[Paper Review] A complete $g$-vector for convex polytopes

Jonathan Fine|arXiv (Cornell University)|Jan 10, 2010

Advanced Combinatorial Mathematics3 references3 citations

TL;DR

This paper introduces a complete $g$-vector for convex polytopes that encodes the entire flag vector via a generalized toric $g$-vector extension using operators $C$ and $D=IC-CC$. The key contribution is a linear transformation from the flag vector to this extended $g$-vector, with extensive computations showing non-negativity for most components, though a few exceptional negative values are found for higher-dimensional polytopes.

ABSTRACT

We define an extension of the toric (middle perversity intersection homology) $g$-vector of a convex polytope $X$. The extended $g(X)$ encodes the whole of the flag vector $f(X)$ of $X$, and so is called complete. We find that for many examples that $g_k(X)\geq 0$ for most $k$ (independent of $X$).

Motivation & Objective

• To define a complete $g$-vector that fully encodes the flag vector of any convex polytope.
• To extend the classical toric $g$-vector using a basis of words in $C$ and $D=IC-CC$, enabling linear reconstruction of the flag vector.
• To investigate the non-negativity of the extended $g_k(X)$ components across various polytopes and dimensions.
• To identify and characterize exceptional Fibonacci index patterns where $g_k(X) < 0$, especially beyond dimension 4.
• To formulate a combinatorial problem on effective index sets that propagate non-negativity under polytope operations like $CX$, $IX$, and $BX$.

Proposed method

• Define $C$ as the cone operator and $D = IC - CC$ as a difference operator on formal sums of polytopes.
• Use the generalized Dehn-Sommerville relation to show that values of linear functionals on $W(\mathrm{pt})$ for words $W$ in $C$ and $D$ determine the flag vector uniquely.
• Introduce Fibonacci indices $k = [i_0,i_1; j_0,j_1]$ to parametrize the basis words $(C,D)^k$, with a partial order on these indices via shifting and splitting relations.
• Define $g_k(X)$ as the linear functional satisfying $g_k((C,D)^ u(\mathrm{pt})) = 1$ if $k \leq \nu$, and 0 otherwise, forming a lower-triangular transformation to the flag vector.
• Apply the transformation to compute $g_k(X)$ for specific polytopes like bipyramids and iterated cones using computational tools such as polymake.
• Use duality to define $g^ u_k(X) = g_k(X^\vee)$, enabling the study of effective index sets under duality and polytope operations.

Experimental results

Research questions

• RQ1 For which Fibonacci indices $k$ does $g_k(X) < 0$ hold for some convex polytope $X$?
• RQ2 Can the exceptional $k$ with negative $g_k(X)$ be systematically characterized across all dimensions?
• RQ3 Is there a combinatorial rule or recursive structure that ensures $g_k(X) \geq 0$ for non-exceptional $k$?
• RQ4 Can effective index sets $S_d$ be constructed such that non-negativity of $g_k(X)$ for $k \in S_d$ implies non-negativity for $CX$, $IX$, and $BX$?
• RQ5 How do the $g_k(X)$ values behave under duality, and what does this imply for the structure of the complete $g$-vector?

Key findings

• The complete $g$-vector linearly reconstructs the full flag vector of any convex polytope via a lower-triangular transformation from the $(C,D)$-basis.
• For dimensions $d \leq 4$, all computed $g_k(X) \geq 0$ for all Fibonacci indices $k$.
• In dimension 5, the only negative value occurs at $k = [00;11]$ for $X = BIC^3(\mathrm{pt})$, with $g_k(X) < 0$.
• In dimension 6, the only negative value occurs at $k = [00;21]$ for $X = B^4C^2(\mathrm{pt})$, with $g_k(X) < 0$.
• In dimension 7, three exceptional $k$ values $[00;13]$, $[00;31]$, and $[10;11]$ yield negative $g_k(X)$ for $X = B^2ICIC^2(\mathrm{pt})$.
• In dimension 8, four exceptional $k$ values $[00;23]$, $[00;41]$, $[10;21]$, and $[000;011]$ produce negative $g_k(X)$ for $X = B^4CIC^2(\mathrm{pt})$.