1
$\begingroup$

I am looking for a way to compute the determinant of a homogeneous vector bundle over any homogeous variety. I am aware of how these computations work for the $A_n$ case (i.e., for flag varieties), but I miss the theory for the general case.

Maybe there are some general exacts sequence as in the grassmannian case from which one can deduce the formulae, but I was not able to find anywhere anything about them.

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ What exactly do you want to compute? $\endgroup$ Commented Nov 3, 2024 at 15:05
  • 2
    $\begingroup$ Welcome new contributor. The dual Abelian group of the Picard group is generated by effective curve classes of torus orbit closures (for a sufficiently general one-parameter subgroup of a maximal torus). Thus you are reduced to computing degrees on these toric curves. One way to do this is to apply Atiyah-Bott localization. $\endgroup$ Commented Nov 3, 2024 at 15:55
  • $\begingroup$ In practise, what I am looking for is a way to compute the weight associated to the determinant line bundle of a given (irreducible) homogeneous vector bundle, of which I know the weight. $\endgroup$ Commented Nov 4, 2024 at 13:17

1 Answer 1

Reset to default
1
$\begingroup$

Irreducible homogeneous vector bundles correspond to irreducible representations of the Levi quotient of the corresponding parabolic group. The determinant of the bundle corresponding to the representation $V$ is the line bundle whose weight is the sum of all weights appearing in $V$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.