Extended WY Representation

• Extended WY Representation is a combinatorial framework that produces explicit polynomial representatives for orbit closures in type-A symmetric spaces using divided-difference operators.
• It constructs both cohomology and equivariant K-theory classes via seed polynomials and recurrence relations, guaranteeing nonnegative coefficients and stability under group extensions.
• The method unifies orthogonal and symplectic cases by leveraging solid and dashed edge rules, thereby bridging geometric orbit classification with Schubert calculus.

The Extended WY (Wyser–Yong) Representation refers to explicit polynomial representatives for the cohomology and $K$-theory classes of orbit closures in type-A symmetric spaces, specifically for the symmetric pairs $(GL_n, O_n)$ (orthogonal case) and $(GL_{2n}, Sp_{2n})$ (symplectic case). These polynomials, denoted $\Upsilon_\pi$ and their $K$-theoretic analogues $\Upsilon^K_\pi$, extend earlier constructions for $(GL_{p+q}, GL_p \times GL_q)$ and deliver universal, combinatorially rich representatives for orbit closures under symmetry constraints imposed by $O_n$ or $Sp_{2n}$. The representation is dictated by divided-difference operators and specific recurrence relations, with proven combinatorial, stability, and equivariant $K$-theoretic properties (Wyser et al., 2013).

1. Symmetric Pair Setting and Orbit Classification

For type-A symmetric pairs, the Extended WY Representation is formulated on the full flag varieties $Fl_n = GL_n/B$ and $Fl_{2n} = GL_{2n}/B$, where $B$ is the Borel subgroup of upper triangular matrices.

• For $(GL_n, O_n)$, $O_n$ orbits in $Fl_n$ are indexed by involutions $\pi \in S_n$ (i.e., $\pi^2 = \mathrm{id}$). The closure $Y_\pi \subset Fl_n$ corresponds to the involution $\pi$. The unique closed orbit is given by the long involution $w_0=(1\, n)(2\, n-1)\cdots$; the open dense orbit corresponds to the identity.
• For $(GL_{2n}, Sp_{2n})$, $Sp_{2n}$ orbits in $Fl_{2n}$ are indexed by fixed-point-free involutions $\pi \in S_{2n}$ (products of $n$ disjoint transpositions). The closure $X_\pi\subset Fl_{2n}$ corresponds to $\pi$. The unique closed orbit is associated with $w_0(i)=2n+1-i$; the open orbit to $(1\,2)(3\,4)\cdots(2n-1\,2n)$.

Weak order graphs encode covering relations by transpositions $s_i$. In $(GL_n, O_n)$, each such edge is called "solid" or "dashed" depending on cohomological push-pull relations: solid edges correspond to the classic divided-difference operator $\partial_i(f) = \frac{f - s_i f}{x_i-x_{i+1}}$, while dashed edges introduce an additional scaling by $1/2$.

2. Construction of WY-Polynomials

The core combinatorial objects are the $\Upsilon_\pi$ (cohomology) and $\Upsilon^K_\pi$ ($K$-theory) polynomials. Construction begins with explicit "seed" polynomials for closed orbits:

• Orthogonal seed:

$$\Upsilon_{w_0;(GL_n, O_n)} = \prod_{1\le i\le j\le n-i} (x_i + x_j)$$

• Symplectic seed:

$$\Upsilon_{w_0;(GL_{2n}, Sp_{2n})} = \prod_{1\le i < j \le 2n-i} (x_i + x_j)$$

Given any involution $\pi$, $\Upsilon_\pi$ is constructed by descending from the closed orbit along a saturated path in weak order. At each step, apply $\partial_i$ for a solid edge, and $\tfrac12 \partial_i$ for a dashed edge (orthogonal case). Formally, this defines:

$$\Upsilon_{\pi;(G,K)} = \left(\cdots \tfrac12^{\epsilon_{i_2}} \partial_{i_2} \right) \circ \left(\tfrac12^{\epsilon_{i_1}}\partial_{i_1}\right)\, \Upsilon_{w_0;(G,K)}$$

where $\epsilon_{i_j}$ tracks dashed edges. This construction is independent of path by [(Wyser et al., 2013), Theorem 1.4].

Transition relations are thus:

$$\Upsilon_{s_i\cdot\pi} = \begin{cases} \partial_i\,\Upsilon_{\pi} & \text{(solid edge)} \ \tfrac12 \partial_i\,\Upsilon_{\pi} & \text{(dashed edge)} \end{cases}$$

3. Combinatorial and Functorial Properties

WY-polynomials enjoy key structural properties:

• Nonnegativity: $\Upsilon_\pi$ decomposes as a non-negative integer combination of classical Schubert polynomials $\mathfrak{S}_w$, with coefficients $c_{\pi,w} \in \mathbb Z_{\ge 0}$:

$$\Upsilon_{\pi} = \sum_{w} c_{\pi,w} \mathfrak{S}_w(x_1,\dots,x_N)$$

Monomial coefficients are thus also nonnegative.

• Stability under $GL_N \hookrightarrow GL_{N+1}$: Label-preserving inclusions of involutions induce natural compatibilities between flag variety embeddings; the stable limit of WY-polynomials is well defined: $$\Upsilon_{\pi';(GL_{N+1},K)} = \Upsilon_{\pi;(GL_N,K)}$$.

4. Equivariant $K$-Theory Analogues

The WY framework extends to equivariant $K$-theory using the Demazure operator:

$D_i(f) = \frac{x_{i+1}f - x_i s_i f}{x_{i+1}-x_i}$

For $(GL_{2n}, Sp_{2n})$, the $K$-theoretic seed is

$$\Upsilon^{K}_{w_0;(GL_{2n}, Sp_{2n})} = \prod_{1\leq i<j\leq 2n-i}(1 - x_i x_j)$$

Strings of $D_i$'s along saturated weak-order paths produce well-defined $\Upsilon^K_\pi$, representing structure-sheaf classes in $K^0_S(Fl_{2n})$. These retain the stability property. For $(GL_n, O_n)$, only the seed polynomial $\prod_{1\leq i \leq j \leq n-i}(1 - x_i x_j)$ is given in closed form; other cases are computed via explicit multigraded $K$-polynomial methods in small rank [(Wyser et al., 2013), §4].

5. Explicit Examples and Comparison with Classical Settings

• Cohomology, $(GL_4, O_4)$: The involutions in $S_4$ index ten orbits. The closed orbit class is

$$\Upsilon_{(1\,4)(2\,3)} = 4\,x_1 x_2 (x_1 + x_2)(x_1 + x_3)$$

The remaining WY-polynomials are obtained from the seed by divided-difference operators. Full tables for low rank appear in [(Wyser et al., 2013), Table 1].

• Cohomology, $(GL_6, Sp_6)$: There are 15 fixed-point-free involutions indexing orbit closures. Each is produced via divided-difference steps from the seed.
• Equivariant $K$-theory: For $(GL_6, Sp_6)$, explicit $K$-polynomials for all 15 orbits are listed.
• Comparison: Unlike the classical $(GL_{p+q}, GL_p\times GL_q)$ case, the extended WY framework for orthogonal and symplectic cases features a unique closed orbit and "dashed" edges in the orthogonal setting. The polynomial construction principle—seed plus repeated operator action—remains universal.

6. Mathematical and Geometric Significance

Extended WY Representations provide a pathway to explicit, path-independent, and functorial representatives of orbit closure classes in the cohomology and $K$-theory of type-A symmetric flag varieties. They encode rich combinatorics that mirror geometry: nonnegativity connects to positivity in cohomology, stability ensures functorial behavior under group extensions, and explicit operator calculus enables computation in small ranks. The availability of $K$-theoretic analogues in the symplectic setting amplifies the applicability to representation theory, intersection theory, and the study of degeneracy loci.

These constructions generalize and unify previous approaches for arbitrary symmetric pairs in type A, providing concrete links between orbit geometry, Schubert calculus, and equivariant $K$-theory. The formalism is vital for computational applications, enumeration of orbit closures, and further generalizations to other types and equivariant contexts.

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For a comprehensive and technical treatment of the Extended WY Representation, detailed proofs, explicit low-rank tables, and illustration of all operator identities, see Wyser–Yong, "Polynomials for symmetric orbit closures in the flag variety" (Wyser et al., 2013).