Littelmann path model: Difference between revisions

Revision as of 03:15, 3 January 2009

Revision as of 17:36, 10 April 2009

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* The celebrated [[Littlewood-Richardson rule]] that describes both tensor product decompositions and branching from <math>\mathfrak{gl}</math>_''m''+''n'' to <math>\mathfrak{gl}</math>_''m'' <math>\oplus</math> <math>\mathfrak{gl}</math>_''n'' in terms of lattice permutations of skew tableaux.

Attempts at finding similar algorithms without overcounting for the other classical Lie algebras had only been partially successful.<ref>Numerous authors have made contributions, including the physicist R. C. King, and the mathematicians S. Sundaram, [[I. M. Gelfand]], [[A,. Zelevinsky]] and A. Berenstein. The surveys of {{harvtxt|King|1990}} and {{harvtxt|Sundaram|1990}} give variants of [[Young tableaux]] which can be used to compute weight multiplicities, branching rules and tensor products with fundamental representations for the remaining classical Lie algebras. {{harvtxt|Berenstein|Zelevinsky|2001}} discuss how their method using [[convex polytope]]s, proposed in 1988, is related to Littelmann paths and crystal bases. </ref>

Littlemann's contribution was to give a unified combinatorial model that applied to all symmetrizable [[Kac-Moody algebra]]s and provided explicit subtraction-free combinatorial formulas for weight multiplicities, tensor product rules and [[branching rule]]s. He accomplished this by introducing the vector space ''V'' over '''Q''' generated by the [[weight lattice]] of a [[Cartan subalgebra]]; on the vector space of piecewise-linear paths in ''V'' connecting the origin to a weight, he defined a pair of ''root operators'' for each [[root lattice|simple root]] of <math>\mathfrak{g}</math>.