Sasha Pevzner, University of Minnesota, Twin Cities
Authors: Ayah Almousa, Michael Perlman, Alexandra Pevzner, Victor Reiner, Keller VandeBogert
2023 AWM Research Symposium
Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]

We give explicit, characteristic-free constructions of $\mathrm{GL}(V)$-equivariant minimal free resolutions of all isotypic components of the polynomial ring $S = \mathrm{Sym}(V)$ over its $d^{th}$ Veronese subalgebra $S^{(d)}$. These isotypic components come from an action of $\mathbb{Z}/d\mathbb{Z}$ on $S$ for which $S^{(d)}$ is the ring of invariants. The free modules appearing in the resolutions are (base changes of) Schur modules associated to ribbon or border strip diagrams, and the differential comes from a simple degree lowering map on a certain tensor algebra. We use these resolutions to compute $\mathrm{Hom}$ and $\mathrm{Tor}$ between these modules. This is based on joint work with Ayah Almousa, Michael Perlman, Victor Reiner, and Keller VandeBogert.

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