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Laplacians on shifted multicomplexes

Jan Snellman

TL;DR

The paper develops a Laplacian theory for finite multicomplexes by defining $L'_d = \partial_{d+1}\partial_{d+1}^*$ and relating it to the conventional Laplacians, enabling spectral computation from constituent simplicial components. For shifted (strongly stable) multicomplexes, it provides a concrete combinatorial formula showing that the Laplacian eigenvalues are nonnegative integers, mirroring results for shifted simplicial complexes. The framework connects to commutative algebra via the Eliahou–Kervaire resolution of strongly stable monomial ideals, and it extends to arithmetical-function truncations under Dirichlet convolution, yielding spectra tied to prime-factor data. This synthesis links combinatorics, topology, and number theory, offering explicit spectral descriptions across these domains.

Abstract

We define the Laplacian operator on finite multicomplexes and give a formula for its spectra in the case of shifted multicomplexes.

Laplacians on shifted multicomplexes

TL;DR

The paper develops a Laplacian theory for finite multicomplexes by defining and relating it to the conventional Laplacians, enabling spectral computation from constituent simplicial components. For shifted (strongly stable) multicomplexes, it provides a concrete combinatorial formula showing that the Laplacian eigenvalues are nonnegative integers, mirroring results for shifted simplicial complexes. The framework connects to commutative algebra via the Eliahou–Kervaire resolution of strongly stable monomial ideals, and it extends to arithmetical-function truncations under Dirichlet convolution, yielding spectra tied to prime-factor data. This synthesis links combinatorics, topology, and number theory, offering explicit spectral descriptions across these domains.

Abstract

We define the Laplacian operator on finite multicomplexes and give a formula for its spectra in the case of shifted multicomplexes.

Paper Structure

This paper contains 8 sections, 9 theorems, 51 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Laplacians of multicomplexes
  3. The homology theory of Björner and Vrećica
  4. Definition of Laplacians on multicomplexes
  5. The spectrum of the Laplacian
  6. Shifted multicomplexes and their Laplacian spectra
  7. Strongly stable monomial ideals
  8. Truncations of arithmetical functions with Dirichlet convolution

Key Result

Lemma 2

Theorems & Definitions (16)

  • Definition 1
  • Lemma 2
  • Lemma 3
  • Definition 4
  • Lemma 5
  • proof
  • Definition 6
  • Lemma 7
  • Theorem 8: Reiner and Duval
  • Theorem 9
  • ...and 6 more