Type: Seminar

Modified: Thu 8 Jul 2021 1710; Thu 8 Jul 2021 1711

Distribution: World

Expiry: 8 Jul 2022

CalTitle1: SMRI Algebra and Geometry Online: Kumar -- Root components for tensor product of affine Kac-Moody Lie algebra modules

Auth: hannahb@10.48.16.40 (hbry8683) in SMS-SAML

SMRI Algebra and Geometry Online ’Root components for tensor product of affine Kac-Moody Lie algebra modules’ Shrawan Kumar (University of North Carolina) Friday Jul 23, 2021 11:00am-12:30pm (AEST) Register: https://uni-sydney.zoom.us/meeting/register/tZ0sc--qrzkpHdd5eE6IgQUYtXWfnEssOCIC This is a joint work with Samuel Jeralds. Let ð”¤ be an affine Kac-Moody Lie algebra and let Î», Âµ be two dominant integral weights for ð”¤. We prove that under some mild restriction, for any positive root Î², V(Î») âŠ— V(Âµ) contains V(Î» + Âµ â€“ Î²) as a component, where V(Î») denotes the integrable highest weight (irreducible) ð”¤-module with highest weight Î». This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V(Î») âŠ— V(Âµ). Then, we prove the corresponding geometric results including the higher cohomology vanishing on the ð’¢-Schubert varieties in the product partial flag variety ð’¢/ð’« Ã— ð’¢/ð’« with coefficients in certain sheaves coming from the ideal sheaves of ð’¢-sub Schubert varieties. This allows us to prove the surjectivity of the Gaussian map. Note: These seminars will be recorded, including participant questions (participants only when asking questions), and uploaded to the SMRI YouTube Channel https://www.youtube.com/c/SydneyMathematicalResearchInstituteSMRI Other upcoming SMRI events can be found here: https://mathematical-research-institute.sydney.edu.au/seminars/