setWeights(PolynomialRing,DynkinType,List) -- attach (Lie theoretic) weights to the variables of a ring

Synopsis

Function: setWeights
Usage:

T = setWeights(R,D,L)
Inputs:
- R, a polynomial ring,
- D, an instance of the type DynkinType,
- L, a list, a table of weights given as lists of integers
Outputs:
- T, a tally, the highest weights decomposition of L
Consequences:
- Two keys are created in the hashTable of R: LieWeights, with value a matrix whose rows are the weights in the list L; GroupActing, with value the DynkinType D.

Description

Let $G$ be a semisimple Lie group of Dynkin type D which acts on R compatibly with the grading. Let $T\subset G$ be a maximal torus and assume the variables in R are weight vectors for the action of $T$. This function is used to assign a weight to each variable of R. Knowing these weights allows Macaulay2 to return the weight of monomials of R upon request.

For more information on inputting the Dynkin type of the group the user should consult the documentation of the WeylGroups package.

Weights are expressed with respect to the basis of fundamental weights in the weight lattice associated to the root system of the given type D. In this package, each weight $w$ is represented by a list of integers, namely the coefficients of $w$ in the basis of fundamental weights.

In the following example, the polynomial ring R is the symmetric algebra over $V$, where $V=\mathbb{C}^4$ and is acted upon by the group $SL_4 (\mathbb{C})$.

i1 : R=QQ[x_1..x_4]

o1 = R

o1 : PolynomialRing

i2 : D=dynkinType{{"A",3}}

o2 = DynkinType{{A, 3}}

o2 : DynkinType

i3 : W={{1,0,0},{-1,1,0},{0,-1,1},{0,0,-1}}

o3 = {{1, 0, 0}, {-1, 1, 0}, {0, -1, 1}, {0, 0, -1}}

o3 : List

i4 : setWeights(R,D,W)

o4 = Tally{{1, 0, 0} => 1}

o4 : Tally

Ways to use this method: