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Partial Algebraic Shifting

Antony Della Vecchia, Michael Joswig, Fabian Lenzen

TL;DR

The paper develops a unified framework for partial algebraic shifting in the exterior algebra, tying nongeneric shifting to the Bruhat decomposition and revealing that full shifting arises as a special case corresponding to the longest element $w_0$. It shows that partial shifts with appropriately large permutations can preserve Betti numbers of simplicial complexes by exploiting near-cone structure, while providing a robust combinatorial-geometric lens through Bruhat theory to relate algebraic and combinatorial shifting. The introduction of the partial shift graph PSG$(n,k,m)$ establishes acyclicity and clarifies the ordering of shifts, and concrete examples (e.g., RP$^2_6$) illustrate the limits and topological consequences of partial shifting. The work opens several directions, including extensions to other groups, connections to symmetric shifting, and a deeper understanding of how partial shifts influence topology and combinatorial invariants in broader settings.

Abstract

We study algebraic shifting of uniform hypergraphs and finite simplicial complexes in the exterior algebra with respect to matrices which are not necessarily generic. Several questions raised by Kalai (2002) are addressed. For instance, it turns out that the combinatorial shifting of Erdős$\unicode{x2013}$Ko$\unicode{x2013}$Rado (1961) arises as a special case. Moreover, we identify a sufficient condition for partial shifting to preserve the Betti numbers of a simplicial complex; examples show that this condition is sharp.

Partial Algebraic Shifting

TL;DR

The paper develops a unified framework for partial algebraic shifting in the exterior algebra, tying nongeneric shifting to the Bruhat decomposition and revealing that full shifting arises as a special case corresponding to the longest element . It shows that partial shifts with appropriately large permutations can preserve Betti numbers of simplicial complexes by exploiting near-cone structure, while providing a robust combinatorial-geometric lens through Bruhat theory to relate algebraic and combinatorial shifting. The introduction of the partial shift graph PSG establishes acyclicity and clarifies the ordering of shifts, and concrete examples (e.g., RP) illustrate the limits and topological consequences of partial shifting. The work opens several directions, including extensions to other groups, connections to symmetric shifting, and a deeper understanding of how partial shifts influence topology and combinatorial invariants in broader settings.

Abstract

We study algebraic shifting of uniform hypergraphs and finite simplicial complexes in the exterior algebra with respect to matrices which are not necessarily generic. Several questions raised by Kalai (2002) are addressed. For instance, it turns out that the combinatorial shifting of ErdősKoRado (1961) arises as a special case. Moreover, we identify a sufficient condition for partial shifting to preserve the Betti numbers of a simplicial complex; examples show that this condition is sharp.

Paper Structure

This paper contains 8 sections, 24 theorems, 63 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Acknowledgement.
  3. Algebraic shifting
  4. Bruhat decomposition
  5. Combinatorial shifting
  6. The partial shift graph
  7. Partial Shifting to Near Cones
  8. Outlook

Key Result

Lemma 4

For every $S \subseteq \binom{[n]}{k}$ and $g \in \mathrm{GL}(n,{\mathbb E})$, we have $\lvert\Delta_g(S)\rvert = \lvert S\rvert$.

Figures (2)

  • Figure 1: Partial shift graph $\mathrm{PSG}(4,2,5)$ of all graphs on four vertices with five edges. It contains precisely one shifted complex (i.e., one sink), which is the node on the left. For each edge $S \to T$ for $S, T \in \binom{[4]}{2}$, the filled circles in the small diagrams on that edge represents the set $\Set{w \in \SymmetricGroup{4}; \Delta_{{\mathfrak r}(w)}(S) = T}$, following the visualization of $\SymmetricGroup{4}$ from BjornerBrenti:2005[Figure 3.65]Stanley:EC1. If a permutation $w$ is missing from all outgoing arrows from a complex $S$, this means that $\Delta_{{\mathfrak r}(w)}(S) = S$. In particular, $e$ is missing from all arrows, and $w_0$ is present in all arrows to the sink.
  • Figure 2: Contracted partial shift graphs $\overline{\mathrm{PSG}}({\mathbb R}{\mathbb P}^2_6)$ with respect to ${\mathbb Q}$ (left) and $\operatorname{GF}(2)$ (right). The complexes $A$, $B$, $C$ and $D$ are explained in \ref{['tab:shifted partial shifts of RP2']}.

Theorems & Definitions (74)

  • Definition 1
  • Example 2
  • Remark 3: The role of the fields ${\mathbb E}$ and ${\mathbb F}$
  • Lemma 4
  • proof
  • Remark 5: Compound matrices and exterior powers; see Kalai02
  • Remark 6: Shifting via generic initial ideals; see HerzogTerai:1999
  • Remark 7: Shifting via Grassmannians
  • Lemma 8: Fiedler:2013
  • Lemma 9
  • ...and 64 more