Schur functions and generating functions for Kostka, Littlewood-Richardson and Kronecker coefficients, and their stretched versions
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TALK
Prof. Ronald C. King (University of Southhampton, UK), 23 May 2012, 2:00 pm, RISC seminar room
Schur functions and generating functions for Kostka, Littlewood-Richardson and Kronecker coefficients, and their stretched versions
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 Littlewood-Richardson 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 quasi-polynomial 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 quasi-polynomials. This represents ongoing joint work with Trevor Welsh.
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 Littlewood-Richardson 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 quasi-polynomial 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 quasi-polynomials. This represents ongoing joint work with Trevor Welsh.