Littelmann path model: Difference between revisions

Revision as of 02:08, 24 November 2013

Revision as of 22:46, 11 March 2014

Line 1:

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 Lustzig using quantum groups.

Line 15:

* [[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).

* The celebrated [[Littlewood-RichardsonLittlewood–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>

Littelmann'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.

Line 30:

== 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

Line 38:

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

Let (''H''_α) 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''.

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

Line 44:

:<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_r(t)= \pi(t) + r(t) \alpha,\,\,\,\,\,\, \pi_l(t) = \pi(t) - l(t)\alpha</math>

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

Line 77:

==Applications==

===Character formula===

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.

===Generalized Littlewood-Richardson rule===

===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

Line 102:

* [[Crystal basis]]

==Notes==

{{reflist}}

== References ==

*{{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|idisbn =ISBN 0821832328}}

*{{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 }}

*{{citation|title=Introduction to Quantum Groups and Crystal Bases|first=Jin|last= Hong|first2=Seok-Jin|last2= Kang|year= 2002|idisbn=ISBN 0821828746

|series=Graduate Studies in Mathematics|volume=42|publisher=American Mathematical Society}}

*{{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}}