[GAP Forum] naming convention for basis of highest weight modules

[Willem de Graaf]

Dear R.N. Tsai,

You asked about the way the basis elements of a highest weight module
are printed. This is described in the manual section: 64.14-4
IsWeightRepElement.

A basis vector is printed as mon*v0. Here 'v0' is the highest weight vector
and 'mon' is a monomial in the root vectors corresponding to the negative
roots (sometimes called lowering operators).

Example:

gap> L:= SimpleLieAlgebra("A", 2, Rationals);;
gap> V:= HighestWeightModule( L, [1,1] );;
gap> v0:=Basis(V)[1];
1*v0
gap> y:= ChevalleyBasis(L)[2];;
gap> y[1]^(y[2]^v0);
y1*y2*v0


If you have questions about this, then please ask.

Best wishes,

Willem de Graaf

On Thu, Dec 20, 2012 at 7:49 AM, R.N. Tsai <r_n_tsai at yahoo.com> wrote:
> Dear gap-forum,
>
> Is the naming convention for the basis of highest weight modules described anywhere?
>
> For example :
>
> gap>L:=SimpleLieAlgebra("A",2,Rationals);;
> gap>M1:=HighestWeightModule(L,[1,0]);;
> gap>B1:=Basis(M1);;
> gap>M2:=TensorProductOfAlgebraModules(M1,M1);;
> gap>B2:=Basis(M2);;
>
> gap> BasisVectors(B1);
> [ 1*v0, y1*v0, y3*v0 ]
> gap> BasisVectors(B2);
> [ 1*(1*v0<x>1*v0), 1*(1*v0<x>y1*v0), 1*(1*v0<x>y3*v0), 1*(y1*v0<x>1*v0), 1*(y1*v0<x>y1*v0), 1*(y1*v0<x>y3*v0),
>   1*(y3*v0<x>1*v0), 1*(y3*v0<x>y1*v0), 1*(y3*v0<x>y3*v0) ]
>
> 1*v0,y1*v0,y3*v0 seems an odd choice without knowing where it came from.
>
> Thanks,
> r.n.
>
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