With epsilon invertible and delta arbitrary, Rook-Brauer homology is isomorphic to symmetric group homology in all degrees and Motzkin homology vanishes in positive degrees.
Cohomology of rook-Brauer algebras and their subalgebras
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This paper studies the (co)homology of rook-Brauer algebras and their subalgebras. Our main results focus on the cohomology of rook-Brauer algebras (which is related to the cohomology of symmetric groups), the cohomology of Motzkin algebras (for which we obtain a vanishing result in positive degrees) and the cohomology of walled Brauer algebras (which is related to the cohomology of products of symmetric groups). Along the way we collect some cohomological analogues of known results for Temperley--Lieb algebras, Brauer algebras and rook algebras.
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Homology of Rook-Brauer Algebras and Motzkin Algebras
With epsilon invertible and delta arbitrary, Rook-Brauer homology is isomorphic to symmetric group homology in all degrees and Motzkin homology vanishes in positive degrees.