Given a quantized enveloping algebra U, corresponding to a root
datum R of rank r and a sequence w of non-negative
integers of length r returns the sequence consisting
of the elements of the canonical basis of the negative part of U
that are of weight ν,
where ν denotes the linear combination
of the simple roots of R defined by w (i.e., ν = w[1]α1 + ... + w[r]αr and α1, ..., αr denote the
simple roots of R).
> R:= RootDatum("F4");
> U:= QuantizedUEA(R);
> c:= CanonicalElements(U, [1,2,1,1]); c;
[
F_1*F_3^(2)*F_9*F_24,
q*F_1*F_3^(2)*F_9*F_24 + F_1*F_3^(2)*F_23,
q^4*F_1*F_3^(2)*F_9*F_24 + F_1*F_3*F_7*F_24,
q^5*F_1*F_3^(2)*F_9*F_24 + q^4*F_1*F_3^(2)*F_23 + q*F_1*F_3*F_7*F_24 +
F_1*F_3*F_21,
q^4*F_1*F_3^(2)*F_9*F_24 + F_2*F_3*F_9*F_24,
q^5*F_1*F_3^(2)*F_9*F_24 + q^4*F_1*F_3^(2)*F_23 + q*F_2*F_3*F_9*F_24 +
F_2*F_3*F_23,
(q^6 + q^2)*F_1*F_3^(2)*F_9*F_24 + q^2*F_1*F_3*F_7*F_24 +
q^2*F_2*F_3*F_9*F_24 + F_2*F_7*F_24,
(q^7 + q^3)*F_1*F_3^(2)*F_9*F_24 + (q^6 + q^2)*F_1*F_3^(2)*F_23 +
q^3*F_1*F_3*F_7*F_24 + q^3*F_2*F_3*F_9*F_24 + q^2*F_1*F_3*F_21 +
q^2*F_2*F_3*F_23 + q*F_2*F_7*F_24 + F_2*F_21,
q^8*F_1*F_3^(2)*F_9*F_24 + q^4*F_1*F_3*F_7*F_24 + q^4*F_2*F_3*F_9*F_24 +
q^2*F_2*F_7*F_24 + F_3*F_4*F_24,
q^9*F_1*F_3^(2)*F_9*F_24 + q^8*F_1*F_3^(2)*F_23 + q^5*F_1*F_3*F_7*F_24 +
q^5*F_2*F_3*F_9*F_24 + q^4*F_1*F_3*F_21 + q^4*F_2*F_3*F_23 +
q^3*F_2*F_7*F_24 + q*F_3*F_4*F_24 + q^2*F_2*F_21 + F_3*F_18
]
> b:= BarAutomorphism(U);
> [ b(u) eq u : u in c ];
[ true, true, true, true, true, true, true, true, true, true ]
All elements of the canonical basis are invariant under the bar-automorphism.
In the next example we show how to use the crystal graph to determine
whether an element of the canonical basis acting on the highest weight
vector of an irreducible module gives zero or not.
> U:= QuantizedUEA(RootDatum("A2"));
> G, p:= CrystalGraph(RootDatum(U), [1,1]);
> e:= Edges(G);
> for edge in e do
> print edge, Label(edge);
> end for;
[1, 2] 1
[1, 3] 2
[2, 4] 2
[3, 5] 1
[4, 6] 2
[5, 7] 1
[6, 8] 1
[7, 8] 2
We see that f
α1f
α2f
α2f
α1(p
1)
= p
8 (where p
i is the i-th path in p). We apply the
same sequence of Kashiwara operators to the identity element
of U.
> Falpha(Falpha(Falpha(Falpha(One(U), 1), 2), 2), 1);
F_1*F_2*F_3
Now the element of the canonical basis with this principal
monomial (see Section
The Canonical Basis)
acting on the highest weight vector of the irreducible
module with highest weight
[1,1] gives a non-zero result.
The weight of this monomial is 2α
1 + 2α
2. All other
elements of the canonical basis of this weight give zero,
as there is only one point of the crystal graph that gives a monomial
of this weight.
> V:= HighestWeightModule(U, [1,1]);
> ce:= CanonicalElements(U, [2,2]);
> ce;
[
F_1^(2)*F_3^(2),
(q^3 + q)*F_1^(2)*F_3^(2) + F_1*F_2*F_3,
q^4*F_1^(2)*F_3^(2) + q*F_1*F_2*F_3 + F_2^(2)
]
> v0:= V.1;
> ce[2]^v0;
V: ( 0 0 0 0 0 0 0 -1/q)
> ce[1]^v0;
V: (0 0 0 0 0 0 0 0)
> ce[3]^v0;
V: (0 0 0 0 0 0 0 0)
V2.28, 13 July 2023