[Paper Review] Schubert Polynomials in Types A and C

David Anderson, William Fulton|arXiv (Cornell University)|Feb 10, 2021

Advanced Combinatorial Mathematics23 references4 citations

TL;DR

This paper introduces enriched Schubert polynomials in types A and C by extending classical double Schubert polynomials with coefficients in a polynomial ring $Λ = \mathbb{Z}[c_1, c_2, \ldots]$, where the $c_k$ encode Chern classes. The construction unifies classical Schubert polynomials, back-stable polynomials, and type C analogues via specialization, and provides a geometric framework for degeneracy locus formulas with irreducible polynomials for vexillary permutations.

ABSTRACT

Enriched versions of type A Schubert polynomials are constructed with coefficients in a polynomial ring in variables $c_1, c_2, \ldots$. Specializing these variables to $0$ recovers the double Schubert polynomials of Lascoux and Schützenberger; specializing them to certain power series recovers the back-stable double Schubert polynomials of Lam, Lee, and Shimozono; specializing them to Schur Q-polynomials relates them to the type C double Schubert polynomials of Ikeda, Mihalcea, and Naruse. Many formulas for classical Schubert polynomials generalize to this setting. They give, and are characterized by, formulas for degeneracy loci.

Motivation & Objective

• To construct enriched Schubert polynomials in type A with coefficients in $\Lambda = \mathbb{Z}[c_1, c_2, \ldots]$, generalizing classical double Schubert polynomials.
• To establish a geometric and algebraic bridge between type A and type C Schubert polynomials via a homomorphism $c_k \mapsto q_k$ to Schur Q-polynomials.
• To extend the theory of back-stable Schubert polynomials to include infinite permutations and provide a unified framework for degeneracy loci.
• To develop a twisted version of the polynomials with a parameter $z$ corresponding to the first Chern class of a line bundle.
• To demonstrate that the enriched polynomials are irreducible for all vexillary permutations, a property not shared by classical Schubert polynomials.

Proposed method

• Construct enriched Schubert polynomials $\mathsf{S}_w$ in the ring $\Lambda[x,y]$, where $\Lambda = \mathbb{Z}[c_1, c_2, \ldots]$ with $c_k$ of degree $k$, generalizing classical double Schubert polynomials.
• Use determinantal formulas for vexillary permutations via Schur polynomials $S_\lambda(c) = \det(c_{\lambda_i + j - i})$, ensuring irreducibility of $\mathsf{S}_w$ for vexillary $w$.
• Apply difference operators and transition formulas from [AF3] to recursively build all Schubert polynomials from the vexillary ones.
• Define twisted enriched Schubert polynomials $\mathbf{S}_w$ by replacing $\Lambda$ with $\bm{\Lambda} = \Lambda[z]$, modeling the first Chern class of a line bundle.
• Establish a homomorphism from $\Lambda$ to the back-stable ring $\overleftarrow{R}$, mapping $\mathsf{S}_w$ to back-stable Schubert polynomials $\overleftarrow{\mathfrak{S}}_w(x;a)$, with $y_i \mapsto -a_i$.
• Characterize the polynomials via their role in degeneracy locus formulas for vector bundles with bilinear pairings $V \times W \to L$ and isotropic filtrations.

Experimental results

Research questions

• RQ1 How can classical Schubert polynomials be enriched with Chern class variables $c_k$ to unify type A and type C Schubert calculus?
• RQ2 What is the geometric meaning of the enriched Schubert polynomials in terms of degeneracy loci for vector bundles with bilinear pairings?
• RQ3 How do the enriched polynomials relate to back-stable Schubert polynomials and what is the significance of the embedding $\Lambda \to \overleftarrow{R}$?
• RQ4 Why are the enriched Schubert polynomials irreducible for all vexillary permutations, unlike their classical counterparts?
• RQ5 What is the compatibility of twisted Schubert polynomials in types A and C, and how does the Chern class $c_k(V - E - F)$ emerge in type C?

Key findings

• The enriched Schubert polynomials $\mathsf{S}_w$ in $\Lambda[x,y]$ are irreducible for all vexillary permutations, a key distinction from classical Schubert polynomials.
• Specializing $c_k \to 0$ recovers the classical double Schubert polynomials of Lascoux and Schützenberger.
• Specializing $c_k \to q_k$ (Schur Q-polynomials) yields the type C double Schubert polynomials of Ikeda, Mihalcea, and Naruse.
• The homomorphism $\Lambda \to \overleftarrow{R}$, sending $y_i \to -a_i$, realizes $\mathsf{S}_w$ as the back-stable Schubert polynomials $\overleftarrow{\mathfrak{S}}_w(x;a)$, embedding the enriched theory into the back-stable framework.
• The twisted enriched Schubert polynomials $\mathbf{S}_w$ in $\Lambda[z][x,y]$ are obtained by replacing $x_i$ with $x_i - z$, and they are compatible between types A and C.
• The ring $\bm{\Gamma} = \bm{\Lambda}/I$ for type C encodes Chern classes $c_k(V - E - F)$, and the map $c_k \mapsto q_k$ sends type A to type C Schubert polynomials.