Let
be a complex finite-dimensional simple Lie algebra. Given a positive integer
and a dominant
weight
, we
define a preorder
on the set
of
-tuples of dominant
weights which add up to
.
Let
be the equivalence relation defined by the preorder and
be the corresponding poset of equivalence classes. We show that if
is a multiple of a
fundamental weight (and
is general) or if
(and
is general),
then
coincides with
the set of
-orbits
in
, where
acts on
as the permutations
of components. If
is of type
and
, we show
that the
-orbit
of the row shuffle defined by Fomin et al. (2005) is the unique maximal element in the
poset.
Given an element of
,
consider the tensor product of the corresponding simple finite-dimensional
-modules. We show
that (for general
,
,
and )
the dimension of this tensor product increases along
. We also show that
in the case when
is a multiple of a fundamental minuscule weight
( and
are general)
or if
is of
type
and
( is
general), there exists an inclusion of tensor products along with the partial order
on
. In
particular, if
is of type
,
this means that the difference of the characters is Schur positive.
