Related papers: Stable Centres II: Finite Classical Groups

We use geometry to prove a number of new identities among the Littlewood-Richardson coefficients for Schubert polynomials (Schubert classes in a flag manifold). For many of these identities, there is a companion result about the Bruhat…

We introduce fusion algebras with not necessarily positive structure constants and without identity element. We prove that they are semisimple when tensored with $\mathbb{C}$ and that their characters satisfy orthogonality relations. Then…

We define an algebra $A$ to be centrally stable if, for every epimorhism $\varphi$ from $A$ to another algebra $B$, the center $Z(B)$ of $B$ is equal to $\varphi(Z(A))$, the image of the center of $A$. After providing some examples and…

We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of…

The paper describes the algebraic structure of the graded algebra of differentially homogeneous polynomials of fixed finite order. We show that it is a finitely generated algebra, and we exhibit a minimal set of generators. Along the way,…

We give half a dozen bases of the Hecke algebra of the symmetric group, and relate them to the basis of Geck-Rouquier, and to the basis of Jones, using matrices of change of bases of the ring of symmetric polynomials.

We study representation stability in the sense of Church, Ellenberg, and Farb \cite{FI-module} through the lens of symmetric function theory and the different symmetric function bases. We show that a sequence, $(F_n)_n$, where $F_n$ is a…

Let $k$ be a noetherian commutative ring and let $G$ be a finite flat group scheme over $k$. Let $G$ act rationally on a finitely generated commutative $k$-algebra $A$. We show that the cohomology algebra $H^*(G,A)$ is a finitely generated…

We consider for two based graphs $G$ and $H$ the sequence of graphs $G_k$ given by the wedge sum of $G$ and $k$ copies of $H$. These graphs have an action of the symmetric group $\Sigma_k$ by permuting the $H$-summands. We show that the…

Let $G$ be a complex simply-connected semisimple Lie group and let $\g=\hbox{\rm Lie}\,G$. Let $\g = \n_- +\hh + \n$ be a triangular decomposition of $\g$. One readily has that $\hbox{\rm Cent}\,U(\n)$ is isomorphic to the ring $S(\n)^{\n}$…

For a tree $G$, we study the changing behaviors in the homology groups $H_i(B_nG)$ as $n$ varies, where $B_nG := \pi_1($UConf$_n(G))$. We prove that the ranks of these homologies can be described by a single polynomial for all $n$, and…

The structure of the centres ${\cal Z}(\Lg)$ and ${\cal Z}(\Mg)$ of the graph algebra ${\cal L}_g(sl_2)$ and the moduli algebra ${\cal M}_g(sl_2)$ is studied at roots of 1. It it shown that ${\cal Z}(\Lg)$ can be endowed with the structure…

We extend the Oprea's result $G_1(\mathbb{S}^{2n+1}/H)=\mathcal{Z}H$ to the 1st generalized Gottlieb group $G_1^f(\mathbb{S}^{2n+1}/H)$ for a map $f\colon A\to \mathbb{S}^{2n+1}/H$. Then, we compute or estimate the groups…

The flag-major index "fmaj" and the classical length function "$\ell$" are used to construct two $q$-analogs of the generating polynomial for the hyperoctahedral group~$B_n$ by number of positive and negative fixed points (resp. pixed…

Let $D_n$ be the dihedral group with $2n$ elements, and suppose $n$ is greater than one. We call ring system a finite $D_n$-symmetric set of points in $\mathbb{R}^2$. Ring systems have been used as models for planets surrounded by rings,…

A $nil$-closed, noetherian, unstable algebra $K$ over the Steenrod Algebra is determined, up to isomorphism, by the functor $\text{Hom}_{\mathcal{K}\text{f.g.}}(K,H^*(\_))$, which is a presheaf on the category $\mathcal{VI}$ of finite…

We prove homological stability for a twisted version of the Houghton groups and their multidimensional analogues. Based on this, we can describe the homology of the Houghton groups and that of their multidimensional analogues over constant…

To a set $\mathcal{B}$ of 4-subsets of a set $\Omega$ of size $n$ we introduce an invariant called the `hole stabilizer' which generalises a construction of Conway, Elkies and Martin of the Mathieu group $M_{12}$ based on Loyd's…

We study subgroups $H_U$ of the R. Thompson group $F$ which are stabilizers of finite sets $U$ of numbers in the interval $(0,1)$. We describe the algebraic structure of $H_U$ and prove that the stabilizer $H_U$ is finitely generated if and…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (a)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…