# ALGEBRAIC COMBINATORICS

Integral Schur–Weyl duality for partition algebras
Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 371-399.

Let $\mathbf{V}$ be a free module of rank $n$ over a commutative ring $𝕜$. We prove that tensor space ${\mathbf{V}}^{\otimes r}$ satisfies Schur–Weyl duality, regarded as a bimodule for the action of the group algebra of the Weyl group of $\mathrm{GL}\left(\mathbf{V}\right)$ and the partition algebra ${𝒫}_{r}\left(n\right)$ over $𝕜$. We also prove a similar result for the half partition algebra.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.214
Classification: 16G99,  20B30
Keywords: Schur–Weyl duality, partition algebras, symmetric groups, invariant theory.
Bowman, Chris 1; Doty, Stephen 2; Martin, Stuart 3

1 Department of Mathematics University of York Heslington, York, YO10 5DD, UK
2 Department of Mathematics and Statistics Loyola University Chicago Chicago, IL 60660 USA
3 DPMMS, Centre for Mathematical Sciences Wilberforce Road Cambridge, CB3 0WB, UK
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Bowman, Chris; Doty, Stephen; Martin, Stuart. Integral Schur–Weyl duality for partition algebras. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 371-399. doi : 10.5802/alco.214. https://alco.centre-mersenne.org/articles/10.5802/alco.214/

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