The root subsystem of the root system R generated by the roots
αa1, ..., αak where a={a1, ..., ak} is a set of
integers.
The root subsystem of the root system R generated by the roots
αs1, ..., αsk where s=[s1, ..., sk] is a sequence
of integers.
In this version the roots must be simple in the root subsystem (i.e. none of
them may be a summand of another), otherwise an error is signalled.
The simple roots will appear in the subsystem in the given order.
Returns true if and only if the root system R1 is a subset of the root system R2.
If true, returns an injection as sequence of roots as second return value.
DirectSum(R1, R2) : RootSys, RootSys -> RootSys
The direct sum of the root systems R1 and R2. The root space of the result
is the direct sum of the root spaces of R1 and R2.
The union of the root systems R1 and R2. The root systems must have the
same root space, which will also be the root space of the result.
> R := RootSystem("A1A1");
> R1 := sub<R|[1]>;
> R2 := sub<R|[2]>;
> R1 + R2;
Root system of dimension 4 of type A1 A1
> R1 join R2;
Root system of dimension 2 of type A1 A1
> R1 := RootSystem("A3T2B4T3");
> R2 := RootSystem("T3G2T4BC3");
> R1 + R2;
Root system of dimension 24 of type A3 B4 G2 BC3
> R1 join R2;
Root system of dimension 12 of type A3 B4 G2 BC3
IndecomposableSummands(R) : RootDtm -> [], RootDtm, Map
The set of irreducible direct summands of the semisimple root system R.
The dual of the root system R, obtained by swapping the roots
and coroots.
The root system consisting of all indivisible roots of the root system R.
> R1 := RootSystem("H4");
> R2 := RootSystem("B4");
> R1 + Dual(R2);
Root system of type H4 C4
> R := RootSystem("BC2");
> I := IndivisibleSubsystem(R); I;
I: Root system of type B2
> I subset R;
true [ 1, 2, 3, 5, 7, 8, 9, 11 ]
V2.28, 13 July 2023