Some homological properties of category $\mathcal{O}$, V
Representation Theory
2023-02-27 v3
Abstract
We compute projective dimension of translated simple modules in the regular block of the BGG category in terms of Kazhdan-Lusztig combinatorics. This allows us to determine which projectives can appear at the last step of a minimal projective resolution for a translated simple module, confirming a conjecture by Johan K{\aa}hrstr\"om. We also derive some inequalities, in terms of Lusztig's -function, for possible degrees in which the top (or socle) of a translated simple module can live. Finally, we relate Kostant's problem with decomposability and isomorphism of translated simple modules, addressing yet another conjecture by Johan K{\aa}hrstr\"om.
Keywords
Cite
@article{arxiv.2007.00342,
title = {Some homological properties of category $\mathcal{O}$, V},
author = {Hankyung Ko and Volodymyr Mazorchuk and Rafael Mrđen},
journal= {arXiv preprint arXiv:2007.00342},
year = {2023}
}