli space of Gieseker semi-stable sheaves on the complex projective plane supported on sextic curves and having Euler characteristic one. We determine locally free resolutions of length one for all such sheaves. We decompose…

Goldman and Turaev found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in…

We extend to semi-abelian categories the notion of characteristic subobject, which is widely used in group theory and in the theory of Lie algebras. Moreover, we show that many of the classical properties of characteristic subgroups of a…

We give some higher dimensional analogues of the Durfee square formula and point out their relation to dissections of multipartitions. We apply the results to write certain affine Lie algebra characters in terms of Universal Chiral…

I present a generalized notion of obstruction in \v{C}ech cohomology on semi-modules, which allows one to characterize non-disturbing contextual behaviors with any semi-field. This framework generalizes the usual \v{C}ech cohomology used in…

Character formulas for Lie superalgebras have been shown to have important applications to number theory and combinatorics. We prove the Kac-Wakimoto character formula for the general linear Lie superalgebra gl(m|n). This formula…

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of…

We survey recent advances in the theory of moduli spaces of stable sheaves on hyperk\"ahler manifolds of dimension greater than $2$. We start by recalling the well-known theory in dimension $2$, i.e.~for $K3$ surfaces, emphasizing the…

These are expanded notes from graduate courses about Lie algebras and Chevalley groups held at the University of Stuttgart. In the 1950s Chevalley showed how linear groups over arbitrary fields could be obtained~ -- ~by a uniform procedure~…

We construct an action of a Lie algebra on the homology groups of moduli spaces of stable sheaves on K3 surfaces under some technical conditions. This is a generalization of Nakajima's construction of sl_2-action on the homology groups. In…

We develop a theory of modulus sheaves with transfers, which generalizes Voevodsky's theory of sheaves with transfers. This paper and its sequel are foundational for the theory of motives with modulus, which is developed in [KMSY20].

Recently, a strong exponential character bound has been established in [3] for all elements $g \in \mathbf{G}^F$ of a finite reductive group $\mathbf{G}^F$ which satisfy the condition that the centraliser $C_{\mathbf{G}}(g)$ is contained in…

This article is based on lectures given by the authors in 2005 and 2006. Our first goal is to present an introduction to the orbit method with an emphasis on the character theory of finite nilpotent groups. The second goal (motivated by a…

A semiring scheme generalizes a scheme in such a way that the underlying algebra is that of semirings. We generalize \v{C}ech cohomology theory and invertible sheaves to semiring schemes. In particular, when $X=\mathbb{P}^n_M$, a projective…

By exploiting relationships between the values taken by ordinary characters of symmetric groups we prove two theorems in the modular representation theory of the symmetric group. 1. The decomposition matrices of symmetric groups in odd…

Over the past three decades, there have been several attempts to characterize modules over affine Lie superalgebras. One of the main issues in this regard is dealing with zero-level modules. In this paper, we study these modules and…