Schubert Cycle Expansion in Flag Varieties

• Schubert cycle expansion is the process of expressing (co)homology classes of Richardson varieties as linear combinations of Schubert classes, with coefficients derived from combinatorial and geometric data.
• The expansion employs combinatorial models—such as (p,q)-clans in type A—to parameterize symmetric subgroup orbits and yield explicit, multiplicity-free structure constants.
• This framework bridges geometry, combinatorics, and representation theory by offering subtraction-free formulas that simplify Schubert calculus and suggest extensions to other types like B and D.

Schubert cycle expansion refers to the representation of (co)homology classes of subvarieties of the flag variety—especially Richardson varieties—as linear combinations in the Schubert basis, with coefficients encoding fundamental geometric, combinatorial, or representation-theoretic information. In the context of Richardson varieties stable under spherical Levi subgroups, this expansion is governed by Brion’s theorem, combinatorial models parametrizing symmetric subgroup orbits, and explicit positivity results for the associated structure constants in several cases. The interplay of these elements yields manifestly positive, often multiplicity-free, combinatorial formulas for the expansion coefficients and establishes new connections between representation theory, geometry, and combinatorics.

1. Brion’s Theorem for Spherical Subgroup Orbits

The critical structural input is Brion’s theorem, which describes the expansion of the cohomology class of an orbit closure of a spherical subgroup $H$ on the flag variety $G/B$ in the Schubert basis. For a spherical subgroup $H \leq G$ and an $H$-orbit closure $Y \subset G/B$: $[Y] = \sum_{w \in W(Y)} 2^{D(w)} \cdot S_w$ where:

• $W(Y)$ is the set of Weyl group elements indexing paths in the weak order graph from $Y$ to the dense orbit.
• $D(w)$ is the number of “double edges” (edges associated to noncompact imaginary roots of type II) encountered in (any) reduced expression for $w$.

For a Richardson variety $X^u_v = X^u \cap X_v$ (intersection of a Schubert and an opposite Schubert variety), if $X^u_v$ is stable under a spherical Levi subgroup $L$, it is the closure of an $L$-orbit. Brion’s theorem then holds: $$[X^u_v] = S_{w_0 u} \cdot S_v = \sum_{w \in W(Y)} 2^{D(w)} S_w$$ where $Y$ is the associated $L$-orbit closure. In type A, all edges are simple, so $2^{D(w)} = 1$, and the expansion is multiplicity-free.

2. Clans and the Combinatorial Model for Orbits

In type $A$, the pair $$(G, L) = (GL(p+q,\mathbb{C}), GL(p,\mathbb{C}) \times GL(q,\mathbb{C}))$$ leads to an $L$-orbit combinatorial model parametrized by $(p, q)$-clans. A (p, q)-clan is a string on $n=p+q$ positions filled with “+”, “–”, or a natural number, with each number appearing exactly twice (up to relabeling), and the difference between “+” and “–” is $p-q$. The weak order on the closure relations among $L$-orbit closures is governed by explicit operations on clans, corresponding to Weyl group reflections.

The action of a Weyl group element $w$ on a clan $\gamma$ follows two primary rules: swapping entries at positions $i$ and $i+1$ (unless these are opposite signs, in which case they are replaced with a new matching numeric symbol). Thus, a path in the weak order graph corresponds to a sequence of such operations, and structure constants can be read from the combinatorics of the action.

3. Explicit Schubert Structure Constants for Special Cases

For special permutations $u$ and $v$ (specifically, $u$ the inverse of a Grassmannian permutation with unique descent at $p$, and $v$ inverse to one with unique descent at $q$), the expansion coefficient $c_{w_0 u, v}^w$ is given by: $$c_{w_0 u, v}^{w} = \begin{cases} 1, & w \cdot \gamma(u,v) = \gamma_0\ 0, & \text{otherwise} \end{cases}$$ where $\gamma(u, v)$ is the clan associated to the pair $(u,v)$, $\gamma_0$ is the dense $L$-orbit clan, and $w$ is required to have length equal to the codimension of $X^u_v$. This formula (Theorem 3.11) is positive and combinatorial: to determine the coefficient, one simply checks the action of $w$ on $\gamma(u, v)$ in the clan model.

In type $A$, all edge multiplicities are one, so all coefficients in the expansion are 0 or 1.

4. Conjectures in Types B and D

For types $B$ and $D$ (i.e., $G = SO(2n+1,\mathbb{C})$ or $SO(2n,\mathbb{C})$ with $L$ a spherical Levi subgroup), the authors propose similar combinatorial expansions, conjecturing that:

• For type $B$, coefficients may take values $0$, $1$, or $2$—the value $2$ arises when a specific scenario occurs in the action of $M(W)$ on the clan.
• For type $D$, coefficients remain $0$ or $1$.

The basic philosophy remains: construct a clan model for the $L$-orbits, identify the closure corresponding to the Richardson variety, use Brion-type expansions (with possible double edges), and read off structure constants via combinatorics.

5. Positivity, Applications, and Implications

Key implications include:

• All described structure constants in type $A$ (and conjecturally in types $B$ and $D$) are positive (subtraction-free). This advances the program of providing positive formulas for Schubert structure constants, a central question in Schubert calculus.
• The geometry of $L$-orbit closures translates intersection theoretic computations into combinatorics on clans, linking geometric representation theory (via orbits and equivariant geometry) with explicit combinatorial algorithms.
• This connection offers new perspectives on singularity properties (e.g., rational smoothness and pattern avoidance) and enables the study of Richardson varieties' local geometry.
• Algorithmic computations for Schubert calculus benefit from this explicit expansion, with implications for computational representation theory frameworks, such as ATLAS.

6. Summary Table: Key Structural Ingredients

Aspect	Type $A$ (GL)	Type $B$/$D$ (SO)
Orbit parametrization	$(p,q)$-clans	symmetric clans (Matsuki–Oshima)
Expansion via Brion’s theorem	$[Y]=\sum_w S_w$ (multiplicity-free)	$[Y]=\sum_w 2^{D(w)} S_w$
Expansion coefficient values	$0, 1$	$0, 1$ ($D$); $0, 1, 2$ ($B$)
Coefficient computation	$w \cdot \gamma = \gamma_0$ condition	as above, with possible $2$s
Application to positivity	Unconditional	Conjectural

7. Research Directions and Extensions

• Extension of explicit combinatorial rules for structure constants beyond type $A$ remains an active direction, hinging on verifying conjectures for types $B$ and $D$ or constructing clan analogues.
• The interface between local geometry (e.g., rational smoothness criteria) of Schubert varieties and the combinatorics of clans suggests new approaches for classification and singularity theory.
• Since the expansion provides effective formulas computationally, implementing these rules in computational packages opens Schubert calculus to wider experimentation and large-scale data analysis.
• The reduction to positivity using these expansions may have broader impact in neighboring domains, such as the theory of total positivity or the exploration of representation spaces of real forms.

The expansion of classes of Richardson varieties (stable under spherical Levi subgroups) in the Schubert basis, via Brion’s theorem and combinatorial models of orbits, provides not only subtraction-free structure constants for large classes but also structural bridges between geometry, combinatorics, and representation theory, offering new opportunities for theoretical development and computation in Schubert calculus (Wyser, 2012).