The combinatorial invariance problem for Kazhdan–Lusztig–Vogan polynomials

KLV theory developed from the work of Lusztig–Vogan (opens in a new tab) and Vogan (opens in a new tab) on orbit-closure singularities and characters of real reductive groups. Its combinatorial invariance problem concerns which combinatorial data determine the associated polynomials, and which structure-preserving identifications preserve their values.

Several established results provide starting points.

1. Rational smoothness from orbit combinatorics. In settings including the action of $Sp_{2n}$ on the flag variety of $SL_{2n}$, Hultman (opens in a new tab) gives Bruhat-graph criteria for rational smoothness. Under his hypotheses, global rational smoothness is characterized by rank symmetry of the corresponding principal interval. This recovers an important geometric property from combinatorial information.

2. KLV invariance for fixed-point-free involutions. Abdallah–Hultman, Theorem 1.2 (opens in a new tab), prove a direct combinatorial invariance theorem for $(SL_{2n},Sp_{2n})$. Let $F_{2n}$ be the fixed-point-free involutions in $S_{2n}$, with the induced Bruhat order. In their indexing convention, an isomorphism

   $$f:[u,w_0]\longrightarrow[u',w_0]$$

   of upper intervals implies

   $$P_{v,u}(q)=P_{f(v),u'}(q)\qquad(v\geq u).$$

   Their proof uses special partial matchings: order-compatible operations pairing vertices with adjacent vertices, with fixed points allowed. The theorem uses the full upper intervals; it does not assert invariance for arbitrary endpoint intervals.

3. Interval patterns and local geometry in type AIII. Clans are combinatorial labels for $GL_p\times GL_q$-orbits on the flag variety of $GL_{p+q}$. Woo–Wyser–Yong, Theorem 4.5 (opens in a new tab), show that their interval pattern embeddings of clans induce isomorphisms between the associated Mars–Springer slices, which model local orbit-closure singularities. Consequently, these embeddings preserve local intersection cohomology and the corresponding KLV polynomials. The pattern-embedding hypothesis retains more structure than an abstract poset isomorphism.

4. Matching-based recursions and kernels. Marietti, Theorem 6.2 (opens in a new tab), recovers the fixed-point-free KLV $R$- and $Q$-polynomials through abstract recursions based on special partial matchings. Caselli–Marietti, Theorem 3.3 (opens in a new tab), give an additional symmetry condition under which the resulting $R$-polynomials form a poset kernel—the auxiliary data governing a KLS recursion. They verify this condition in the relevant symplectic KLV setting. These results apply under specific compatibility hypotheses on the matching data.

I tried some computations with Atlas in type AIII to explore additional combinatorial data that might support invariance. The terminology and numbering used below are as follows.

A block is a linked family of representation parameters at a fixed infinitesimal character, organized by cross actions and Cayley transformations. Each vertex in the computation represents one parameter. I selected blocks at infinitesimal character $\rho$, the half-sum of the chosen positive roots. This is regular integral: it lies on no root hyperplane and has integral pairings with coroots.

Atlas returns an ordered parameter list. I write $\xi_i$ for the parameter at zero-based position $i$. For the selected $SU(4,3)$ block, there are 665 parameters, numbered $0,\ldots,664$. These integers are labels from the recorded computation.

The parameters carry the block’s Bruhat partial order, denoted $\leq_{\mathcal B}$. The notation

$$[i,j]=[\xi_i,\xi_j]
=\{\xi\in\mathcal B:\xi_i\leq_{\mathcal B}\xi\leq_{\mathcal B}\xi_j\}$$

therefore means a parameter interval in this partial order. It does not mean all integer labels between $i$ and $j$. A Hasse diagram records the cover relations: an edge joins two comparable parameters when no parameter lies strictly between them.

Atlas also assigns a length to each parameter. In the tested blocks, each Hasse cover increases length by one. The length difference of an interval is

$$\ell(\xi_j)-\ell(\xi_i).$$

Its endpoint KLV polynomial is $P_{\xi_i,\xi_j}(q)$, the polynomial attached to its lower and upper parameters.

For each simple root, the root type records the relevant complex, real, or imaginary behavior, including ascent/descent, parity, compactness, and type-I/type-II distinctions where applicable. A simple-root cross action is an involution on parameters. Cayley correspondences, where defined, relate parameters of different Cartan types and can have multiple images. These operations and their types describe elementary rank-one Hecke-module data beyond the Hasse diagram. Trapa’s lectures (opens in a new tab) give a detailed account.

In this experiment, the one-step boundary of an interval $I$ consists of parameters outside $I$ connected to $I$ by one cross action or an allowed Cayley correspondence. Thus “one step” refers to these particular operations. If several transitions reach the same external parameter, it is retained as one shared boundary vertex.

The compared local data contain:

• The interval’s order and the relative lengths of all interval and boundary vertices.
• The complete simple-root types of those vertices.
• All labelled cross/Cayley relations with at least one endpoint in the interval.
• The actual identifications among shared boundary endpoints.

Relations between two boundary vertices are omitted. An isomorphism must preserve the marked interval and all retained data. Polynomial values are compared afterwards and are never used to find the isomorphism.

A whole-block symmetry preserves the corresponding structure on the entire parameter block. I excluded a pair whenever such a symmetry carries one interval to the other, allowing Dynkin diagram reversal. Comparisons were made within each block.

The tests covered ten selected regular-integral AIII blocks: the $SU(p,q)$ forms with

$$3\leq p+q\leq6,\qquad p\geq q\geq1,$$

together with $SU(5,2)$ and $SU(4,3)$.

There were 77,761 intervals, including singleton intervals, of which 17,704 had nonconstant endpoint polynomials. After identifying intervals related by whole-block symmetries, 306 pairs of distinct global symmetry orbits had isomorphic retained local data. All occurred in $SU(4,3)$, and their endpoint polynomials agreed in every case.

Four pairs involved nonconstant polynomials, all equal to $1+q$:

• First parameter interval $[288,550]$; second parameter interval $[289,552]$. Length difference: 3. Each interval contains 10 parameters and has 36 one-step boundary parameters.
• First parameter interval $[182,559]$; second parameter interval $[183,560]$. Length difference: 4. Each interval contains 15 parameters and has 48 one-step boundary parameters.
• First parameter interval $[285,601]$; second parameter interval $[286,601]$. Length difference: 4. Each interval contains 17 parameters and has 54 one-step boundary parameters.
• First parameter interval $[182,601]$; second parameter interval $[183,601]$. Length difference: 5. Each interval contains 24 parameters and has 63 one-step boundary parameters.

For example, the first pair compares two decorated configurations, each containing ten interval vertices and 36 boundary vertices. Four boundary vertices on each side are shared by multiple interior vertices, and those identifications are preserved. The endpoint polynomials satisfy

$$P_{\xi_{288},\xi_{550}}(q)
=
P_{\xi_{289},\xi_{552}}(q)
=1+q.$$

All four local isomorphisms preserve the simple-root labels individually. None of these pairs is related by a whole-block symmetry of the specified kind, even when diagram reversal is allowed. I also checked all eight endpoint coefficients separately using KL_column.

The parameter numbering refers to Atlas revision 4d3e9449062a07c1c85f4e6df215eb6ccc0eeae9; the corresponding inputs, parameter lists, and isomorphism maps are retained with the computation records.

These are partial results for one candidate choice of combinatorial data. The absence of counterexamples provides limited evidence beyond whole-block symmetry, but the effective nonconstant sample consists of only four pairs with polynomial $1+q$. It does not establish a general invariance criterion or determine which retained data are necessary.