n environment to decompose the i-th Hochschild cohomology on some homogeneous varieties X=G/P by Hoschild-Kostant-Rosenberg.
Hochschild-Kostant-Rosenberg_decomposition / src / HochschildKostantRosenberg_decomposition / __pycache__ / minimal_irreducible_homogeneous_variety.cpython-311.pyc
383 viewsLicense: GPL3
ubuntu2204
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Return the dimension of ``self`.
INPUT:
- ``self`` -- minimal_irreducible_homogeneous_variety; the minimal irreducible homogeneous variety X=G/P.
OUTPUT:
- ``Output`` -- Integer; the dimension of X=G/P
ALGORITHM:
Thanks to the post by Pieter Belmans concerning the dimension of partial flag varieties
from Jun 14th, 2017 on his blog (cf. to [Blog_PieterBelmans]_). The link is
https://pbelmans.ncag.info/blog/2017/06/14/dimensions-of-partial-flag-varieties/
(Date: Apr 21st, 2021).
For the Borel group B ⊂ G: dim B = # of positive roots + rank (which accounts for the center)
For G: dim G = # of roots in the root system + rank (which accounts for the center)
i.e. # of roots in the root system = # of positive roots + # negative roots
and # of positive roots = # negative roots
so # of roots in the root system = 2 * # of positive roots
For the Borel group B ⊂ G: dim B = # of positive roots + rank (which accounts for the center)
For P: dim P = dim B + # of negative roots which are added to construct P
REFERENCE:
[Blog_PieterBelmans] https://pbelmans.ncag.info/blog/
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