Abstract
We show that the homology of the partition algebras, interpreted as
appropriate Tor-groups, is isomorphic to that of the symmetric groups in a range
of degrees that increases with the number of nodes. Furthermore, we show that
when the defining parameter δ of the partition algebra is invertible, the homology
of the partition algebra is in fact isomorphic to the homology of the symmetric
group in all degrees. These results parallel those obtained for the Brauer algebras
in the authors’ earlier work, but with significant differences and difficulties in the
inductive resolution and high acyclicity arguments required to prove them. Our
results join the growing literature on homological stability for algebras, which
now encompasses the Temperley-Lieb, Brauer and partition algebras, as well as
the Iwahori-Hecke algebras of types A and B.
appropriate Tor-groups, is isomorphic to that of the symmetric groups in a range
of degrees that increases with the number of nodes. Furthermore, we show that
when the defining parameter δ of the partition algebra is invertible, the homology
of the partition algebra is in fact isomorphic to the homology of the symmetric
group in all degrees. These results parallel those obtained for the Brauer algebras
in the authors’ earlier work, but with significant differences and difficulties in the
inductive resolution and high acyclicity arguments required to prove them. Our
results join the growing literature on homological stability for algebras, which
now encompasses the Temperley-Lieb, Brauer and partition algebras, as well as
the Iwahori-Hecke algebras of types A and B.
| Original language | English |
|---|---|
| Journal | Pacific Journal of Mathematics |
| Publication status | Accepted/In press - 5 Dec 2023 |