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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.

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-MoodyKac–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 LustzigLusztig using quantum groups.

Developed as a bridge between the theory of [[crystal basis|crystal bases]] arising from the work of [[Kashiwara]] and [[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.

==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|19461953}}</ref><ref>{{harvnb|Humphreys|1994}}</ref>

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

* For two highest weights λ, μ, find the decomposition of their tensor product ''L''(λ) <math>\otimes </math> ''L''(μ) into irreducible representations.

* 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 '''&lambda;λ''', determine the [[branching rule]] for decomposing the restriction of ''L''('''&lambda;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>

* [[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).

(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.)

* 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.

AnswersAttempts toat thesefinding questionssimilar 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 integersalgorithms ''without overcounting''. His work generalized the celebrated [[Littlewood-Richardson rule]] for the [[special linear group|special linear Lie algebra]] <math>\mathfrak{g}</math> &nbsp;=&nbsp;''sl''_n 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,Lie therealgebras had previously only been partialpartially success in finding algorithms with no overcountingsuccessful.<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|19881990}} and {{harvtxt|Sundaram|19881990}} give variants of [[Young tableaux]] which can be used to compute weight mutliplicitiesmultiplicities, branching rules and tensor products with fundamental representations for the remaining classical groupsLie algebras. {{harvtxt|Berenstein|Zelevinsky|2001}} discuss how their method using [[Convex polytope|convex polytopes]], proposed in 1988, is related to Littelmann paths and crystal bases.</ref>

{{harvtxt|Berenstein|Zelevinsky|2001}} discuss how their method using [[convex polytope]]s, proposed in 1988, is related to Littelmann paths and crystal bases. </ref>

LittlemannLittelmann's contribution was to give a unified combinatorial model that applied to all symmetrizable [[Kac-MoodyKac–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>.

The combinatorial data could be encoded in a coloured directed graph, with labels given by the simple roots.

Littelmann's main motivation<ref>{{harvnb|Littelmann|20071997}}</ref> was to reconcile two different aspects of representation theory:

* The standard monomial theory of Lakshmibai and Seshadri arising from the geometry of [[Schubert variety|Schubert varieties]].

*[[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}}.

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.

== Definitions ==

Let ''P'' be the [[weight lattice]] in the dual of a [[Cartan subalgebra]] of the [[semisimple Lie algebra]] <math>\mathfrak{g}</math>.

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>

such that &pi;π(0)&nbsp;=&nbsp;0 and &pi;π(1) is a [[weight lattice|weight]].

Let (''H''_α) be the basis of <math>\mathfrak{h}</math> consisting of "coroot" vectoctsvectors, 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''.

Define non-decreasing self-mappings ''l'' and ''r'' of [0,1] <math>\cap</math> '''Q''' by

Line 39 ⟶ 46:

:<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>

Thus ''l''(''t'') = 0 until the last time that ''h''(''s'') = ''M'' and ''r''(''t'') = 1 after the first time that ''h''(''s'') = ''M''.

Define new paths π_l and π_r by

:<math>\pi_lpi_r(t) = \pi(t) + lr(t) \alpha,\,\,\,\,\,\, \pi_rpi_l(t) = \pi(t) - rl(t) \alpha.</math>

The '''root operators''' ''e''_α and ''f''_α are defined on a basis vector [π] by

* <math>\displaystyle{ e_\alpha [\pi] = [\pi_r]} </math> if ''r'' (0) = 0 and 0 otherwise;

* <math> \displaystyle{f_\alpha [\pi] = [\pi_l]} </math> if ''l'' (1) = 1 and 0 otherwise.

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* ''s''_α [π] = [π] if ''k'' = 0;

* ''s''_α [π]= ''f''_α^''k'' [π] if ''k'' > 0;

* ''s''_α [π]= ''e''_α^{– ''k''} [π] if ''k'' >< 0.

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 successivlysuccessively applying the operators ''f''_α 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''_α.

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 successivly applying the operators ''f''_α 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''_α.

* The Littelmann graphs of two paths are isomorphic as coloured, directed graphs if and only if the paths have the same end point.

Line 72 ⟶ 78:

==Applications==

====Character formula====

If π(1) = λ, the multiplicity of the weight μ in ''L''(λ) is the number of paths from π to vertices σ in the Littelmann graph <math> \mathcal{G}_\pi </math> with σ(1) = μ.

====Character formula====

====Generalized Littlewood-Richardson rule====

If π(1) = λ, the multiplicity of the weight μ in ''L''(λ) is the number of paths from π to vertices σ in the Littelmann graph <math> \mathcal{G}_\pi </math> with σ(1) = μ.

====Generalized Littlewood-RichardsonLittlewood–Richardson rule====

Let π and σ be paths in the positive Weyl chamber with π(1) = λ and σ(1) = μ. Then

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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.

====Branching rule====

If <math>\mathfrak{g}_1</math> is the Levi component of a parabolic subalgebra of <math>\mathfrak{g}</math> with weight lattice ''P''₁ <math>\supset </math> ''P'' then

:<math> L(\lambda)|_{\mathfrak{g}_1} = \bigoplus_{\sigma} LL_{\mathfrak{g}_1}(\sigma(1)),</math>

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>.

== See also ==

* [[Crystal basis]]

==Notes==

{{reflist}}

== References ==

{{refbegin}}

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{{refend}}

[[Category:Representation theory]]

[[Category:Lie algebras]]

[[Category:CombinatoricsAlgebraic combinatorics]]