AbstractIn this article, we study, in the context of complex representations of symmetric groups, some aspects of the Heisenberg product, introduced by Marcelo Aguiar, Walter Ferrer Santos and Walter Moreira in 2017.
When applied to irreducible representations, this product gives rise to the Aguiar coefficients.
We prove that these coefficients are in fact also branching coefficients for representations of connected complex reductive groups.
This allows to use geometric methods already developed in a previous article, notably based on notions from geometric invariant theory, and to obtain some stability results on Aguiar coefficients, generalising some of the results concerning them given by Li Ying.
All ingredients of the previous chapters are combined in order to build a gauge invariant theory of the interactions among the elementary particles. We start with a unified model of the weak and the electromagnetic interactions. The gauge symmetry is spontaneously broken through the BEH mechanism and we identify the resulting BEH boson. Then we describe the theory known as quantum chromodynamics (QCD), a gauge theory of the strong interactions. We present the property of confinement which explains why the quarks and the gluons cannot be extracted out of the protons and neutrons to form free particles. The last section contains a comparison of the theoretical predictions based on this theory with the experimental results. The agreement between theory and experiment is spectacular.
AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.
Abstract
Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic
$p>0$
. We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when
$p=2$
.
The Heisenberg product seen as a branching problem for connected reductive groups, stability properties