bott-theorem
A Python script to compute cohomology of irreducible homogeneous vector bundles on rational homogeneous varieties.
The content of this project is essentially the function coh
, all the rest is preparatory to it.
The arguments of coh
are a string and a list. The string determines the Dynkin type of the homogeneous vartiety and it must be either A, B, C, D, E or F. The list represents the coefficients of the weight of the bundle of which we compute cohomology, expressed in a basis of fundamental weights. The coefficients must be integers.
The output is a string [x,y] where x is the dimension of the unique nonzero cohomology group, and y is the degree of such cohomology.
Example: a vector bundle on G(2,4)
Consider the Grassmannian G(2,4) with tautological bundle U and quotient bundle Q.
The following code computes the cohomology of the tensor product of U^* with Q(-2), which is zero in every degree except for 1, where it is one-dimensional.
input:
import bott
print(bott.coh('A', [1, -2, 1]))
output:
[1, 1]