Littelmann path model

For path models in statistics, see Path analysis (statistics).

In mathematics, the Littelmann path model is a combinatorial device due to Peter Littelmann for computing multiplicities without overcounting in the representation theory of symmetrisable Kac–Moody algebras. Its most important application is to complex semisimple Lie algebras or equivalently compact semisimple Lie groups, the case described in this article. Multiplicities in irreducible representations, tensor products and branching rules can be calculated using a coloured directed graph, with labels given by the simple roots of the Lie algebra.

Developed as a bridge between the theory of crystal bases arising from the work of Kashiwara and Lusztig on quantum groups 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 weight as well as a pair of root operators acting on paths for each 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 groups going back to Hermann Weyl include:^[1]^[2]

For a given dominant weight λ, find the weight multiplicities in the irreducible representation L(λ) with highest weight λ.
For two highest weights λ, μ, find the decomposition of their tensor product L(λ) $\otimes$ L(μ) into irreducible representations.
Suppose that ${\mathfrak {g}}_{1}$ is the Levi component of a parabolic subalgebra of a semisimple Lie algebra ${\mathfrak {g}}$ . For a given dominant highest weight λ, determine the branching rule for decomposing the restriction of L(λ) to ${\mathfrak {g}}_{1}$ .^[3]

(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 explicit character formulas,^[4] followed by later combinatorial formulas of Hans Freudenthal, Robert Steinberg and Bertram Kostant; see 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 tableaux for the general linear Lie algebra ${\mathfrak {gl}}$ _n or the special linear Lie algebra ${\mathfrak {sl}}$ _n:^[5]^[6]^[7]^[8]

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–Richardson rule that describes both tensor product decompositions and branching from ${\mathfrak {gl}}$ _m+n to ${\mathfrak {gl}}$ _m $\oplus$ ${\mathfrak {gl}}$ _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.^[9]

Littelmann's contribution was to give a unified combinatorial model that applied to all symmetrizable Kac–Moody algebras and provided explicit subtraction-free combinatorial formulas for weight multiplicities, tensor product rules and branching rules. 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 simple root of ${\mathfrak {g}}$ . The combinatorial data could be encoded in a coloured directed graph, with labels given by the simple roots.

Littelmann's main motivation^[10] was to reconcile two different aspects of representation theory:

The standard monomial theory of Lakshmibai and Seshadri arising from the geometry of Schubert varieties.
Crystal bases arising in the approach to quantum groups of Masaki Kashiwara and George Lusztig. Kashiwara and Lusztig constructed canonical bases for representations of deformations of the universal enveloping algebra of ${\mathfrak {g}}$ depending on a formal deformation parameter q. In the degenerate case when q = 0, these yield crystal bases together with pairs of operators corresponding to simple roots; see 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 Hong & Kang (2002, p. xv). In the case of complex semisimple Lie algebras, there is a simplified self-contained account in Littelmann (1997) relying only on the properties of root systems; this approach is followed here.

Definitions

Let P be the weight lattice in the dual of a Cartan subalgebra of the semisimple Lie algebra ${\mathfrak {g}}$ .

A Littelmann path is a piecewise-linear mapping

\pi :[0,1]\cap {\mathbf {Q}}\rightarrow P\otimes _{{{\mathbf {Z}}}}{\mathbf {Q}}

such that π(0) = 0 and π(1) is a weight.

Let (H_α) be the basis of ${\mathfrak {h}}$ consisting of "coroot" vectors, dual to basis of ${\mathfrak {h}}$ * formed by simple roots (α). For fixed α and a path π, the function $h(t)=(\pi (t),H_{\alpha })$ has a minimum value M.

Define non-decreasing self-mappings l and r of [0,1] $\cap$ Q by

l(t)=\min _{{t\leq s\leq 1}}(1,h(s)-M),\,\,\,\,\,\,r(t)=1-\min _{{0\leq s\leq t}}(1,h(s)-M).

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

\pi _{r}(t)=\pi (t)+r(t)\alpha ,\,\,\,\,\,\,\pi _{l}(t)=\pi (t)-l(t)\alpha

The root operators e_α and f_α are defined on a basis vector [π] by

$\displaystyle {e_{\alpha }[\pi ]=[\pi _{r}]}$ if r (0) = 0 and 0 otherwise;
$\displaystyle {f_{\alpha }[\pi ]=[\pi _{l}]}$ if l (1) = 1 and 0 otherwise.

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.

Properties

Let ${\mathcal {A}}$ 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

the ${\mathcal {A}}$ -module generated by [π] depends only on π(1) = λ and has a Q-basis consisting of paths [σ];
the multiplicity of the weight μ in the integrable highest weight representation L(λ) is the number of paths σ with σ(1) = μ.

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_α acts as follows:

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 ${\mathcal {G}}_{\pi }$ is defined to be the coloured, directed graph having as vertices the non-zero paths obtained by successively 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.

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.

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 ${\mathcal {G}}_{\pi }$ with σ(1) = μ.

Generalized Littlewood–Richardson rule

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

L(\lambda )\otimes L(\mu )=\bigoplus _{\eta }L(\lambda +\tau (1)),

where τ ranges over paths in ${\mathcal {G}}_{\sigma }$ such that π $\star$ τ lies entirely in the positive Weyl chamber and the concatenation π $\star$ τ (t) is defined as π(2t) for t ≤ 1/2 and π(1) + τ( 2t – 1) for t ≥ 1/2.

Branching rule

If ${\mathfrak {g}}_{1}$ is the Levi component of a parabolic subalgebra of ${\mathfrak {g}}$ with weight lattice P₁ $\supset$ P then

L(\lambda )|_{{\mathfrak {g}}_{1}}=\bigoplus _{\sigma }L_{{\mathfrak {g}}_{1}}(\sigma (1)),

where the sum ranges over all paths σ in ${\mathcal {G}}_{\pi }$ which lie wholly in the positive Weyl chamber for ${\mathfrak {g}}_{1}$ .

Notes

↑ Weyl 1946
↑ Humphreys 1994
↑ Every complex semisimple Lie algebra ${\mathfrak {g}}$ is the complexification of the Lie algebra of a compact connected simply connected semisimple Lie group. The subalgebra ${\mathfrak {g}}_{1}$ corresponds to a maximal rank closed subgroup, i.e. one containing a maximal torus.
↑ 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).
↑ Littlewood 1950
↑ Macdonald 1979
↑ Sundaram 1990
↑ King 1990
↑ 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 King (1990) and 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. Berenstein & Zelevinsky (2001) discuss how their method using convex polytopes, proposed in 1988, is related to Littelmann paths and crystal bases.
↑ Littelmann 2007

References

Ariki, Susumu (2002), Representations of Quantum Algebras and Combinatorics of Young Tableaux, University Lecture Series, 26, American Mathematical Society, ISBN 0821832328
Berenstein, Arkady; Zelevinsky, Andrei (2001), "Tensor product multiplicities, canonical bases and totally positive varieties", Invent. Math., 143: 77–128, Bibcode:2001InMat.143...77B, doi:10.1007/s002220000102
Hong, Jin; Kang, Seok-Jin (2002), Introduction to Quantum Groups and Crystal Bases, Graduate Studies in Mathematics, 42, American Mathematical Society, ISBN 0821828746
King, Ronald C. (1990), "S-functions and characters of Lie algebras and superalgebras", Institute for Mathematics and Its Applications, IMA Vol. Math. Appl., Springer-Verlag, 19: 226–261, Bibcode:1990IMA....19..226K
Humphreys, James E. (1994), Introduction to Lie Algebras and Representation Theory (2 ed.), Springer-Verlag, ISBN 0-387-90053-5
Littelmann, Peter (1994), "A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras", Invent. Math., 116: 329–346, Bibcode:1994InMat.116..329L, doi:10.1007/BF01231564
Littelmann, Peter (1995), "Paths and root operators in representation theory", Ann. of Math., Annals of Mathematics, 142 (3): 499–525, doi:10.2307/2118553, JSTOR 2118553
Littelmann, Peter (1997), "Characters of Representations and Paths in ${\mathfrak {h}}$ _R*", Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 61: 29–49, doi:10.1090/pspum/061/1476490 [instructional course]
Littlewood, Dudley E. (1950), "The Theory of Group Characters and Matrix Representations of Groups", Nature, Oxford University Press, 146 (3709): 699, Bibcode:1940Natur.146..699H, doi:10.1038/146699a0
Macdonald, Ian G. (1979), Symmetric Functions and Hall Polynomials, Oxford University Press
Mathieu, Olivier (1995), Le modèle des chemins, Exposé No. 798, Séminaire Bourbaki (astérique), 37
Sundaram, Sheila (1990), "Tableaux in the representation theory of the classical Lie groups", Institute for Mathematics and Its Applications, IMA Vol. Math. Appl., Springer-Verlag, 19: 191–225, Bibcode:1990IMA....19..191S
Weyl, Hermann (1946), The classical groups, Princeton University Press

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