Littelmann path model: Difference between revisions

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In [[mathematics]], the '''Littelmann path model''' is a [[combinatorics|combinatorial device]] due to [[Peter Littelmann]] for computing multiplicities ''without overcounting'' in the [[representation theory]] of symmetrisable [[Kac–Moody algebra]]s. Its most important application is to complex [[semisimple Lie algebra]]s or equivalently compact [[semisimple Lie group]]s, the case described in this article. Multiplicities in [[Weyl character formula|irreducible representations]], tensor products and [[branching rule]]s can be calculated using a [[graph theory|coloured directed graph]], with labels given by the [[root system|simple roots]] of the Lie algebra.

Developed as a bridge between the theory of [[crystal basis|crystal bases]] arising from the work of [[Masaki Kashiwara|Kashiwara]] and [[George Lusztig|Lusztig]] on [[quantum group]]s and the [[standard monomial theory]] of [[C. S. Seshadri]] and Lakshmibai, Littelmann's path model associates to each irreducible representation a rational [[vector space]] with basis given by paths from the origin to a [[root lattice|weight]] as well as a pair of '''root operators''' acting on paths for each [[root lattice|simple root]]. This gives a direct way of recovering the algebraic and combinatorial structures previously discovered by Kashiwara and Lusztig using quantum groups.

==Background and motivation==

Some of the basic questions in the representation theory of complex semisimple Lie algebras or compact semisimple [[Lie groupsgroup]]s going back to [[Hermann Weyl]] include:<ref>{{harvnb|Weyl|1953}}</ref><ref>{{harvnb|Humphreys|1994}}</ref>

* For a given [[dominant weight]] '''λ''', find the weight multiplicities in the [[Weyl character formula|irreducible representation]] ''L''(λ) with highest weight λ.

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* Suppose that <math>\mathfrak{g}_1</math> is the [[Levi decomposition|Levi component]] of a [[parabolic subalgebra]] of a semisimple Lie algebra <math>\mathfrak{g}</math>. For a given dominant highest weight '''λ''', determine the [[branching rule]] for decomposing the restriction of ''L''('''λ''') to <math>\mathfrak{g}_1</math>.<ref>Every complex semisimple Lie algebra <math>\mathfrak{g}</math> is the [[complexification]] of the Lie algebra of a compact connected simply connected semisimple Lie group. The subalgebra <math>\mathfrak{g}_1</math> corresponds to a maximal rank closed subgroup, i.e. one containing a maximal torus.</ref>

(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|1953|p=230,312<!-- Need to fix pages for 2nd ed -->}}. 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 generalizes classical results based on [[Young tableau]]x for the [[general linear group|general linear Lie algebra]] <math>\mathfrak{gl}</math>_''n'' or the [[special linear group|special linear Lie algebra]] <math>\mathfrak{sl}</math>_''n'':<ref>{{harvnb|Littlewood|1950}}</ref><ref>{{harvnb|Macdonald|1998}}</ref><ref>{{harvnb|Sundaram|1990}}</ref><ref>{{harvnb|King|1990}}</ref>

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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 [[convexConvex polytope|convex polytopes]]s, proposed in 1988, is related to Littelmann paths and crystal bases.</ref>

Littelmann'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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*{{citation| last=Littelmann|first= Peter|title=Paths and root operators in representation theory|journal=Ann. of Math.|volume= 142 |year=1995|pages= 499–525| doi=10.2307/2118553| jstor=2118553| issue=3| publisher=Annals of Mathematics}}

*{{citation|first=Peter|last=Littelmann|title=Characters of Representations and Paths in <math>\mathfrak{h}</math>_'''R'''*|pages=29–49|year=1997|journal=Proceedings of Symposia in Pure Mathematics|publisher=American Mathematical Society|volume=61|doi=10.1090/pspum/061/1476490}} [instructional course]

*{{citation|first=Dudley E.|last= Littlewood|authorlink=Dudley E. Littlewood|title= The Theory of Group Characters and Matrix Representations of Groups |url=https://books.google.com/books?id=C6xHHEaI-GcC |year=1977 |publisher=American Mathematical Society |isbn=978-0-8218-7435-6 |edition=2nd |origyearorig-date=1950 |volume=357 |series=AMS Chelsea Publishing Series |ref={{harvid|Littlewood|1950}}}}

*{{citation|first=Ian G.|last=Macdonald|authorlink=I. G. Macdonald|title=Symmetric Functions and Hall Polynomials |url=https://books.google.com/books?id=srv90XiUbZoC |year=1998 |orig-date=1979 |publisher=Clarendon Press |isbn=978-0-19-850450-4 |edition=2nd |series=Oxford mathematical monographs}}

*{{citation|last=Mathieu|first= Olivier|title=Le modèle des chemins, Exposé No. 798 |series= Séminaire Bourbaki (astérique)|volume= 37|url=http://www.numdam.org/numdam-binitem/fitem?id=SB_1994-1995__37__209_0/|year=1995}}

*{{citation|first=Sheila|last=Sundaram|title=Tableaux in the representation theory of the classical Lie groups|pages= 191–225|

series=IMA Vol. Math. Appl.|journal=Institute for Mathematics and Its Applications|volume= 19|publisher=Springer-Verlag |year=1990|bibcode=1990IMA....19..191S}}

*{{citation|first=Hermann|last=Weyl|authorlink=Hermann Weyl|title=The Classical Groups: Their Invariants and Representations (PMS-1) |url=https://books.google.com/books?id=2twDDAAAQBAJ |date=2016 |publisher=Princeton University Press |isbn=978-1-4008-8390-5 |volume=45 |series=Princeton Landmarks in Mathematics and Physics |edition=2nd |origyearorig-date=1953 |ref={{harvid|Weyl|1953}}}}

{{refend}}