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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.
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| 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 Lustzig using quantum groups.
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| ==Background and motivation==
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| Some of the basic questions in the representation theory of complex semisimple Lie algebras or compact semisimple Lie groups going back to [[Hermann Weyl]] include:<ref>{{harvnb|Weyl|1946}}</ref><ref>{{harvnb|Humphreys|1994}}</ref>
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| * 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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| * For two highest weights λ, μ, find the decomposition of their tensor product ''L''(λ) <math>\otimes </math> ''L''(μ) into irreducible representations.
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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>
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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.)
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| 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 generalizes classical results based on [[Young tableau]]x for the [[general linear group|general linear Lie algebra]] <math>\mathfrak{gl}</math><sub>''n''</sub> or the [[special linear group|special linear Lie algebra]] <math>\mathfrak{sl}</math><sub>''n''</sub>:<ref>{{harvnb|Littlewood|1950}}</ref><ref>{{harvnb|Macdonald|1979}}</ref><ref>{{harvnb|Sundaram|1990}}</ref><ref>{{harvnb|King|1990}}</ref>
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| * [[Issai Schur]]'s result in his 1901 dissertation that the weight multiplicities could be counted in terms of column-strict Young tableaux (i.e. weakly increasing to the right along rows, and strictly increasing down columns).
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| * The celebrated [[Littlewood-Richardson rule]] that describes both tensor product decompositions and branching from <math>\mathfrak{gl}</math><sub>''m''+''n''</sub> to <math>\mathfrak{gl}</math><sub>''m''</sub> <math>\oplus</math> <math>\mathfrak{gl}</math><sub>''n''</sub> in terms of lattice permutations of skew tableaux.
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| 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>
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| 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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| The combinatorial data could be encoded in a coloured directed graph, with labels given by the simple roots.
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| Littelmann's main motivation<ref>{{harvnb|Littelmann|2007}}</ref> was to reconcile two different aspects of representation theory:
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| * The standard monomial theory of Lakshmibai and Seshadri arising from the geometry of [[Schubert variety|Schubert varieties]].
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| *[[crystal basis|Crystal bases]] arising in the approach to [[quantum group]]s of [[Masaki Kashiwara]] and [[George Lusztig]]. Kashiwara and Lusztig constructed canonical bases for representations of deformations of the [[universal enveloping algebra]] of <math>\mathfrak{g}</math> depending on a formal deformation parameter ''q''. In the degenerate case when ''q'' = 0, these yield [[crystal basis|crystal bases]] together with pairs of operators corresponding to simple roots; see {{harvtxt|Ariki|2002}}.
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| Although differently defined, the crystal basis, its root operators and crystal graph were later shown to be equivalent to Littelmann's path model and graph; see {{harvtxt|Hong|Kang|2002|p=xv}}. In the case of complex semisimple Lie algebras, there is a simplified self-contained account in {{harvtxt|Littelmann|1997}} relying only on the properties of [[root system]]s; this approach is followed here.
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| == Definitions ==
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| Let ''P'' be the [[weight lattice]] in the dual of a [[Cartan subalgebra]] of the [[semisimple Lie algebra]] <math>\mathfrak{g}</math>.
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| A '''Littelmann path''' is a piecewise-linear mapping
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| :<math>\pi:[0,1]\cap \mathbf{Q} \rightarrow P\otimes_{\mathbf{Z}}\mathbf{Q}</math>
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| such that π(0) = 0 and π(1) is a [[weight lattice|weight]].
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| Let (''H''<sub> α</sub>) be the basis of <math>\mathfrak{h}</math> consisting of "coroot" vectors, dual to basis of <math>\mathfrak{h}</math>* formed by [[root system|simple roots]] (α). For fixed α and a path π, the function <math>h(t)= (\pi(t), H_\alpha)</math> has a minimum value ''M''.
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| Define non-decreasing self-mappings ''l'' and ''r'' of [0,1] <math>\cap</math> '''Q''' by
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| :<math> l(t) = \min_{t\le s\le 1} (1,h(s)-M),\,\,\,\,\,\, r(t) = 1 - \min_{0\le s\le t} (1,h(s)-M).</math>
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| Thus ''l''(''t'') = 0 until the last time that ''h''(''s'') = ''M'' and ''r''(''t'') = 1 after the first time that ''h''(''s'') = ''M''.
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| Define new paths π<sub>l</sub> and π<sub>r</sub> by
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| :<math>\pi_r(t)= \pi(t) + r(t) \alpha,\,\,\,\,\,\, \pi_l(t) = \pi(t) - l(t)\alpha</math>
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| The '''root operators''' ''e''<sub>α</sub> and ''f''<sub>α</sub> are defined on a basis vector [π] by
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| * <math>\displaystyle{ e_\alpha [\pi] = [\pi_r]} </math> if ''r'' (0) = 0 and 0 otherwise;
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| * <math> \displaystyle{f_\alpha [\pi] = [\pi_l]} </math> if ''l'' (1) = 1 and 0 otherwise.
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| The key feature here is that the paths form a basis for the root operators like that of a [[monomial representation]]: when a root operator is applied to the basis element for a path, the result is either 0 or the basis element for another path.
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| ==Properties==
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| Let <math>\mathcal{A}</math> be the algebra generated by the root operators. Let π(''t'') be a path lying wholly within the positive [[Weyl chamber]] defined by the simple roots. Using results on the path model of [[C. S. Seshadri]] and Lakshmibai, Littelmann showed that
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| *the <math>\mathcal{A}</math>-module generated by [π] depends only on π(1) = λ and has a '''Q'''-basis consisting of paths [σ];
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| *the multiplicity of the weight μ in the integrable highest weight representation ''L''(λ) is the number of paths σ with σ(1) = μ.
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| There is also an action of the [[Weyl group]] on paths [π]. If α is a simple root and ''k'' = ''h''(1), with ''h'' as above, then the corresponding reflection ''s''<sub>α</sub> acts as follows:
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| * ''s''<sub>α</sub> [π] = [π] if ''k'' = 0;
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| * ''s''<sub>α</sub> [π]= ''f''<sub>α</sub><sup>''k''</sup> [π] if ''k'' > 0;
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| * ''s''<sub>α</sub> [π]= ''e''<sub>α</sub><sup> – ''k''</sup> [π] if ''k'' < 0.
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| If π is a path lying wholly inside the positive Weyl chamber, the '''Littelmann graph''' <math>\mathcal{G}_\pi</math> is defined to be the coloured, directed graph having as vertices the non-zero paths obtained by successively applying the operators ''f''<sub>α</sub> to π. There is a directed arrow from one path to another labelled by the simple root α, if the target path is obtained from the source path by applying ''f''<sub>α</sub>.
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| * The Littelmann graphs of two paths are isomorphic as coloured, directed graphs if and only if the paths have the same end point.
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| The Littelmann graph therefore only depends on λ. Kashiwara and Joseph proved that it coincides with the "crystal graph" defined by Kashiwara in the theory of crystal bases.
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| ==Applications==
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| ===Character formula===
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| If π(1) = λ, the multiplicity of the weight μ in ''L''(λ) is the number of vertices σ in the Littelmann graph <math> \mathcal{G}_\pi </math> with σ(1) = μ.
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| ===Generalized Littlewood-Richardson rule===
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| Let π and σ be paths in the positive Weyl chamber with π(1) = λ and σ(1) = μ. Then
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| :<math> L(\lambda) \otimes L(\mu) = \bigoplus_\eta L(\lambda + \tau(1)),</math>
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| where τ ranges over paths in <math>\mathcal{G}_\sigma</math> such that π <math>\star</math> τ lies entirely in the positive Weyl chamber and
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| the ''concatenation'' π <math>\star</math> τ (t) is defined as π(2''t'') for ''t'' ≤ 1/2 and π(1) + τ( 2''t'' – 1) for ''t'' ≥ 1/2.
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| ===Branching rule===
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| If <math>\mathfrak{g}_1</math> is the Levi component of a parabolic subalgebra of <math>\mathfrak{g}</math> with weight lattice ''P''<sub>1</sub> <math>\supset </math> ''P'' then
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| :<math> L(\lambda)|_{\mathfrak{g}_1} = \bigoplus_{\sigma} L(\sigma(1)),</math> | |
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| where the sum ranges over all paths σ in <math>\mathcal{G}_\pi</math> which lie wholly in the positive Weyl chamber for <math>\mathfrak{g}_1</math>.
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| == See also ==
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| * [[Crystal basis]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| *{{citation|title=Representations of Quantum Algebras and Combinatorics of Young Tableaux|first= Susumu |last=Ariki|series=University Lecture Series|volume=26|publisher=American Mathematical Society|year= 2002|id =ISBN 0821832328}}
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| *{{citation|last=Berenstein|first= Arkady|last2=Zelevinsky|first2= Andrei|title=Tensor product multiplicities, canonical bases and totally positive varieties|journal=Invent. Math.|volume= 143 |year=2001|pages =77–128|doi=10.1007/s002220000102|bibcode = 2001InMat.143...77B }}
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| *{{citation|title=Introduction to Quantum Groups and Crystal Bases|first=Jin|last= Hong|first2=Seok-Jin|last2= Kang|year= 2002|id=ISBN 0821828746
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| |series=Graduate Studies in Mathematics|volume=42|publisher=American Mathematical Society}}
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| *{{citation|first=Ronald C.|last=King|title=S-functions and characters of Lie algebras and superalgebras|pages=226–261| series=IMA Vol. Math. Appl.|journal=Institute for Mathematics and Its Applications|volume= 19|publisher= Springer-Verlag|year=1990|bibcode=1990IMA....19..226K}}
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| *{{citation|first=James E.|last=Humphreys|year=1994|edition=2|isbn= 0-387-90053-5|title=Introduction to Lie Algebras and Representation Theory|publisher=Springer-Verlag}}
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| * {{citation|last=Littelmann|first=Peter|title=A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras|journal= Invent. Math.|volume= 116|year=1994|pages=329–346|doi=10.1007/BF01231564|bibcode = 1994InMat.116..329L }}
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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}}
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| *{{citation|first=Peter|last=Littelmann|title=Characters of Representations and Paths in <math>\mathfrak{h}</math><sub>'''R'''</sub>*|pages=29–49|year=
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| 1997|journal=Proceedings of Symposia in Pure Mathematics|publisher=American Mathematical Society|volume=61}} [instructional course]
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| *{{citation|first=Dudley E.|last= Littlewood|authorlink=Dudley E. Littlewood|title= The Theory of Group Characters and Matrix Representations of Groups|journal=Nature|volume=146|issue=3709|pages=699|publisher= Oxford University Press|year= 1950|doi=10.1038/146699a0|bibcode=1940Natur.146..699H}}
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| *{{citation|first=Ian G.|last=Macdonald|authorlink=I. G. Macdonald|title=Symmetric Functions and Hall Polynomials|publisher=Oxford University Press|year=1979}}
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| *{{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-bin/fitem?id=SB_1994-1995__37__209_0|year=1995}}
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| *{{citation|first=Sheila|last=Sundaram|title=Tableaux in the representation theory of the classical Lie groups|pages= 191–225|
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| series=IMA Vol. Math. Appl.|journal=Institute for Mathematics and Its Applications|volume= 19|publisher=Springer-Verlag |year=1990|bibcode=1990IMA....19..191S}}
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| *{{citation|first=Hermann|last=Weyl|authorlink=Hermann Weyl|title=The classical groups|year=1946|publisher=Princeton University Press}}
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| [[Category:Representation theory]]
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| [[Category:Lie algebras]]
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| [[Category:Algebraic combinatorics]]
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