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Equivariant resolutions over Veronese rings

Ayah Almousa, Michael Perlman, Alexandra Pevzner, Victor Reiner, Keller VandeBogert

Abstract

Working in a polynomial ring $S=\mathbf{k}[x_1,\ldots,x_n]$ where $\mathbf{k}$ is an arbitrary commutative ring with $1$, we consider the $d^{th}$ Veronese subalgebras $R=S^{(d)}$, as well as natural $R$-submodules $M=S^{(\geq r, d)}$ inside $S$. We develop and use characteristic-free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple $GL_n(\mathbf{k})$-equivariant minimal free $R$-resolutions for the quotient ring $\mathbf{k}=R/R_+$ and for these modules $M$. These also lead to elegant descriptions of $\mathrm{Tor}^R_i(M,M')$ for all $i$ and $\mathrm{Hom}_R(M,M')$ for any pair of these modules $M,M'$.

Equivariant resolutions over Veronese rings

Abstract

Working in a polynomial ring where is an arbitrary commutative ring with , we consider the Veronese subalgebras , as well as natural -submodules inside . We develop and use characteristic-free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple -equivariant minimal free -resolutions for the quotient ring and for these modules . These also lead to elegant descriptions of for all and for any pair of these modules .
Paper Structure (19 sections, 31 theorems, 182 equations)

This paper contains 19 sections, 31 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Representation theory preliminaries
  3. Some generalities on equivariant resolutions
  4. Review of polynomial $GL(V)$ representations
  5. Schur-Weyl duality
  6. Ribbon Schur Functors
  7. Concatenation and near-concatenation
  8. Resolving concatenations by near-concatenations
  9. The complex of ribbons and proof of Theorem \ref{['skew-Schur-functor-theorem']}
  10. The complex of ribbons
  11. Resolution of the Veronese modules
  12. Tor and Koszulity
  13. A symmetric function identity
  14. A curious symmetry
  15. Poset homology
  16. ...and 4 more sections

Key Result

Theorem 1.1

Fix $d, r \geq 1$ and let $R={S^{(d)}}$ and $M={S^{({\geq r},{d})}}$ as above. Then one has an explicit $GL(V)$-equivariant minimal $R$-free resolution of $M$ of the form whose $i^{th}$ resolvent $R \otimes_{\mathbf{k}} {\mathbb{S}}^{\sigma(d^i,r)}(V)$ has $R$-basis elements in degree $di+r$.

Theorems & Definitions (78)

  • Theorem 1.1
  • Corollary 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Example 1.9
  • Proposition 2.1
  • ...and 68 more