Elements of the Canonical Basis

CanonicalElements(U, w): AlgQUE, SeqEnum -> SeqEnum: 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 \(\nu\), where \(\nu\) denotes the linear combination of the simple roots of \(R\) defined by \(w\) (i.e., \(\nu = w[1]\alpha_1+ \cdots +w[r]\alpha_r\) and \(\alpha_1,\ldots, \alpha_r\) denote the simple roots of \(R\)).

Example: Canonical Basis Elements (ex-60f4ee)

> 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.

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Example: Canonical Basis and Crystal Graphs (ex-7c2231)

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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 \(f_{\alpha_1}f_{\alpha_2}f_{\alpha_2}f_{\alpha_1}(p_1) = p_8\) (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 \(2\alpha_1+2\alpha_2\). 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)

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Elements of the Canonical Basis#

Elements of the Canonical Basis