Properties of the Solomon homomorphism

A Quiver Presentation for Solomon's Descent Algebra

The descent algebra $Σ(W)$ is a subalgebra of the group algebra $\Q W$ of a finite Coxeter group $W$, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of $W$. Thus $Σ(W)$ is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct $Σ(W)$ as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of $S$, the set of simple reflections in $W$. From this construction we obtain some general information about the quiver of $Σ(W)$ and an algorithm for the constructi…

Twisted descent algebras and the Solomon-Tits algebra

The purpose of the present article is to define and study a new class of descent algebras, called twisted descent algebras. These algebras are associated to the Barratt-Joyal theory of twisted bialgebras in the same way than classical descent algebras are associated to classical bialgebras. The formal properties of twisted descent algebras seem particularly meaningful in view of applications to discrete probabilities, to the geometry of Coxeter groups and buildings, and to symmetric group combinatorics.

A Satake homomorphism for the $\bmod \, p$ derived Hecke algebra

We explore the structure of the derived spherical Hecke algebra of a $p$-adic group, a graded associative algebra whose degree $0$ subalgebra is the classical spherical Hecke algebra. Working with $\mathbb Z/ p^a$ coefficients, we establish a Satake homomorphism relating this graded algebra to the corresponding graded algebra for the torus. We investigate the image of this homomorphism in degree $1$, as well as other properties, such as transitivity with respect to inclusion of Levi subgroups.

关于Buchstaber–Rees“Frobenius n-同态”理论及其推广

On the Buchstaber--Rees theory of "Frobenius $n$-homomorphisms" and its generalization

本文综述了Buchstaber–Rees n-同态理论及其推广，提出以特征函数刻画交换环间线性映射的乘法性质，并由此引入p|q-同态新概念，简化了原有理论。

Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups

We give an explicit approach for Bockstein homomorphisms of the Hochschild cohomology of a group algebra and of a block algebra of a finite group and we show some properties. To give explicit definitions for these maps we use an additive decomposition and a Product Formula for the Hochschild cohomology of group algebras given by Siegel and Witherspoon in 1999. For k an algebraically closed field of characteristic p and G a finite group we prove an additive decomposition and a Product Formula for the cohomology algebra of a defect group of a block ideal of kG with coefficients in the source alg…

Functoriality properties of the dual group

Let $G$ be a connected reductive group. In a previous paper, arxiv:1702.08264, is was shown that the dual group $G^\vee_X$ attached to a $G$-variety $X$ admits a natural homomorphism with finite kernel to the Langlands dual group $G^\vee$ of $G$. Here, we prove that the dual group is functorial in the following sense: if there is a dominant $G$-morphism $X\to Y$ or an injective $G$-morphism $Y\to X$ then there is a canonical homomorphism $G^\vee_Y\to G^\vee_X$ which is compatible with the homomorphisms to $G^\vee$.