Dual Polar Graphs of Hermitian Polar Spaces

• The paper establishes the algebraic and combinatorial structure of dual polar graphs with explicit eigenvalue formulas and intersection arrays derived from Hermitian geometry.
• It employs geometric embeddings, apartment characterizations, and Terwilliger algebra decompositions to elucidate distance-regularity and metric properties.
• The results imply significant applications in finite geometry, coding theory, and representation theory through q-polynomial association schemes and related algebraic frameworks.

A dual polar graph of a Hermitian polar space is a distance-regular graph whose vertices correspond to maximal totally isotropic subspaces (generators) of a finite-dimensional vector space $V$ over $\mathrm{GF}(q^2)$ equipped with a nondegenerate Hermitian form. Adjacency, combinatorial invariants, metric structure, and algebraic properties of these graphs are determined by the underlying Hermitian geometry. This structure provides a paradigmatic example of a $Q$-polynomial association scheme with deep connections to buildings, representation theory, and finite geometry.

1. Construction of Hermitian Dual Polar Graphs

Given $V = \mathrm{GF}(q^2)^n$ with nondegenerate Hermitian form $h$, the Hermitian polar space $H(n-1, q^2)$ of rank $d = \lfloor n/2 \rfloor$ consists of all totally isotropic subspaces under $h$. The set of $d$-dimensional totally isotropic subspaces (generators) forms the vertex set $\Omega$ of the dual polar graph $\Gamma$. Two generators $X,Y$ are adjacent if $\dim(X \cap Y) = d-1$.

The resulting graph $\Gamma$ is distance-regular of diameter $d$. The distance between generators $X$ and $Y$ is $d(X, Y) = d - \dim(X \cap Y)$. The number of generators is $|\Omega| = \prod_{i=0}^{d-1} (q^{i+2}+1)$ for the case $n = 2d$ (Witt index $d$) (Ihringer et al., 2015, Qiao et al., 2017). The intersection array and eigenvalues are explicitly determined by $q$ and $d$.

2. Apartments and Metric Characterization

Apartments in the dual polar graph correspond to subsets indexed by frames: sets of $2d$ points $\{p_1, \ldots, p_{2d}\}$ partitioned into $d$ pairs so that within each pair the points are not collinear but every other distinct pair is collinear. The family of maximal singular subspaces selecting one point from each pair is an apartment, combinatorially forming an $n$-cube $H_d$ as an induced subgraph (Pankov, 2010).

Pankov's metric characterization (Theorem 2) states that any isometric embedding of an $m$-dimensional hypercube $H_m$ into $\Gamma$ arises from an apartment (possibly within a parabolic subspace corresponding to a lower-rank polar space), and any isometric copy of $H_d$ is a full apartment. The classification (Theorem 3) further asserts that every isometric embedding of a smaller dual polar graph into a larger arises from embedding into a parabolic determined by a singular subspace of requisite codimension. This rigidifies the geometry: all possible cube subgraphs with isometric distances are accounted for by geometric apartments (Pankov, 2010).

3. Combinatorial and Eigenvalue Structure

The Hermitian dual polar graph $\Gamma$ is $Q$-polynomial and distance-regular, with intersection numbers: $$\begin{aligned} k &= q \cdot [d]_{q^2} = q (q^{2d}-1)/(q^2-1), \ c_i &= [i]_{q^2}, \ b_{i-1} &= q \cdot [d-i+1]_{q^2}, \ a_i &= k - b_i - c_i = (q+1)([i]_{q^2} - q [i-1]_{q^2}), \quad i=1,\ldots,d, \end{aligned}$$ where $[m]_{q^2} = (q^{2m}-1)/(q^2-1)$. The eigenvalues are

$$\theta_i = q \cdot [d-i]_{q^2} - [i]_{q^2}, \qquad i=0,1,\ldots,d$$

with multiplicities given by differences of $q^2$-Gaussian coefficients (Qiao et al., 2017).

$\Gamma$ is geometric: each edge belongs to a unique Delsarte clique of size $a_1+2 = q+2$. The smallest eigenvalue is $\theta_d = -k/(a_1+1) = -(q^{2d-1}+1)/(q(q+1))$, and the graph satisfies the Krein condition with equality for a "light tail" idempotent at the minimal eigenvalue, characterizing the family among geometric $2$-bounded distance-regular graphs (Koolen et al., 2015).

4. Algebraic, Module, and Polynomial Structures

The Terwilliger (subconstituent) algebra $\mathbb{T}(x)$ generated by adjacency and dual adjacency matrices acting on the standard module decomposes into irreducible thin modules. On each irreducible, the restrictions of adjacency and dual adjacency satisfy the Leonard pair tridiagonal relations, with split sequences and eigenparameters matching dual $q$-Krawtchouk type (Worawannotai, 2012).

A fundamental correspondence is established with the quantum algebra $U_q(\mathfrak{sl}_2)$: $\mathbb{T}(x)$ is generated by two commuting $U_q(\mathfrak{sl}_2)$ actions up to its center, and the spectrum and recurrences of the adjacency operator correspond to those of dual $q$-Krawtchouk polynomials. In the Hermitian case, the generalized Terwilliger algebra with respect to base vertex and maximal clique admits a primary $2d$-dimensional module which realizes a rank-one nil-DAHA of type $(C^\vee_1, C_1)$, precisely encoding the dual $q$-Krawtchouk polynomials and their non-symmetric versions (Lee et al., 2017).

5. Clique Classification and Metric Dimension

Erdős–Ko–Rado-type theorems describe maximal $\{0,1,\ldots,t\}$-cliques in the dual polar graph: for $0 \le t < \frac{8d}{5}-2$, every maximal $t$-clique is the set of all generators containing a fixed totally isotropic subspace of codimension $t/2$ (for even $t$) or $(t+1)/2$ (for odd $t$). The size and structure of these cliques meet the Hoffman bound with equality, and there is a complete classification for large $q$ (Ihringer et al., 2015).

The metric dimension $\mu(\Gamma)$, or smallest resolving set, satisfies the upper bound

$$\mu(\Gamma) \leq q^{d-1/2}\frac{q^{d-1/2}+1}{q^{1/2}+1}$$

where $q$ is the base field size and $d$ is the Witt index of the Hermitian form (Bailey et al., 2017). No matching lower bounds are known for arbitrary $d$.

6. Characterization Theorems and Isomorphism Criteria

A detailed characterization (Qiao–Koolen) shows that under the conditions of non-bipartiteness, geometricity, diameter $D\geq 4$, $c_2 \geq 3$, and smallest eigenvalue $\leq -k/2$, the only possible distance-regular graph with parameters matching $a_i = c_i(q+1)$ for $i=1,2$ and a specific relation for $c_3$ is the Hermitian dual polar graph $^2A_{2D-1}(q)$. These numerical identities, together with spectral information, rigidly determine the graph's isomorphism class (Qiao et al., 2017).

The light tail condition at the minimal eigenvalue provides a characterization coinciding precisely with the Hermitian dual polar graphs among distance-regular $2$-bounded graphs with the prescribed intersection array and eigenvalue structure (Koolen et al., 2015).

7. Broader Significance and Applications

Dual polar graphs of Hermitian spaces are central objects in the theory of finite geometries, combinatorics, and algebraic graph theory. Their apartments encode the cube geometry of Tits buildings, and the combinatorial embedding structure is completely dictated by the underlying geometry. Their highly regular structure underlies the classification of geometric distance-regular graphs and is instrumental in the paper of Leonard systems, $q$-polynomial association schemes, and related objects in algebraic combinatorics and representation theory (Pankov, 2010, Qiao et al., 2017, Worawannotai, 2012).

These graphs serve as fertile ground for connections to quantum algebra, representation theory (via $U_q(\mathfrak{sl}_2)$ and DAHA), and the theory of special polynomials, especially dual $q$-Krawtchouk families and their non-symmetric generalizations (Lee et al., 2017, Worawannotai, 2012). Advanced applications include metric dimension calculations, clique size bounds, and explicit control over spectral properties relevant to coding theory and extremal set theory (Bailey et al., 2017, Ihringer et al., 2015).