DEFORMED GRAPHICAL ZONOTOPAL ALGEBRAS

NEW

Problem: Over a field $\mathbf k$ of characteristic $0$, let $G$ be a finite tree and let

For $f\in\mathcal A_G$, the deformed graphical zonotopal algebra $\mathcal C_G^f$ is filtered by word-length in the generators $Y_v=f(X_v)$, and its Hilbert sequence is

Problem 5.2 asks whether, for trees, the Hilbert stratification of $\mathcal A_G$ is by coordinate subspaces in the coordinates $a_2,\dots,a_d$. The phrase is slightly informal; the standard interpretation at least implies that two points with the same vanishing pattern of coordinates lie in the same Hilbert stratum.

Result: The coordinate-subspace assertion is false.

Take $G=K_{1,5}$, the star with five leaves. Write edge variables $x_1,\dots,x_5$, so

has dimension $2^5=32$. The leaf generators are $x_i$, and the central generator is

Since $s$ is already in the span of the leaf generators, the filtration is generated in degree $1$ by $x_1,\dots,x_5$ and

where $e_r$ is the elementary squarefree symmetric sum of degree $r$, and

A direct filtration computation gives:

for $A,B,C,D\ne0$.

Indeed, $F_1=\langle 1,x_1,\dots,x_5,q\rangle$, so $\dim F_1=7$. Modulo ordinary degree $\le2$, $F_2$ is spanned by

The five middle vectors contribute a $5$-dimensional space; adding the two symmetric vectors raises the dimension by $2$ or $1$ according as

is nonzero or zero. Thus $\dim F_2=23$ or $22$. Also $F_3=R$: ordinary degree $\le3$ is generated by the $x_i$’s, the elements $x_ix_jq$ span degree $4$, and $x_iq^2=6A^2e_5$ gives degree $5$.

Now compare

and

Both have all coordinates $a_2,a_3,a_4,a_5$ nonzero. For $f_1$,

so $\mathscr H_G^{f_1}=(1,6,15,10)$. For $f_2$, the same expression is nonzero, so

Thus the Hilbert stratification separates two points in the same coordinate torus $(\mathbf k^\*)^4$. Hence it cannot consist of coordinate subspaces.

The exponential point, interpreted in $\mathcal A_G$ as the truncation of $e^u-1$, has $A=B=C=D=1$, and the determinant equals $45-9-40=-4\ne0$; so for this particular counterexample $e^u$ is general. The counterexample disproves the coordinate-subspace assertion, but not the separate global exponential question.

Citation: Definitions and Problem 5.2: Boris Shapiro, Ilya Smirnov, Arkady Vaintrob, “Deformed graphical zonotopal algebras,” arXiv:2204.11331. The counterexample above is the resolution supplied here.