Popular branchings and their dual certificates

Let G be a digraph where every node has preferences over its incoming edges. The preferences of a node extend naturally to preferences over branchings, i.e., directed forests; a branching B is popular if B does not lose a head-to-head election (where nodes cast votes) against any branching. Such popular branchings have a natural application in liquid democracy. The popular branching problem is to decide if G admits a popular branching or not. We give a characterization of popular branchings in terms of dual certificates and use this characterization to design an efficient combinatorial algorithm for the popular branching problem. When preferences are weak rankings, we use our characterization to formulate the popular branching polytope in the original space and also show that our algorithm can be modified to compute a branching with least unpopularity margin. When preferences are strict rankings, we show that “approximately popular” branchings always exist.

A reciprocal branching problem for automorphic representations and global Vogan packets

Let G be a group and let H be a subgroup of G. The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G, determine the occurrence of an irreducible representation σ of H in the restriction of π to H. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation σ of H, find an irreducible representation π of G such that σ occurs in the restriction of π to H. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan–Gross–Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [13]. The method may be applied to other classical groups as well.

The Heisenberg product seen as a branching problem for connected reductive groups, stability properties

In 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.

The branching problem for generalized Verma modules, with application to the pair (so(7), LieG2)

We consider the branching problem for generalized Verma modules Mλ(𝔤, 𝔭) applied to couples of reductive Lie algebras [Formula: see text]. Our analysis of the problem is based on projecting character formulas to quantify the branching, and on the action of the center of [Formula: see text] to construct explicitly singular vectors realizing the [Formula: see text]-top level of the branching. We compute explicitly the top part of the branching for the pair [Formula: see text] for both strongly and weakly compatible with i( Lie G2) parabolic subalgebras and a large class of inducing representations.