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#W canbas.gd QuaGroup Willem de Graaf
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## Declarations for operations, attributes connected to canonical bases.
##
#1
## Functions for computing elements of the canonical basis of
## a quantized uea.
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#C IsCanonicalBasisOfQuantumUEA( <obj> )
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DeclareCategory( "IsCanonicalBasisOfQuantumUEA", IsObject );
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#O PBWElements( <B>, <rt> )
##
## Here <B> is the canonical basis of a quantized uea, and `rt' a list
## of non-negative integers representing a positive element of the
## root lattice. This function returns the elements of <B>, of weight <rt>
## (written as linear combinations of the elements of the PBW-basis).
##
DeclareOperation( "PBWElements", [ IsCanonicalBasisOfQuantumUEA, IsList ] );
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#O MonomialElements( <B>, <rt> )
##
## This does the same as `PBWElements', except that the elements are
## written as linear combinations of monomials in the generators.
##
DeclareOperation( "MonomialElements", [IsCanonicalBasisOfQuantumUEA,IsList ]);
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#O Strings( <B>, <rt> )
##
## Adapted strings corresponding to the elements of the canonical basis.
## Here a string like `[ 1, 2, 2, 3 ]' corresponds to the string
## $\tilde{F}_1^2\tilde{F}_2^3 \cdot 1$, where $\tilde{F}_i$ is the
## $i$-th Kashiwara operator. If an element like that occurs on the
## `k'-th position of the output of `Strings( <B>, <rt> )' then
## the crystal element of the `k'-th element of `PBWElements( <B>, <rt> )'
## is given by this string. (BETTER description!!).
##
DeclareOperation( "Strings", [IsCanonicalBasisOfQuantumUEA,IsList ]);
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#A CrystalBasis( <V> )
##
## Here <V> is a finite-dimensional left module over a quantized enveloping
## algebra. This function returns a basis of <V> that is also a basis
## over ${\bf Z}[q]$ of the ${\bf Z}[q]$-lattice generated by all
## $\tilde{F}_{\alpha_{i_1}}\cdots \tilde{F}_{\alpha_{i_r}}\cdot v_0$.
## REM: only works if the basis vectors of Basis(<V>) are all weight
## vectors! (Usually the case).
##
DeclareAttribute( "CrystalBasis", IsAlgebraModule );
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#C IsCrystalVector( <obj> )
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## Category of crystal vectors.
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DeclareCategory( "IsCrystalVector", IsObject );
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#A PBWMonomial( <v> )
##
## A crystal vector always has the form `m^v0', where v0 is a highest
## weight vector, and `m' a PBW-monomial. This function returns that
## monomial.
DeclareAttribute( "PBWMonomial", IsCrystalVector );
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#A CrystalVectors( <V> )
##
## Here <V> is a finite-dimensional left module over a quantized enveloping
## algebra.
## Let <B> be the crystal basis of <V> (i.e., output by `CrystalBasis').
## This function returns a list of cosets of the basis elements of
## <B> mod $q L(\lambda)$.
##
DeclareAttribute( "CrystalVectors", IsAlgebraModule );
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#O StringMonomial( <mon> )
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## Computes the "string" corresponding to the PBW-monomial <mon>.
##
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DeclareOperation( "StringMonomial", [ IsQEAElement ] );
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#O PrincipalMonomial( <p> )
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## Computes the principal monomial of an element of the canonical basis.
##
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DeclareOperation( "PrincipalMonomial", [ IsQEAElement ] );
