getWeights(RingElement) -- retrieve the (Lie theoretic) weight of a monomial

Synopsis

Function: getWeights
Usage:

w = getWeights(r)
Inputs:
- r, a ring element,
Outputs:
- w, a list,

Description

If r is a monomial, this will return the weight of r which is simply the sum of the weights of the variables in the support of r each counted with multiplicity equal to its exponent. If r is a polynomial, then the weight of the leading monomial is returned (which may differ from the weights of the other monomials of r).

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];

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

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

i4 : setWeights(R,D,W);

i5 : getWeights(x_1^4*x_2*x_4^6)

o5 = {3, 1, -6}

o5 : List

Caveat

The weight of 0 is undefined, so an error is returned if r is 0.

Ways to use this method: