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]