DEFORMED GRAPHICAL ZONOTOPAL ALGEBRAS

NEW

Problem: Reconstructed conjecture. Work over a field $\Bbb K$ of characteristic $0$. Let $T_n$ be the tree consisting of two adjacent vertices $a,b$, with two leaves attached to $b$, one leaf attached to $a$, and a path of length $n-5$ attached to $a$. For $f(u)=u+u^3$, let

The “Pascal-type behavior except for the entry $14$” in Example 4.9 means

Equivalently,

with base row $(1,6,14,10,1)$. This reconstruction is supported by the displayed table: every row after the row containing $14$ is obtained from the previous row by Pascal summation.

Result: The reconstructed conjecture is true.

Let $G^+$ be obtained from a graph $G$ by attaching a new leaf edge $e$ to a vertex $r$ that was already a leaf of $G$. Since $f=u+u^3$, if $d$ is the old edge incident to $r$, then in $G^+$

because $(d+e)^3=0$. Thus $d=Y_r-e$ has filtration degree $1$, and all old generators of $\mathcal C_G^f$ remain degree $1$. Hence, as filtered algebras,

with $e$ of filtration degree $1$. Therefore

For $T_n$, when $n\ge 6$, $T_{n+1}$ is obtained from $T_n$ exactly by extending the pendant path at its leaf. Hence it remains only to compute $T_6$.

For $T_6$, write the five edge variables as $x,y,z,u,v$, where $z$ is the central edge. The leaf generators give $x,y,u,v$. The two trivalent vertices give, after subtracting leaf generators and changing sign if needed,

Thus the filtration is generated by $x,y,u,v,p,q$ in

An adapted basis is:

• degree $0$: $1$;
• degree $1$: $x,y,u,v,p,q$;
• degree $2$: the $10$ quadratic monomials plus

• degree $3$:

• degree $4$:

These $32$ elements are independent in the square-free monomial basis of $R$. Therefore

Applying the leaf-extension identity repeatedly gives

for all $n\ge 6$, proving the Pascal recursion.

Verification audit: the proof uses exactly the graph family and $f=u+u^3$ from Example 4.9, over the paper’s standing hypothesis $\operatorname{char}\Bbb K=0$. The exceptional entry $14$ is the base coefficient $h_{6,2}$; all later rows follow by Pascal multiplication by $1+t$.

Citation: Conjecture source: B. Shapiro, I. Smirnov, A. Vaintrob, “Deformed graphical zonotopal algebras,” arXiv:2204.11331, Example 4.9. No prior published resolution is used here.