v_g-Theorem: Realizing g-Vectors as f-Vectors

• The v_g-Theorem is a combinatorial result that connects g-vectors from Veronese constructions on formal power series with the f-vectors of simplicial complexes.
• It employs admissible vectors and combinatorial recurrences to ensure that the sequence of g-vector differences becomes nonnegative and monotone, analogous to classical g-theorem properties.
• Its applications extend to Hilbert series of graded algebras, edgewise subdivisions of complexes, and establishing M-sequence properties in algebraic combinatorics.

The $v_g$-Theorem, as formulated in Kubitzke–Welker, is an enumerative g-theorem that concerns the transformation properties and combinatorial realization of $g$-vectors arising from Veronese constructions on formal power series and graded algebras. Given any sequence of integers with a generating function expressible as a rational function with nonnegative $h$-vector coefficients, the $v_g$-Theorem establishes that, for large Veronese indices, the vector of successive differences (the $g$-vector) of the numerator polynomial becomes the $f$-vector of a simplicial complex. This result yields consequences analogous to the unimodality part of the classical $g$-conjecture, notably in contexts lacking direct geometric or polytope structure. The theorem impacts the theory of Hilbert series of Veronese algebras and subdivisions of complexes, placing itself as an essential bridge between algebraic combinatorics, commutative algebra, and the combinatorial topology of simplicial complexes (Kubitzke et al., 2011).

1. Formal Definitions and Notation

Let $(a_n)_{n \geq 0} \subset \mathbb{Z}$ be an integer sequence, with generating series

$$a(t) = \sum_{n \geq 0} a_n t^n = \frac{h_0(a) + h_1(a) t + \cdots + h_\lambda(a) t^\lambda}{(1-t)^d}$$

where $h(a) = (h_0(a), h_1(a), \dots, h_\lambda(a))$ is the $h$-vector, $d, \lambda \in \mathbb{N}$, $h_0(a)=1$, and $h_i(a)\geq0$ for $i>0$. The $g$-vector, defined as

$$g(a) = (g_0(a), g_1(a), \ldots, g_{\lfloor\lambda/2\rfloor}(a)), \quad g_0(a)=1,\quad g_i(a)=h_i(a)-h_{i-1}(a) \quad (i\geq1)$$

collects the "first-half differences" of the $h$-vector. For an integer $r\geq1$, the $r^\text{th}$ Veronese series is

$$a^{\langle r\rangle}(t) = \sum_{n \geq 0} a_{nr} t^n = \frac{h_0(a^{\langle r\rangle}) + \cdots + h_{\lambda'}(a^{\langle r\rangle}) t^{\lambda'}}{(1-t)^d}$$

where $h(a^{\langle r\rangle})$ and $g(a^{\langle r\rangle})$ are the respective transformed vectors. For a $(d-1)$-dimensional simplicial complex $\Delta$, its $f$-vector is

$$f(\Delta) = (f_{-1}(\Delta), f_0(\Delta), \ldots, f_{d-1}(\Delta))$$

where $f_i$ denotes the number of $i$-dimensional faces.

2. Statement and Content of the $v_g$-Theorem

The core assertion—the $v_g$-Theorem—is that under mild positivity assumptions, for $r\geq\max(d,\lambda)$, there exists a simplicial complex $\Delta_r$ of dimension $\lfloor d/2\rfloor$ such that

$$(g_0(a^{\langle r\rangle}), \ldots, g_{\lfloor d/2\rfloor}(a^{\langle r\rangle})) = (f_{-1}(\Delta_r), f_0(\Delta_r), \ldots, f_{\lfloor d/2\rfloor-1}(\Delta_r))$$

Thus, for large Veronese parameter $r$, the differences $g_i(a^{\langle r\rangle})$ realize the face-vector of an actual simplicial complex, so the sequence

$$1 = g_0 \leq g_1 \leq \cdots \leq g_{\lfloor d/2\rfloor}$$

is non-decreasing and all further combinatorial consequences of the classical $g$-theorem for polytopes or homology spheres hold for the Veronese series. This ensures unimodality-type inequalities, and M-sequence properties, are satisfied in this purely algebraic context (Kubitzke et al., 2011).

3. Linear Transformation Formulas and Admissibility

The transformation of the $h$-vector under the Veronese construction is given by

$$h_i(a^{\langle r\rangle}) = \sum_{j=0}^\lambda C(r-1,d, ir - j) h_j(a)$$

where

$$C(r,d,m) = \#\{(u_1,\ldots, u_d)\in \{0,1,\ldots,r\}^d : u_1+\cdots+u_d = m\}$$

One frames $h(a^{\langle r\rangle})$ as a sum over column vectors $C_k^{r,d}$ and passes to first differences for the $g$-vector. The crucial inductive apparatus is the concept of "admissible vectors" (in the sense of Murai), guaranteeing that nonnegative integer combinations of these remain $f$-vectors of simplicial complexes. All coordinates of such vectors are shown to be nonnegative and monotone, and the combinatorial recurrences and symmetries of $C(r,d,i)$ further secure the proof.

4. Applications: Hilbert Series and Subdivisions

Principal applications include:

• Veronese algebras: For a standard graded $k$-algebra $A = \bigoplus_{n \geq 0} A_n$ of Krull dimension $d$ with Hilbert series of the stated rational form, the $r$th Veronese subalgebra $A^{\langle r\rangle}$ acquires a Veronese Hilbert series so that, for all $r \geq \max(d, \lambda)$ and $A$ Cohen–Macaulay, the new $g$-vector admits an $f$-vector realization (Kubitzke et al., 2011).
• Edgewise subdivision: For a $(d-1)$-dimensional simplicial complex $\Delta$, its $r$th edgewise subdivision $\Delta(r)$ has a Stanley–Reisner ring whose Hilbert series is the $r$th Veronese of $k[\Delta]$, implying via the $v_g$-Theorem that the $g$-vector of $\Delta(r)$ is an $f$-vector for $r\geq d$.
• M-sequences: These transformations imply that, for large $r$, the $g$-vector produced is always an M-sequence—a significant algebraic combinatorial property.

5. Context in the Landscape of $g$-Theorems

The classical $g$-theorem (Billera-Lee, Stanley) characterizes the $f$-vectors of simplicial polytopes in terms of $g$-vectors satisfying M-sequence inequalities, with the $g$-conjecture extending this assertion to all simplicial homology spheres. The $v_g$-Theorem situates itself as an "enumerative g-theorem," functioning independent of geometric realization: starting from any nonnegative $h$-vector in rational form, the resulting Veronese $g$-vector realizes a genuine combinatorial object for large enough index $r$. This fundamentally extends the methodology for establishing unimodality and $g$-type inequalities within algebraic combinatorics, independent of geometry or topology.

6. Implications and Further Directions

This suggests that the Veronese transformation, acting on generating functions with nonnegative $h$-vectors, regularizes the $g$-vector sufficiently to guarantee its realization as a true simplicial complex structure. A plausible implication is strengthened connections between combinatorial commutative algebra and enumerative combinatorics via linear algebraic manipulations of series. The techniques and admissibility criteria, framed in terms of combinatorial recurrences and induction, offer avenues for extension to other transformations or classes of graded algebraic structures.

7. Summary Table: Key Constructs

Construct	Definition/Formula	Role in v₉-Theorem
$h$-vector	$(h_0(a), h_1(a), \ldots, h_\lambda(a))$	Encodes coefficients in rational generating fn.
$g$-vector	$$g_0=1,\,g_i = h_i - h_{i-1},\, 1\leq i\leq \lfloor\lambda/2\rfloor$$	Successive differences, combinatorial invariant
Veronese series	$a^{\langle r\rangle}(t) = \sum a_{nr} t^n$	Transformed sequence under $r$th Veronese
$f$-vector	$(f_{-1}, f_0, \ldots, f_{d-1})$	Counts faces of each dimension in complex

The $v_g$-Theorem of Kubitzke–Welker establishes that, for high-order Veronese transformations, the $g$-vector constructed from nonnegative algebraic data stabilizes into the $f$-vector of a simplicial complex, embedding enumerative and combinatorial properties directly into the algebraic context (Kubitzke et al., 2011).