Geometric approaches to computing Kostka numbers and Littlewood-Richardson coefficients

Using tools from combinatorics, convex geometry and symplectic geometry, we study the behavior of the Kostka numbers and Littlewood-Richardson coefficients (the type A weight multiplicities and Clebsch-Gordan coefficients). We sh w that both are given by piecewise polynomial functions in the entries of the partitions and compositions parametrizing them, and that the domains of polynomiality form a complex of cones. Interesting factorization patterns are found in the polyomials giving the Kostka numbers. The case of A3 is studied more carefully and involves computer proofs. We relate the description of the domains of polynomiality for the weight multiplicity function to that of the domains for the Duistermaat-Heckman measure from symplectic geometry (a continuous analogue of the weight multiplicity function). As an easy consequence of this work, one obtains simple proofs of the fact the Kostka numbers, and Littlewood-Richardson numbers are given by polynomial functions in the nonnegative integer variable N. Both these results were known previously but have non-elementary proofs involving fermionic formulas for Kostka-Foulkes polynomials and semi-invariants of quivers. Also investigated is a new q-analogue of the Kostant partition function, which is shown to be given by polynomial functions over the relative interiors of the cells of a complex of cones.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2004

Author dc:creator

• Rassart, Étienne, 1975-

Advisor dc:contributor.advisor

• Sara Billey.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/16632

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/16632