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Hook algebras and determinantal varieties

  • Steven V Sam
  • Keller VandeBogert
  • Jerzy Weyman
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Abstract

In characteristic zero, we prove that the Tor algebras of the generic, symmetric, and skew-symmetric determinantal rings are Koszul with respect to a natural regrading. In particular, their Eilenberg--Moore spectral sequence degenerates, which allows us to explicitly determine the Poincaré series of their residue fields and answers a problem raised by Avramov. To do this, we construct the universal hook algebra and prove that it has a quadratic Gröbner basis and thus is Koszul. This leads to a new standard monomial construction of the irreducible polynomial GL-representations, alongside the Schur and Weyl functors, which is of independent interest.

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Cite

BibTeXRIS
BibTeX
@misc{sam2026hook,
  title = {{Hook algebras and determinantal varieties}},
  author = {Steven V Sam and Keller VandeBogert and Jerzy Weyman},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12675},
  eprint = {2610.12675},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12675},
}
RIS
TY  - UNPB
TI  - Hook algebras and determinantal varieties
AU  - Steven V Sam
AU  - Keller VandeBogert
AU  - Jerzy Weyman
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12675
UR  - https://arxiv.org/abs/2610.12675
AB  - In characteristic zero, we prove that the Tor algebras of the generic, symmetric, and skew-symmetric determinantal rings are Koszul with respect to a natural regrading. In particular, their Eilenberg--Moore spectral sequence degenerates, which allows us to explicitly determine the Poincaré series of their residue fields and answers a problem raised by Avramov. To do this, we construct the universal hook algebra and prove that it has a quadratic Gröbner basis and thus is Koszul. This leads to a new standard monomial construction of the irreducible polynomial GL-representations, alongside the Schur and Weyl functors, which is of independent interest.
ER  -