Leibniz Universität Hannover

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.