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Spanning subspace configurations

Brendon Rhoades

Abstract

A {\em spanning configuration} in the complex vector space $\mathbb{C}^k$ is a sequence $(W_1, \dots, W_r)$ of linear subspaces of $\mathbb{C}^k$ such that $W_1 + \cdots + W_r = \mathbb{C}^k$. We present the integral cohomology of the moduli space of spanning configurations in $\mathbb{C}^k$ corresponding to a given sequence of subspace dimensions. This simultaneously generalizes the classical presentation of the cohomology of partial flag varieties and the more recent presentation of a variety of spanning line configurations defined by the author and Pawlowski. This latter variety of spanning line configurations plays the role of the flag variety for the Haglund-Remmel-Wilson Delta Conjecture of symmetric function theory.

Spanning subspace configurations

Abstract

A {\em spanning configuration} in the complex vector space is a sequence of linear subspaces of such that . We present the integral cohomology of the moduli space of spanning configurations in corresponding to a given sequence of subspace dimensions. This simultaneously generalizes the classical presentation of the cohomology of partial flag varieties and the more recent presentation of a variety of spanning line configurations defined by the author and Pawlowski. This latter variety of spanning line configurations plays the role of the flag variety for the Haglund-Remmel-Wilson Delta Conjecture of symmetric function theory.

Paper Structure

This paper contains 19 sections, 21 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Combinatorics
  4. Gröbner theory
  5. Demazure characters
  6. Cohomology
  7. Quotient rings
  8. A slightly unusual ${\mathfrak{S}}_n$-action
  9. Orbit harmonics
  10. The structure of $S_{\alpha,k}$
  11. Mixed reduction
  12. Permuted column reduced echelon form
  13. The mixed reduction algorithm
  14. Geometric consequences of mixed reduction
  15. Cohomology presentation
  16. ...and 4 more sections

Key Result

Theorem \oldthetheorem

Let $k \leq n$ be positive integers.

Figures (1)

  • Figure 1: A point in $X_{(2,1,2),3}$.

Theorems & Definitions (49)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • ...and 39 more