Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Degeneracy locus formulas for amenable Weyl group elements

Harry Tamvakis

Abstract

We define a class of amenable Weyl group elements in the Lie types B, C, and D, which we propose as the analogues of vexillary permutations in these Lie types. Our amenable signed permutations index flagged theta and eta polynomials, which generalize the double theta and eta polynomials of Wilson and the author. In geometry, we obtain corresponding formulas for the cohomology classes of symplectic and orthogonal degeneracy loci.

Degeneracy locus formulas for amenable Weyl group elements

Abstract

We define a class of amenable Weyl group elements in the Lie types B, C, and D, which we propose as the analogues of vexillary permutations in these Lie types. Our amenable signed permutations index flagged theta and eta polynomials, which generalize the double theta and eta polynomials of Wilson and the author. In geometry, we obtain corresponding formulas for the cohomology classes of symplectic and orthogonal degeneracy loci.

Paper Structure

This paper contains 22 sections, 34 theorems, 214 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie type A
  4. Lie type C
  5. Lie types B and D
  6. Raising operators
  7. Alternating properties in types A, B, and C
  8. Alternating properties in type D
  9. Amenable elements: Type A theory
  10. Definitions and main theorem
  11. Type A degeneracy loci
  12. Amenable elements: Type C theory
  13. Definitions and main theorem
  14. Flagged theta polynomials
  15. Symplectic degeneracy loci
  16. ...and 7 more sections

Key Result

Lemma 1

Suppose that $p,r,s\in {\mathbb Z}$. For all $i\geq 1$, we have

Figures (1)

  • Figure 1: The tableau $T$ on the skew diagram $\widehat{\lambda}/\lambda$.

Theorems & Definitions (85)

  • Lemma 1
  • Example 1
  • Lemma 2: MT6
  • Lemma 3
  • Lemma 4
  • Definition 1
  • Definition 2
  • Definition 3
  • Example 2
  • Proposition 1
  • ...and 75 more