12,925,103 research outputs found
Group Marriage Problem
Let be a permutation group acting on and
be a system of subsets of . When
is there an element so that for each ? If
such exists, we say that has a -marriage subject to .
An obvious necessary condition is the {\it orbit condition}: for any , for some . Keevash (J. Combin. Theory Ser. A 111(2005),
289--309) observed that the orbit condition is sufficient when 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 is a direct product of symmetric groups. We extend the notion of
orbit condition to that of -orbit condition and prove that if is the
alternating group \Alt([n]) or the cyclic group where , then
satisfies the -orbit condition subject to \V if and only if
has a -marriage subject to
Global branching laws by global Okounkov bodies
Let be a complex semisimple group, and let be a
semisimple subgroup. We show that the branching cone of the pair ,
which (asymptotically) parametrizes all pairs of irreducible
finite-dimensional -representations which occur as subrepresentations of
a finite-dimensional irreducible -representation , can be identified
with the pseudo-effective cone, \overline{\mbox{Eff}}(Y), of some GIT
quotient of the flag variety of the group . Moreover, we prove
that the quotient is a Mori dream space.
As a consequence, the global Okounkov body of , with respect
to some admissible flag of subvarieties of , is fibred over the branching
cone of , and the fibre over a point
carries information about (the asymptotics of) the multiplicity of in .
Using the global Okounkov body , we easily derive a
multi-dimensional generalization of Okounkov's result about the log-concavity
of asymptotic multiplicities
- β¦