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Bruhat Intervals in the Infinite Symmetric Group are Cohen-Macaulay

Nathaniel Gallup, Leo Gray

Abstract

We show that the (non-Noetherian) Stanley-Reisner ring of the order complex of certain intervals in the Bruhat order on the infinite symmetric group $S_\infty$ of all auto-bijections of $\mathbb{N}$ is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. This gives an infinite-dimensional version of results due to Edelman, Björner, and Kind and Kleinschmidt for finite symmetric groups $S_n$.

Bruhat Intervals in the Infinite Symmetric Group are Cohen-Macaulay

Abstract

We show that the (non-Noetherian) Stanley-Reisner ring of the order complex of certain intervals in the Bruhat order on the infinite symmetric group of all auto-bijections of is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. This gives an infinite-dimensional version of results due to Edelman, Björner, and Kind and Kleinschmidt for finite symmetric groups .
Paper Structure (8 sections, 30 theorems, 13 equations)

This paper contains 8 sections, 30 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Nested-Shellable Simplicial Complexes
  3. Lexicographically Nested-Shellable Posets
  4. Bruhat Order on the Infinite Symmetric Group
  5. Cover Relations in the Bruhat Order
  6. Saturated Chains in the Bruhat Order
  7. An $\mathbb{N}$-Grading of Intervals in $S_\infty$
  8. Lexicographically Nested-Shellable Intervals of $S_\infty$

Key Result

Theorem A

If $\Delta$ is a nested-shellable simplicial complex, its Stanley-Reisner ring $k[\Delta]$ is Cohen-Macaulay in the sense of flat directed limits.

Theorems & Definitions (82)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1
  • proof : Proof of Theorem \ref{['thm: main thm nested implies CM']}
  • Remark 1
  • Definition 2
  • Remark 2
  • Remark 3
  • Definition 3
  • ...and 72 more