Torus quotients of some flag varieties

• Published in: Proceedings of the Indian Academy of Sciences. Mathematical sciences 2023-09, Vol.133 (2), Article 30
• Main Authors: Dake, Somnath, Senthamarai Kannan, S, Venkata Subrahmanyam, K
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics
• ISSN: 0973-7685; 0973-7685
• DOI: 10.1007/s12044-023-00744-4

Consider the Plücker embedding of the Grassmannian of three dimensional subspaces of a seven dimensional space, G 3 , 7 , in projective space P ( ⋀ 3 C 7 ) . The maximal torus of diagonal matrices in S L ( 7 , C ) acts on G 3 , 7 and so it is natural to seek geometric properties of the GIT quotient of G 3 , 7 with respect to this action. Using the notation L ( ω 3 ) for the line bundle on G 3 , 7 coming from its Plücker embedding, it is known that the twisted line bundle L ( 7 ω 3 ) descends to the GIT quotient. We show that the GIT quotient is projectively normal for this polarization. We compute the generators of the ring of torus invariants by computing the Hilbert basis of certain polyhedral cones. In this paper, we also begin a study of the torus quotients of S L ( 5 , C ) / Q , when Q is the intersection of two maximal parabolic subgroups in S L ( 5 , C ) containing a Borel subgroup. We describe the GIT quotient explicitly in some cases and prove geometric properties of the embedding of the quotient when the line bundle descends to it.