Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The miracle of integer eigenvalues

Richard Kenyon, Maxim Kontsevich, Oleg Ogievetsky, Cosmin Pohoata, Will Sawin, Senya Shlosman

Abstract

For partially ordered sets $X$ we consider the square matrices $M^{X}$ with rows and columns indexed by linear extensions of the partial order on $X$. Each entry $\left( M^{X}\right)_{PQ}$ is a formal variable defined by a pedestal of the linear order $Q$ with respect to linear order $P$. We show that all the eigenvalues of any such matrix $M^{X}$ are $\mathbb{Z}$-linear combinations of those variables.

The miracle of integer eigenvalues

Abstract

For partially ordered sets we consider the square matrices with rows and columns indexed by linear extensions of the partial order on . Each entry is a formal variable defined by a pedestal of the linear order with respect to linear order . We show that all the eigenvalues of any such matrix are -linear combinations of those variables.
Paper Structure (14 sections, 2 theorems, 49 equations, 1 figure)

This paper contains 14 sections, 2 theorems, 49 equations, 1 figure.

Table of Contents

  1. The statement of the main result
  2. The filter semigroup
  3. Faces
  4. Filters
  5. Proof of the main result
  6. The filter decomposition
  7. Conjugation of $M_{F}^{X}$-s to upper-triangular
  8. Appendices
  9. Pedestals
  10. Pedestal polynomials
  11. Examples
  12. 1. Partition (3,2).
  13. 2. Partition (3,1,1).
  14. 3. Partition (3,2,1).

Key Result

Theorem 1

For every poset $X$ the matrix $M^{X}$ is non-degenerate, and all its eigenvalues are linear combinations of the variables $a_{\varepsilon}$ with integer coefficients.

Figures (1)

  • Figure 1: The central real hyperplane arrangement $A_{4}$ in $\mathbb{R}^{4},$ projected to $\mathbb{R}^{3}$ along the line $x=y=z=t$ and intersected with the sphere $\mathbb{S}^{2}\subset\mathbb{R}^{3}.$ It is a partition of $\mathbb{S}^{2}$ into 24 equal triangles, each with the angles $\left( \frac{\pi}{2},\frac{\pi}{3},\frac{\pi}{3}\right) .$ The types of convex unions of the triangles are: the sphere, the hemisphere, the region between two great semicircles, an elementary triangle -- or e-triangle, a pair of e-triangles with a common side, a triangle made from three e-triangles, a 'square' formed by four e-triangles with a common $\frac{\pi}{2}$-vertex, a triangle made from a 'square' and a fifth adjacent e-triange, a triangle formed by six e-triangles with a common $\frac{\pi}{3}$-vertex. The number of corresponding convex shapes are 1, 12, 60, 24, 36, 48, 6, 24, 8, with total being 219. This is precisely the number of partial orders on the set of four distinct elements, see the sequence A001035 in OEIS Sl.

Theorems & Definitions (7)

  • Theorem 1
  • Claim 2
  • Claim 3
  • Lemma 4
  • Remark 5
  • Remark 6
  • Remark 7