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 and Root Operators (ex-cccc8c)

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> 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 ]

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

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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: Crystal Graph of a Representation (ex-3f9751)

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> 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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