Speaker: 
Ron King, University of Southampton 
Title: 
Schur functions and generating functions for Kostka, LittlewoodRichardson and Kronecker coefficients, and their stretched versions. 
Date: 
23.05. 2012 14:0015:30 
Location: 
RISC Seminar room 
Abstract: 
Some basic properties of the symmetric functions known as Schur
functions will be presented including the ubiquitous Cauchy identity. The
expansions of Schur functions as sums of monomials define Kostka
coefficients, The evaluation of outer and inner products of Schur
functions define LittlewoodRichardson coefficients and Kronecker
coefficients, respectively. Generating functions will be derived for all
of these coefficients as well as certain reduced Kronecker coefficients.
The explicit evaluation of the various coefficients from their generating
functions involves the extraction of constant terms in their multivariable
expansions. It is here that algorithms such as that supplied by the Omega
package are ripe for exploitation. Some illustrative examples will be
presented, as well as some computational limitations of this approach. In
each case the coefficients are labelled by partitions. Replacing each
partition by one in which each part is scaled by some parameter gives rise
to corresponding stretched coefficients. Generating functions for these
stretched versions will be given and a number of computational methods of
attacking their evaluation will be outlined. The polynomial or
quasipolynomial nature of the dependence of the coefficients on the
stretching parameter will be illustrated along with some open questions
regarding the nature of these polynomials and quasipolynomials. This
represents ongoing joint work with Trevor Welsh. 
VCAL file: 
