Oberseminar zur Algebra und Algebraischen Kombinatorik
Montag, 20.4.2009, ab 16:30 Uhr in Raum A 410
Littlewood-Richardson coefficients, the hive model and Horn
inequalities
Prof.em. Dr. Ronald C. King (Southampton)
Littlewood-Richardson (LR)
coefficients arise as integer multiplicities
in the decomposition of products of Schur functions. The hive model is
introduced
as a means of evaluating these coefficients combinatorially. They are
shown to be
non-zero if and only if a set of essential Horn inequalities are
satisfied, and that the
saturation of any one such inequality leads to a factorisation of the
corresponding
LR-coefficients. Stretched LR-coefficients are defined by scaling all
parts of the
partitions labelling the three relevant Schur functions. It is known
that stretched
LR-coefficients are polynomial in the stretching parameter. If time
permits some
properties of these polynomials will be discussed, together with some
open problems.
Kolloquium der Fakultät für Mathematik und Physik
Dienstag, 21.4.2009, ab 17:15 Uhr im Kleinen Physik-Hörsaal F 342
Paraboson and parafermion statistics and their connection
with
Schur function
series restricted by column or row lengths
Prof.em. Dr. Ronald C. King (Southampton)
In a recent study of paraboson Fock space representations of
orthosymplectic
Lie superalgebras, Lievens, Stoilova and Van der Jeugt (LSVdJ) made a
remarkable
conjecture regarding the generating function for the sum of all Schur
functions specified
by partitions for which the corresponding Young diagrams have a
restricted number of
rows. At first sight this looks quite different from the corresponding
formula given by
Macdonald for the situation in which the number of columns is
restricted. It will be shown
that Macdonald's formula may be recast in a form analogous to that of
the LSVdJ
conjecture, allowing the latter to be proved by a conjugacy argument.
Conversely,
it will be shown that Macdonald’s formula gives the characters of
certain parafermion
Fock space representations of orthogonal Lie algebras. Finally, it will
be pointed out
that both results, the LSVdJ conjecture and Macdonald’s formula, owe
their origin to
the existence of certain Howe dual pairs of groups.