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→Background and motivation: noted connections between the three classic problems |
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(Note that the first problem, of weight multiplicities, is the special case of the third in which the parabolic subalgebra is a Borel subalgebra. Moreover, the Levi branching problem can be embedded in the tensor product problem as a certain limiting case.)
Answers to these questions were first provided by Hermann Weyl and [[Richard Brauer]] as consequences of [[Weyl character formula|explicit character formulas]],<ref>{{harvnb|Weyl|1946|p=230,312}}. The "Brauer-Weyl rules" for restriction to maximal rank subgroups and for tensor products were developed independently by Brauer (in his thesis on the representations of the orthogonal groups) and by Weyl (in his papers on representations of compact semisimple Lie groups).</ref> followed by later combinatorial formulas of [[Hans Freudenthal]], [[Robert Steinberg]] and [[Bertram Kostant]]; see {{harvtxt|Humphreys|1994}}. An unsatisfactory feature of these formulas is that they involved alternating sums for quantities that were known a priori to be non-negative. Littelmann's method expresses these multiplicities as sums of non-negative integers ''without overcounting''. His work generalized the celebrated [[Littlewood-Richardson rule]] for the [[special linear group|special linear Lie algebra]] <math>\mathfrak{g}</math> = ''sl''<sub>n</sub> which gave a method for computing tensor product decompositions using [[Young tableaux]].<ref>{{harvnb|Littlewood|1950}}</ref><ref>{{harvnb|Macdonald|1979}}</ref> For the other classical groups, there had previously only been partial success in finding algorithms with no overcounting.<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|1988}} and {{harvtxt|Sundaram|1988}} give variants of [[Young tableaux]] which can be used to compute weight mutliplicities, branching rules and tensor products with fundamental representations for the remaining classical groups.
{{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>.
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