Homology of Rook-Brauer Algebras and Motzkin Algebras
Khoa Ta
TL;DR
The paper proves that, when $\epsilon$ is invertible in a unital commutative ring $R$ and for any $\delta\in R$, the homology of the Rook-Brauer algebras $\mathscr{RB}_n(R,\delta,\epsilon)$ (interpreted as $\mathrm{Tor}$-groups) is isomorphic to the homology of the symmetric group $\mathfrak{S}_n$ in all degrees, while the Motzkin algebras $\mathcal{M}_n(R,\delta,\epsilon)$ have vanishing homology in positive degrees. The authors achieve this using inductive resolution, a method that extends Shapiro-type lemmas to diagram algebras when flatness over subalgebras may fail. They construct explicit resolutions of quotients $\mathscr{RB}_n/J_X$ and $\mathcal{M}_n/J_X$ by carefully chosen submodules and demonstrate Tor-vanishing after tensoring with the trivial module. The results establish homological stability in both families, aligning RB-algebra homology with that of $\mathfrak{S}_n$ and showing Motzkin-homology stability/vanishing, thereby generalizing and simplifying prior works which required stronger invertibility assumptions on $\delta$ as well as $\epsilon$.
Abstract
Using the technique of inductive resolution introduced in arXiv:2303.07979, we prove that the homology of Rook-Brauer Algebra, interpreted as appropriate Tor-group, is isomorphic to that of symmetric group for all degrees under the assumption that $ε$ in $R$ is invertible; furthermore, we also prove the homology of the Motzkin algebras vanishes in positive degrees under the same assumption. These results thereby establish homological stability of both algebras.