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#W paths.gd QuaGroup Willem de Graaf
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## Declarations for LS-paths, and the path model.
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#C IsLSPath( <p> )
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## The category of LS-paths.
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DeclareCategory( "IsLSPath", IsObject );
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#O DominantLSPath( <R>, <wt> )
##
## Here <R> is a root system, and <wt> a dominant weight in the weight
## lattice of <R>. This function returns the LS-path that is the line
## from the origin to <wt>.
##
DeclareOperation( "DominantLSPath", [ IsRootSystem, IsList ] );
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#O Falpha( <o>, <i> )
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## Here <o> is an object, which can be (A) an LS-path, (B) a monomial in the
## negative part of a quantized enveloping algebra, or (C) an element of a
## module over a quantized enveloping algebra. Furthermore,
## <i> an integer. This function returns the object obtained from <o> by
## (A) applying the root operator $f_{\alpha_i}$, where $\alpha_i$ is the
## <i>-th simple root, or (B,C) applying the Kashiwara operator
## $\tilde{F}_i$, corresponding to the <i>-th simple root.
##
DeclareOperation( "Falpha", [ IsObject, IsInt ] );
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##
#O Ealpha( <p>, <i> )
##
## Here <o> is an object, which can be (A) an LS-path, (B) a monomial in the
## negative part of a quantized enveloping algebra, or (C) an element of a
## module over a quantized enveloping algebra. Furthermore,
## <i> an integer. This function returns the object obtained from <o> by
## (A) applying the root operator $e_{\alpha_i}$, where $\alpha_i$ is the
## <i>-th simple root, or (B,C) applying the Kashiwara operator
## $\tilde{E}_i$, corresponding to the <i>-th simple root.
##
DeclareOperation( "Ealpha", [ IsObject, IsInt ] );
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#A WeylWord( <p> )
##
## Here <p> is an LS-path in the orbit (under the root operators)
## of a dominant LS-path $p_0$ ending in the dominant weight $\lambda$.
## This means that the first direction of <p> is of the form
## $s_1\cdots s_m(\lambda)$. This function returns the reduced
## word $s_1\cdots s_m$.
##
DeclareAttribute( "WeylWord", IsLSPath );
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#A EndWeight( <p> )
##
## Here <p> is an LS-path; this function returns the weight that is
## the endpoint of <p>.
##
DeclareAttribute( "EndWeight", IsLSPath );
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#A LSSequence( <p> )
##
## returns the two sequences (of weights and rational numbers) that
## define the LS-path <p>.
##
DeclareAttribute( "LSSequence", IsLSPath );
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#F CrystalGraph( <R>, <wt> )
#F CrystalGraph( <V> )
##
## This function returns a record describing the crystal graph corresponding
## to the input
## Denote the output by <r>; then <r>`.points' is the list of
## points of the graph. Furthermore, <r>`.edges' is a list
## of edges of the graph; this is a list of lists of the form
## `[ [ i, j ], u ]'. This mean that point `i' (i.e., the point
## on position `i' in <r>.`points') is connected to
## point `j', and that the edge has label `u'.
##
## The input can have two formats. First it can be a root system
## <R> together with a dominant weight <wt>. In this case <r>.`points'.
## is a set of LS-paths.
##
DeclareGlobalFunction( "CrystalGraph" );
