Group Marriage Problem

Let $G$ be a permutation group acting on $[n]=\{1, ..., n\}$ and $\mathcal{V}=\{V_{i}: i=1, ..., n\}$ be a system of $n$ subsets of $[n]$. When is there an element $g \in G$ so that $g(i) \in V_{i}$ for each $i \in [n]$? If such $g$ exists, we say that $G$ has a $G$-marriage subject to $\mathcal{V}$. An obvious necessary condition is the {\it orbit condition}: for any $\emptyset \not = Y \subseteq [n]$, $\bigcup_{y \in Y} V_{y} \supseteq Y^{g}=\{g(y): y \in Y \}$ for some $g \in G$. Keevash (J. Combin. Theory Ser. A 111(2005), 289--309) observed that the orbit condition is sufficient when $G$ is the symmetric group \Sym([n]); this is in fact equivalent to the celebrated Hall's Marriage Theorem. We prove that the orbit condition is sufficient if and only if $G$ is a direct product of symmetric groups. We extend the notion of orbit condition to that of $k$-orbit condition and prove that if $G$ is the alternating group \Alt([n]) or the cyclic group $C_{n}$ where $n \ge 4$, then $G$ satisfies the $(n-1)$-orbit condition subject to \V if and only if $G$ has a $G$-marriage subject to $\mathcal{V}$

Global branching laws by global Okounkov bodies

Let $G'$ be a complex semisimple group, and let $G \subseteq G'$ be a semisimple subgroup. We show that the branching cone of the pair $(G, G')$, which (asymptotically) parametrizes all pairs $(W, V)$ of irreducible finite-dimensional $G$-representations $W$ which occur as subrepresentations of a finite-dimensional irreducible $G'$-representation $V$, can be identified with the pseudo-effective cone, \overline{\mbox{Eff}}(Y), of some GIT quotient $Y$ of the flag variety of the group $G \times G'$. Moreover, we prove that the quotient $Y$ is a Mori dream space. As a consequence, the global Okounkov body $\Delta(Y)$ of $Y$, with respect to some admissible flag of subvarieties of $Y$, is fibred over the branching cone of $(G, G')$, and the fibre $\Delta(Y)_{(W, V)}$ over a point $(W, V)$ carries information about (the asymptotics of) the multiplicity of $W$ in $V$. Using the global Okounkov body $\Delta(Y)$, we easily derive a multi-dimensional generalization of Okounkov's result about the log-concavity of asymptotic multiplicities