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Ribbon Schur Functors

Keller VandeBogert

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.

Ribbon 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 . 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.
Paper Structure (26 sections, 39 theorems, 318 equations, 2 figures)

This paper contains 26 sections, 39 theorems, 318 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Ribbon Schur Functors
  3. Koszul Algebras and Backelin's Theorem
  4. Generalized Ribbon Schur Functors and the Concatenation/near-concatenation criterion for Koszulness
  5. Multi-Schur Functors
  6. Organization of Paper
  7. Augmented Bar Constructions
  8. Distributivity and Submodule Lattices
  9. A Generalization of Schur Modules for Ribbon Diagrams
  10. Standard Operations Between Compositions
  11. Ribbon Schur Modules for Koszul Algebras
  12. Ribbon Schur Modules with Koszul Module Inputs
  13. Multi-Schur Functors
  14. Some Generalities on Filtrations
  15. A Canonical Filtration of Multi-Schur Functors
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $A$ be a quadratic algebra. Then $A$ is Koszul if and only if the collection $Q_2 \otimes V^{\otimes n-2}, V \otimes Q_2 \otimes V^{\otimes n-3}, \cdots , V^{\otimes n-2} \otimes Q_2$ generates a distributive subspace lattice for all $n \geqslant 2$.

Figures (2)

  • Figure 1: The skew shape on the left is not a ribbon, since the bottom two rows have overlap size $2$. The shape on the right is a ribbon.
  • Figure 2: The concatenation/near-concatenation identity (here we are identifying the ribbon diagrams with their corresponding Schur polynomials).

Theorems & Definitions (164)

  • Example 1.1
  • Theorem 1.2: Backelin backelinThesis, see also beilinson1996koszul
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Example 1.8
  • Definition 2.1
  • Remark 2.2
  • ...and 154 more