Ribbon Schur Functors

TL;DR

This paper generalizes classical Schur functors to ribbon diagrams over arbitrary algebras, linking their properties to Koszulness and providing new categorical and algebraic insights.

Contribution

It introduces ribbon Schur functors for arbitrary algebras, constructs categorifying complexes, and relates their exactness to the Koszul property, extending classical theory.

Findings

Exactness of complexes is equivalent to Koszul property of algebra

Complete description of syzygies of Segre products of Koszul modules

Characteristic-free computation of regularity of Schur functors

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 a Schur functor applied to the tautological subbundle on projective space.