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 νdenotes 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.
> 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] 2We see that fα1fα2fα2fα1(p1) = p8 (where pi 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_3Now 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 2α1 + 2α2. 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)