The global dimension of the algebra of the monoid of all partial functions on an $n$-set as the algebra of the EI-category of epimorphisms between subsets
Abstract
We prove that the global dimension of the complex algebra of the monoid of all partial functions on an n-set is $n-1$ for all $n\geq 1$. This is also the global dimension of the complex algebra of the category of all epimorphisms between subsets of an $n$-set. In our proof we use standard homological methods as well as combinatorial techniques associated to the representation theory of the symmetric group. As part of the proof, we obtain a partial description of the Cartan matrix of these algebras.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2017
- DOI:
- 10.48550/arXiv.1801.00357
- arXiv:
- arXiv:1801.00357
- Bibcode:
- 2018arXiv180100357S
- Keywords:
-
- Mathematics - Representation Theory;
- Mathematics - Group Theory;
- 20M30;
- 20M50;
- 20C15
- E-Print:
- 27 pages