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Smith Normal Form of Matrices Associated with Differential Posets

Syed Waqar Ali Shah

Abstract

We prove a conjecture of Miller and Reiner on the existence of Smith normal form for the $DU$-operators for a certain class of $r$-differential posets.

Smith Normal Form of Matrices Associated with Differential Posets

Abstract

We prove a conjecture of Miller and Reiner on the existence of Smith normal form for the -operators for a certain class of -differential posets.

Paper Structure

This paper contains 9 sections, 16 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Recollections
  3. Integral canonical forms
  4. Lifting canonical forms
  5. Tweaking decompositions
  6. Ascensions
  7. Invertible constants
  8. The main theorem
  9. Applications

Key Result

Theorem 2.1

Stanley88 Let $P$ be an $r$-differential poset and $R$ a field of characteristic $0$. Then where $\mathrm{Ch}(A) = \mathrm{Ch}(A,x)$ denotes the characteristic polynomial of the operator $A$, and $\Delta p_n:=p_n - p_{n-1}$ denotes the rank difference. Furthermore, the operators $DU_{n}$ and $UD_{n}$ are diagonalizable.

Theorems & Definitions (38)

  • Conjecture 1.1
  • Theorem 2.1
  • Definition 3.1
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Definition 3.5
  • Corollary 3.6
  • Lemma 4.1
  • proof
  • ...and 28 more