Stretched Littlewood-Richardson and Kostka coefficients Ronald King (R.C.King@maths.soton.ac.uk) Abstract Littlewood-Richardson
and Kostka coefficients both arise in character theory and also in the
study of symmetric functions; the former as tensor product multiplicities
and as coefficients in the expansion of products of Schur functions, and
the latter as weight multiplicities and as transition matrix elements
between Schur functions and the monomial symmetric functions. In each
case these coefficients are specified by means of partitions. If the parts
of the partitions are all multiplied by some fixed positive integer number
$N$, then the values of the coefficients will change as a function of
the stretching parameter $N$. It has been first conjectured and then proved
that a stretched coefficient is non-zero if and only if the corresponding
unstretched coefficient is non-zero. Furthermore a stretched coefficient
is $1$ if and only if the corresponding unstretched coefficient is $1$.
Here we study the behaviour of these coefficients as a function of $N$
for other values $k$ of the unstretched coefficient. It is conjectured
that this behaviour is always polynomial in $N$. The study makes use of
a hive model of both Littlewood-Richardson and Kostka coefficients, thereby
making a connection with the integer points on and within certain rational
convex polytopes. Many examples of the relevant polynomials will be given,
along with further conjectures regarding their general form and that of
their generating functions. |