Strongly Dominant Weight Polytope

• Strongly dominant weight polytope is defined as the convex intersection of the Weyl group orbit of a dominant weight with the closed dominant chamber, yielding a structure combinatorially equivalent to an r-cube.
• Its cube equivalence enables explicit mapping of vertices and faces, yielding precise calculations of f-vectors and h-vectors and simplifying combinatorial analysis.
• The polytope underpins key results in toric geometry and total positivity, with applications ranging from the study of Peterson varieties to Lie algebra character formulas.

A strongly dominant weight polytope is a convex polytope arising in the representation theory of semisimple Lie algebras, constructed as the intersection of the Weyl group orbit polytope of a regular dominant weight with the closure of the dominant Weyl chamber. These polytopes possess rich connections to the geometry of toric varieties, combinatorics of root systems, and the topology of related algebraic varieties. For any crystallographic root system of rank $r$ and strongly dominant weight $\lambda$, the strongly dominant weight polytope $P^\lambda$ is combinatorially equivalent to the $r$-dimensional cube, and underlies several structural results in geometric representation theory and total positivity.

1. Definition and Basic Properties

Let $\Phi$ be a reduced, crystallographic root system of rank $r$ in a real vector space $V$ equipped with a $W$-invariant inner product $(\cdot\mid\cdot)$, where $W$ is the corresponding Weyl group. The set of simple roots is denoted by $\Delta = \{\alpha_1, \dots, \alpha_r\}$, with associated simple coroots $$\alpha_i^\vee = 2\alpha_i / (\alpha_i \mid \alpha_i)$$ and fundamental weights $\omega_1, \dots, \omega_r$ such that $$\langle \omega_j, \alpha_i^\vee \rangle = \delta_{ij}$$.

A weight $\lambda \in V$ is said to be strongly dominant if $(\lambda \mid \alpha_i) > 0$ for all $i=1, \dots, r$. The strongly dominant weight polytope, also called the dominant weight polytope, is defined as

$$P^\lambda := \mathrm{Conv}(W \cdot \lambda) \cap \overline{C_+}$$

where $C_+$ is the open dominant Weyl chamber.

Alternatively, this polytope can be described by the set of inequalities: $$P^\lambda = \left\{ \mu \in V ~\middle|~ \langle \alpha_i, \mu \rangle \ge 0,~ \langle \alpha_i, \lambda - \mu \rangle \ge 0,~ i=1,\dots,r \right\}$$ or, using duals (for a group $G$ of rank $n$): $$P^\lambda = \left\{ \mu \in \mathfrak t^* ~\middle|~ \alpha_i^\vee(\mu) \ge 0,~ \varpi_i^\vee(\lambda - \mu) \ge 0,~ i \in I \right\}$$ This intersection consists of points lying in both the orbit polytope of $\lambda$ under $W$ and the closed dominant chamber, forming a convex, rational polytope of dimension $r$ (Abe et al., 7 Dec 2025, Burrull et al., 2023, Gui et al., 2024).

2. Combinatorial Structure: Cube Equivalence

For every strongly dominant $\lambda$ in a root system of rank $r$, $P^\lambda$ is combinatorially equivalent to the standard $r$-cube $[0,1]^r$ (Burrull et al., 2023, Abe et al., 7 Dec 2025). The combinatorial equivalence is realized as follows:

• Vertices: Indexed by subsets $J \subseteq \{1, \dots, r\}$, with each vertex

$\mu_J = \sum_{j \in J} a_j \varpi_j$

where $\lambda = \sum_{i=1}^r a_i \varpi_i$.

• Facets: Each of the $2r$ facets is aligned along hyperplanes $\alpha_i^\vee(\mu)=0$ or $\varpi_i^\vee(\mu)=\varpi_i^\vee(\lambda)$, for $i=1,\dots, r$.

The face poset of $P^\lambda$ can be indexed by pairs of subsets $(K, J)$ with $K \subseteq J \subseteq \{1, \dots, r\}$, and the number of $k$-faces is $f_k = \binom{r}{k} 2^{r-k}$. The $h$-vector is $(1,1,\dots,1)$, matching the standard cube, and the set of vertices is in bijection with the power set of indices $I$ (Abe et al., 7 Dec 2025). An explicit combinatorial bijection aligns faces of $P^\lambda$ with those of the cube by labeling coordinates and interpreting Weyl group reflections as coordinate flips (Burrull et al., 2023).

3. Toric Geometry and Fan Structure

The normal fan of $P^\lambda$ is the restriction of the Weyl chamber fan:

• The maximal cones are the Weyl chambers $C_w = w(C_+)$ for $w \in W$, and their faces. This fan structure endows $P^\lambda$ with the properties of a smooth, rational, projective toric variety (or a toric orbifold in the non-smooth case) (Gui et al., 2024, Abe et al., 7 Dec 2025).

For the full orbit polytope $\mathrm{Conv}(W \cdot \lambda)$, the associated toric variety admits a $W$-action. The dominant weight polytope $P^\lambda$ represents a fundamental region under this $W$-action. The toric variety $X(P^\lambda)$ inherits the combinatorics and geometry of the cube, and its cohomology ring is related via an explicit ring isomorphism to the $W$-invariants in the cohomology of $X(\mathrm{Conv}(W \cdot \lambda))$ (Gui et al., 2024).

4. Topological and Geometric Applications: Peterson Varieties and Total Positivity

There is a canonical identification, via moment maps, between the polytope $P^\lambda$ and the totally nonnegative part of the Peterson variety, $Y_{\ge 0}$, for the corresponding semisimple Lie group $G$: $$Y_{\ge 0} \xrightarrow{\sim} X(\Sigma)_{\ge 0} \xrightarrow{\mu_{\ge 0}} P^\lambda$$ where $X(\Sigma)$ is a simplicial toric orbifold. $Y_{\ge 0}$ admits a regular CW decomposition with cells indexed by pairs $(K,J)$ as above and is homeomorphic to a topological cube (Abe et al., 7 Dec 2025).

This topological realization confirms that $Y_{\ge 0}$ is contractible, Eulerian, and shellable, and the face numbers match those of $P^\lambda$. Notably, the Betti numbers of Peterson varieties in all classical Lie types agree with those of the cube: $b_{2i}= \binom{r}{i}$, $b_{2i+1}=0$, with Poincaré polynomial $(1+q)^r$ (Burrull et al., 2023, Abe et al., 7 Dec 2025).

5. Orbit Structure, Dynkin Diagrams, and Lattice Points

The $W$-action on $P^\lambda$ and its faces can be classified via the combinatorics of extended Dynkin diagrams. There is a bijection between $W$-orbits of (nonempty) faces of $P^\lambda$ and connected subdiagrams of the extended Dynkin diagram containing the special node $-\lambda$ (Li et al., 2014). Every face is $W$-conjugate to a standard parabolic face, which itself is the convex hull of the orbit of $\lambda$ under a parabolic subgroup.

The affine span of any face is generated by a subset of the roots—this root-parallelism extends to all edges and higher faces. Furthermore, the set of lattice points in $P^\lambda$ is described via Demazure-type formulas, with the generating function expressible as an application of Demazure operators to $x^\lambda$ (Walton, 2021). These generating functions interpolate between Weyl's character formula and Brion's formula for polytope lattice sums.

6. Cohomological and Algebraic Structure

The cohomology ring $H^*(X(P^\lambda))$ encodes the algebraic geometry of the toric variety associated to $P^\lambda$. The Danilov–Jurkiewicz presentation shows $H^*(X(P^\lambda))$ as a quotient of a polynomial ring: $$H^*(X(P^\lambda)) \cong \frac{\mathbb{Q}[x_1, \dots, x_r, y_1, \dots, y_r]}{(x_iy_i,~ \sum\langle q, \omega_i^\vee\rangle x_i + \sum\langle q, -\beta_i\rangle y_i~:~q \in Q)}$$ There is a uniform (type-free) construction of a ring isomorphism: $$H^*(X(P^\lambda)) \cong H^*(X(\mathrm{Conv}(W\cdot \lambda)))^W$$ valid in all finite Coxeter types (Gui et al., 2024).

The associated polytope expansion of the Lie algebra character provides efficient formulas for weight multiplicities and representations, simplifying the combinatorics compared to the classical Kostant partition function. In the strongly dominant case, polytope multiplicities are $\pm 1$ and stack to recover weight multiplicities directly (Walton, 2013).

7. Examples and Special Cases

• Type $A_n$: For $\lambda = (\lambda_1 \ge \cdots \ge \lambda_{n+1})$ in $\mathbb{R}^{n+1}$, $P^\lambda$ corresponds to the classical permutohedron, and the dominant region is a cube (e.g., in $A_2$, a rectangle).
• Type $B_n$: For $$\lambda = (\lambda_1 \ge \cdots \ge \lambda_n \ge 0)$$ in $\mathbb{R}^n$, $P^\lambda$ is the $B$-permutohedron—again, the intersection with the dominant Weyl chamber is a cube.
• Application to Total Nonnegative Spaces: The cell structure and homeomorphism between $Y_{\ge 0}$ and $P^\lambda$ generalize to all Lie types, confirming conjectures on the contractibility and regularity of totally nonnegative sectors of Peterson varieties (Abe et al., 7 Dec 2025).

Object	Combinatorics	Geometry
$P^\lambda$ (strongly dominant)	Cube ($2^r$ vertices)	Toric variety/orbifold
$Y_{\ge 0}$ (Peterson)	Cube ($2^r$ cells)	Regular CW complex, contractible
Normal fan	Weyl chamber fan	Canonical toric structure

The strongly dominant weight polytope synthesizes key structures from representation theory, toric geometry, and total positivity, providing a uniform, type-independent framework for understanding the intersection of combinatorics, geometry, and topology in Lie theory (Abe et al., 7 Dec 2025, Gui et al., 2024, Burrull et al., 2023).