Symmetrization on Hyperbolicity Cones

• Hyperbolicity cones are open convex sets defined by homogeneous polynomials that exhibit only real roots in a specific hyperbolicity direction.
• The symmetrization principle leverages finite group actions and the Reynolds operator to boost polynomial values within the hyperbolicity cone.
• This framework generalizes classical determinant inequalities, such as Hadamard and Fischer inequalities, through the interplay of symmetry and concavity.

The symmetrization principle on hyperbolicity cones asserts an increase in the value of a hyperbolic polynomial when that polynomial is averaged over the orbit of a finite group action that preserves both the polynomial and a prescribed hyperbolicity direction. This result connects the structure of hyperbolicity cones, group invariance, and concavity, leading to generalized inequalities for hyperbolic polynomials, with immediate consequences for principal-minor and Hadamard-type inequalities. The principle highlights the role of symmetry and convexity in optimizing polynomial values within hyperbolicity cones (Zhang, 15 Jan 2026).

1. Hyperbolic Polynomials and Hyperbolicity Cones

Let $P : \mathbb{R}^n \to \mathbb{R}$ be a homogeneous polynomial of degree $k$. Given a direction $e \in \mathbb{R}^n$, $P$ is $e$-hyperbolic if $P(e) > 0$ and, for every $x \in \mathbb{R}^n$, the univariate map $t \mapsto P(x + t e)$ only has real roots. This property allows a canonical factorization: $$P(x + t e) = P(e) \prod_{i=1}^k (t + \lambda_i(P; e, x))$$ where the $\lambda_i(P; e, x)$ are real. The (open) Gårding cone, or hyperbolicity cone, is defined as

$$\Gamma(P; e) = \{ x \in \mathbb{R}^n : \lambda_i(P; e, x) > 0 \,\,\,\,\forall i \}.$$

Gårding's theorem provides that $\Gamma(P; e)$ is a nonempty open convex cone, and that hyperbolicity is preserved together with the cone if the direction is varied inside the cone.

2. Finite Group Actions and the Reynolds Operator

Suppose a finite group $G$ acts linearly on $\mathbb{R}^n$, fixing the hyperbolicity direction ($g \cdot e = e$ for all $g \in G$), and that $P$ is $G$-invariant ($P(g \cdot x) = P(x)$). The Reynolds operator, or group-averaging operator, is defined by

$R_G(x) = \frac{1}{|G|} \sum_{g \in G} g \cdot x.$

This operator projects any $x \in \mathbb{R}^n$ to the space of $G$-fixed points.

Under these conditions, the following properties hold for every $x \in \Gamma(P; e)$:

Condition	Description	Consequence
$G$-invariance of $P$	$P(g \cdot x) = P(x)$	Group orbits preserve value
$G$ fixes $e$	$g \cdot e = e$ for all $g\in G$	Hyperbolicity direction preserved
Reynolds operator	$R_G(x)$ is group-averaged vector	$R_G(x) \in \Gamma(P; e)$

The Reynolds operator thus yields a canonical way of symmetrizing inputs to invariant hyperbolic polynomials.

3. Hyperbolic Symmetrization Principle

The hyperbolic symmetrization principle formally states:

Let $P$ be a homogeneous degree-$k$ polynomial on $\mathbb{R}^n$, hyperbolic with respect to $e$, and invariant under a finite group $G$ fixing $e$. For every $x \in \Gamma(P; e)$:

• For all $g \in G$, $g \cdot x \in \Gamma(P; e)$, thus $R_G(x)\in\Gamma(P; e)$.
• The value under symmetrization increases:

$P(R_G(x)) \geq P(x)$

Equivalently, if $f(x) = P(x)^{1/k}$, then $f$ is concave on $\Gamma(P; e)$ and

$f(R_G(x)) \geq f(x)$

for all $x \in \Gamma(P; e)$.

The concavity of $P^{1/k}$ follows from Gårding's concavity theorem. The result leverages both the properties of the cone and the invariance under group action to ensure that averaging raises the value of the polynomial.

4. Analytical Framework and Proof Outline

The proof employs the following ingredients:

• Gårding’s Concavity Theorem: For $P$ hyperbolic with respect to $e$, the map $x \mapsto P(x)^{1/k}$ is concave on the hyperbolicity cone $\Gamma(P; e)$.
• Group invariance: For any $g\in G$, $f(g \cdot x) = f(x)$.
• Jensen’s Inequality: Concavity and invariance yield

$$f(R_G(x)) \geq \frac{1}{|G|} \sum_{g \in G} f(g \cdot x) = f(x).$$

• Raising to the $k$th power: This yields $P(R_G(x)) \geq P(x)$.

For the symmetric group $S_n$, the principle specializes by invoking the Schur–Horn theorem and Birkhoff's theorem, showing that symmetrizing over coordinate permutations increases $P$ and connecting majorization to hyperbolic inequalities.

5. Principal Examples: Linear Principal Minor Polynomials and Symmetries

A key example is provided by linear principal-minor (lpm) polynomials. For an $n \times n$ symmetric matrix $X=(X_{ij})$ with coefficients $c_J$ for $J \subseteq [n]$, define

$$\mathcal{P}(X) = \sum_{J \subseteq [n]} c_J \det(X_J)$$

where $X_J$ is the principal submatrix indexed by $J$. If $\mathcal{P}$ is PSD-stable (hyperbolic with respect to $I_n$), it is invariant under conjugation by any diagonal sign matrix $D$. The sign-flip subgroup associated with a partition $\Pi = \{S_1, \ldots, S_m\}$ of $[n]$, denoted as $G_\Pi$, acts by flipping signs within blocks: $$G_\Pi = \{\operatorname{diag}(\underbrace{\pm1, \dots, \pm1}_{S_1}, \dots, \underbrace{\pm1, \dots, \pm1}_{S_m})\}.$$ Averaging over this subgroup corresponds to block-diagonal pinching, zeroing all off-block entries: $$\frac{1}{|G_\Pi|} \sum_{D\in G_\Pi} D A D = \pi_\Pi(A).$$ The symmetrization principle yields the hyperbolic Fischer–Hadamard inequalities: $$\mathcal P\bigl(\pi_{\Pi}(A)\bigr) \geq \mathcal P(A)$$ with the classic Hadamard inequality as a special case when $\Pi$ is the finest partition.

6. Connections to Classical Inequalities and Broader Significance

The symmetrization principle provides a unified conceptual framework for a range of determinant and principal-minor inequalities. Classical results such as the Hadamard and Fischer inequalities are subsumed as special cases. The tools of hyperbolicity, concavity, and symmetry collectively underpin the monotonicity of values under pinching and averaging, illuminating the structure shared by these inequalities.

7. Limitations and Generalizations

The validity of the symmetrization principle requires both hyperbolicity (to guarantee convexity of $\Gamma(P; e)$ and concavity of $P^{1/k}$) and invariance under the chosen group action. If either property fails, monotonicity under symmetrization does not necessarily hold. The principle generalizes to any finite group whose action preserves the hyperbolicity direction and leaves the polynomial invariant. Potential extensions include continuous group analogues, such as integration over compact Lie groups, in settings where invariance persists (Zhang, 15 Jan 2026).

A plausible implication is that symmetrization may offer effective strategies in optimization problems constrained to hyperbolicity cones, provided the symmetries of the problem and polynomial are appropriately exploited.