The Path Model#

In this section we describe functions for working with Littelmann’s path model (cf. Section The Path Model). A special role is played by the zero path. The path operators cannot be applied to the zero path. However, on some occasions they do produce the zero path.

DominantLSPath(R, hw): RootDtm, SeqEnum -> PathLS#: Given a root datum \(R\) and a sequence \(hw\) of non-negative integers returns the path that is the straight line from the origin to \(hw\).

Falpha(p, i): PathLS, RngIntElt -> PathLS#: Given a (non-zero) path \(p\) and an integer \(i\) between \(1\) and the rank of the root datum returns the result of applying the path operator \(f_{\alpha_i}\) to \(p\) (where \(\alpha_i\) is the \(i\)-th simple root).

Ealpha(p, i): PathLS, RngIntElt -> PathLS#: Given a (non-zero) path \(p\) and an integer \(i\) between \(1\) and the rank of the root datum returns the result of applying the path operator \(e_{\alpha_i}\) to \(p\) (where \(\alpha_i\) is the \(i\)-th simple root).

WeightSequence(p): PathLS -> SeqEnum#: For a path \(p\) this returns the sequence of weights that, along with the sequence of rational numbers, defines the path (cf. Section The Path Model).

RationalSequence(p): PathLS -> SeqEnum#: For a path \(p\) this returns the sequence of rational numbers that, along with the sequence of weights, defines the path (cf. Section The Path Model).

EndpointWeight(p): PathLS -> ModTupRngElt#: Returns the weight which is the end point of the path \(p\).

Shape(p): PathLS -> ModTupRngElt#: Returns the weight which is the shape of the path \(p\).

WeylWord(p): PathLS -> SeqEnum#: Returns a reduced expression for the element \(\sigma\) of the Weyl group, of shortest length such that \(\sigma(\lambda) = \nu_1\), where \(\lambda\) is the shape of the path \(p\), and \(\nu_1\) is the first weight in the sequence WeightSequence(p). The reduced expression is represented as a sequence of integers between \(1\) and the rank of the root datum. In this sequence the index \(i\) represents the \(i\)-th simple reflection.

IsZero(p): PathLS -> BoolElt#: Returns true if the path \(p\) is the zero path, false otherwise.

p1 eq p2: PathLS, PathLS -> BoolElt#: Returns true if the paths \(p1\) and \(p2\) are equal, false otherwise.

Example: LS Paths (ex-cccc8c)#

> R:= RootDatum("B2");
> p:= DominantLSPath(R, [ 2, 3 ]);
> p;
LS-path of shape (2 3) ending in (2 3)

> Falpha(p, 1);
LS-path of shape (2 3) ending in (0 5)

> Ealpha(Falpha(p, 1), 1);
LS-path of shape (2 3) ending in (2 3)

> p1:= Falpha(Falpha(Falpha(p, 1), 2), 1);
> p1;
LS-path of shape (2 3) ending in (-1  5)

> WeightSequence(p1);
[
    (5 -7),
    (-2  7)
]
> RationalSequence(p1);
[ 0, 1/7, 1 ]
> WeylWord(p1);
[ 2, 1 ]

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So \(s_2s_1(2,3) = (5,-7)\).

CrystalGraph(R, hw): RootDtm, SeqEnum -> GrphDir, SeqEnum#: For a root datum \(R\) and a sequence of non-negative integers \(hw\) (of length equal to the rank of the root datum), this function returns the corresponding crystal graph \(G\), along with a sequence of paths. The graph \(G\) is a directed labelled graph. The labels on the edges are integers between \(1\) and the rank of the root system. If there is an edge from \(i\) to \(j\) with label \(s\), then \(f_{\alpha_s}(p_i) = p_j\), where \(p_i,p_j\) are the \(i\)-th and \(j\)-th elements of the sequence of paths returned by this function (and \(f_{\alpha_s}\) is the root operator corresponding to the \(s\)-th simple root). In other words, the \(i\)-th path is the \(i\)-th point of the graph \(G\).

Example: Cryst Grph (ex-3f9751)#

> R:= RootDatum("G2");
> G, pp:= CrystalGraph(R, [0,1]);
> G;
Digraph
Vertex  Neighbours

1       2 ;
2       3 ;
3       4 ;
4       5 6 ;
5       7 ;
6       8 ;
7       9 ;
8       10 ;
9       11 ;
10      11 ;
11      12 ;
12      13 ;
13      14 ;
14      ;

> e:= Edges(G);
> e[10];
[9, 11]
> Label(e[10]);
1
> Falpha(pp[9], 1) eq pp[11];
true

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