Ribbon Schur functors

VandeBogert, Keller

doi:10.2140/ant.2025.19.771

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Ribbon Schur functors

Keller VandeBogert

Vol. 19 (2025), No. 4, 771–834

DOI: 10.2140/ant.2025.19.771

Abstract

We investigate a generalization of the classical notion of a Schur functor associated to a ribbon diagram. These functors are defined with respect to an arbitrary algebra, and in the case that the underlying algebra is the symmetric/exterior algebra, we recover the classical definition of Schur/Weyl functors, respectively. In general, we construct a family of $3$ -term complexes categorifying the classical concatenation/near-concatenation identity for symmetric functions, and one of our main results is that the exactness of these $3$ -term complexes is equivalent to the Koszul property of the underlying algebra $A$ . We further generalize these ribbon Schur functors to the notion of a multi-Schur functor and construct a canonical filtration of these objects whose associated graded pieces are described explicitly; one consequence of this filtration is a complete equivariant description of the syzygies of arbitrary Segre products of Koszul modules over the Segre product of Koszul algebras. Further applications to the equivariant structure of derived invariants, symmetric function identities, and Koszulness of certain classes of modules are explored at the end, along with a characteristic-free computation of the regularity of the Schur functor $𝕊^{λ}$ applied to the tautological subbundle on projective space.

Keywords

Schur functors, Koszul algebras, free resolutions, sheaf cohomology

Mathematical Subject Classification

Primary: 05E40, 13D02, 14F06

Milestones

Received: 12 December 2023

Revised: 6 April 2024

Accepted: 23 May 2024

Published: 24 March 2025

Authors

Keller VandeBogert
Department of Mathematics University of Notre Dame Notre Dame, IN United States

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Vol. 19, No. 4, 2025