Towards combinatorial characterization of the smoothness of Hessenberg Schubert varieties

This paper establishes a necessary and sufficient combinatorial condition, characterized by pattern avoidance in GKM graphs, for the smoothness of intersections between Schubert and regular semisimple Hessenberg varieties of type $A$, thereby providing a sufficient condition for the smoothness of Hessenberg Schubert varieties.

The Gist

Imagine you are an architect designing a massive, intricate city made entirely of mathematical shapes. This city is called the Flag Variety. It's a place where every building is a "Schubert Variety," and the city itself is built on a grid of permutations (rearrangements of numbers).

Now, imagine you want to build a specific neighborhood within this city. You take a slice of the city (a Schubert variety) and intersect it with a special, curved wall called a Hessenberg Variety. The resulting intersection is your new neighborhood, which the authors call a Hessenberg Schubert Variety.

The big question the authors ask is: Is this new neighborhood smooth?

In math, "smooth" doesn't mean "polished." It means the shape has no sharp corners, jagged edges, or weird pinched points. If a shape is "singular" (not smooth), it's like a crumpled piece of paper or a star-shaped rock with sharp spikes. If it's "smooth," it's like a perfect sphere or a flat plane.

Here is the story of how the authors solved this puzzle, explained with everyday analogies.

1. The Map and the Grid (GKM Graphs)

To check if a shape is smooth, the authors don't look at the shape directly (which is too complex). Instead, they look at a map of the shape, called a GKM graph.

• The Analogy: Imagine the shape is a city. The GKM graph is a subway map.
  • The dots (vertices) on the map are the "fixed points" (like major landmarks or train stations).
  • The lines (edges) connecting them are the paths between them.
• The Rule of Smoothness: The authors discovered a golden rule: A shape is smooth if and only if its subway map is "regular."
  • A "regular" map means every station has the exact same number of tracks leaving it. If one station has 3 tracks and its neighbor has 4, the map is "irregular," and the shape has a jagged corner.

2. The Pattern Detective Work

The authors realized that whether a map is regular depends entirely on the order of the numbers in the permutation that defines the neighborhood. They looked for specific "bad patterns" in the sequence of numbers.

Think of the numbers in a permutation like a line of people waiting for a bus.

• The Classic Case: In the old days (for standard Schubert varieties), mathematicians knew that if the line of people formed a specific "bad pattern" (like a tall person standing between two short people in a specific way), the shape would be jagged.
• The New Challenge: When you add the "Hessenberg wall" (the special curved wall), the rules change. The "bad patterns" become more complex because the wall restricts which people can stand next to whom.

3. The "Forbidden Moves" (Pattern Avoidance)

The authors spent the paper acting like detectives, hunting for 10 specific "forbidden patterns" in the line of people.

• The Metaphor: Imagine you are playing a game of "Musical Chairs" with numbers. The Hessenberg function (the wall) tells you which chairs are available.
• If the numbers arrange themselves in one of these 10 specific "forbidden dances" (patterns like 2-1-4-3 or 2-5-3-1-4), the resulting shape will have a sharp corner.
• If the numbers avoid all these dances, the shape is perfectly smooth.

They found that for some specific types of neighborhoods, you only need to avoid 7 patterns. But for the general case (any neighborhood), you need to avoid 10 patterns.

4. The "Representative" Trick

Here is a clever trick the authors used. Sometimes, the neighborhood you are looking at is messy and hard to analyze. But they proved that every messy neighborhood has a clean, tidy twin (called the representative permutation, $r_w$) that looks exactly the same in terms of its subway map.

• The Analogy: It's like looking at a messy room. Instead of trying to clean the whole room, you find a "clean twin" room that is identical in layout. If the twin room is smooth, your messy room is smooth too.
• This allowed them to simplify the problem: "Don't worry about the messy version; just check the patterns in the clean twin."

5. The Big Conclusion

The paper provides a checklist for mathematicians.

If you want to know if a specific Hessenberg Schubert variety is smooth:

1. Look at the sequence of numbers defining it.
2. Check if that sequence contains any of the 10 forbidden patterns (which depend on the specific "wall" or Hessenberg function you are using).
3. If it contains a pattern: The shape has a jagged corner (it is singular).
4. If it avoids all patterns: The shape is perfectly smooth.

Why Does This Matter?

This isn't just about drawing pretty shapes. These shapes are deeply connected to:

• Physics: Understanding how particles interact.
• Computer Science: Algorithms for sorting and organizing data.
• Symmetry: Understanding the fundamental symmetries of the universe.

By turning a complex geometric problem into a simple "pattern avoidance" game, the authors gave mathematicians a powerful, easy-to-use tool to predict the smoothness of these complex mathematical worlds without having to build them first. They turned a mountain of calculus into a simple game of "Spot the Pattern."

Technical Summary

1. Problem Statement

The paper addresses the geometric problem of characterizing the smoothness of Hessenberg Schubert varieties.

• Context: A Hessenberg Schubert variety ($\Omega_{w,h}$) is defined as the closure of the intersection between a Schubert cell ($\Omega^\circ_w$) and a regular semisimple Hessenberg variety ($Hess(S, h)$).
• The Gap: While the smoothness of standard Schubert varieties ($\Omega_w$) is well-understood via pattern avoidance (specifically avoiding patterns 2143 and 1324), the smoothness of their intersections with Hessenberg varieties is more complex.
• Specific Challenge: The intersection $\Omega_w \cap Hess(S, h)$ is not always irreducible. The authors focus on the condition under which this intersection is smooth, which implies the smoothness of the Hessenberg Schubert variety $\Omega_{w,h}$. The core difficulty lies in the fact that the smoothness of the intersection depends on the specific Hessenberg function $h$, which dictates the structure of the variety.

2. Methodology

The authors employ a combination of algebraic geometry, combinatorics, and graph theory, specifically utilizing GKM (Goresky–Kottwitz–MacPherson) theory.

• GKM Graphs: They utilize the fact that Hessenberg varieties are GKM varieties. The smoothness of a GKM variety is linked to the regularity of its associated GKM graph (where vertices are torus-fixed points and edges represent 1-dimensional orbits).
  • A key result (Proposition 2.11) states that if the intersection $\Omega_w \cap Hess(S, h)$ is smooth and connected, its GKM graph $\Gamma_{w,h}$ must be regular (all vertices have the same degree).
• h-Bruhat Order: The authors introduce the $h$-Bruhat order, a refinement of the standard Bruhat order based on the Hessenberg function $h$. They characterize the set of torus-fixed points of the Hessenberg Schubert variety as an interval in this order.
• Edge Injection Maps: A central technical tool is the construction of an injective map $\phi_{uv}$ between the sets of edges incident to two vertices $u$ and $v$ in the GKM graph, where $u \prec_h v$. This map is used to compare the degrees of vertices.
• Pattern Avoidance: The authors define specific associated patterns (permutation patterns constrained by the Hessenberg function $h$). They investigate how the presence of these patterns affects the injectivity and surjectivity of the edge maps, thereby determining the regularity of the graph.

3. Key Contributions

A. Structural Characterization via h-Bruhat Order

• Theorem A: The authors prove that for a permutation $w$ in the set $G_h$ (permutations satisfying specific conditions relative to $h$), the set of torus-fixed points $\Omega^T_{w,h}$ corresponds exactly to the interval $[w, w_0]_h$ in the $h$-Bruhat order.
• Degree Monotonicity (Theorem B): They establish that in the graph $\Gamma_{w,h}$, the degree of a vertex is non-decreasing with respect to the $h$-Bruhat order. This implies that the graph is regular if and only if the degrees of the minimal element ($w$) and the maximal element ($w_0$) are equal.

B. Combinatorial Characterization of Regularity

The paper provides a complete combinatorial characterization of when the GKM graph $\Gamma_{w,h}$ is regular, distinguishing between two cases:

1. Case 1: $w \in G_h$ (The "Good" Set)

   • Theorem C (Part 1): For $w \in G_h$, the graph $\Gamma_{w,h}$ is regular if and only if $w$ avoids seven specific associated patterns of length 4:
     • $2143_h, 1324_h, 1243_h, 2134_h, 1423_h, 2314_h, 2413_h$.
   • These patterns are defined relative to the incomparability graph of the Hessenberg function $h$.

2. Case 2: Arbitrary $w \in S_n$

   • Since $\Gamma_{w,h}$ is isomorphic to $\Gamma_{r_w, h}$ (where $r_w \in G_h$ is a unique representative), the authors extend the result to all permutations.
   • Theorem C (Part 2): For any $w \in S_n$, $\Gamma_{w,h}$ is regular if and only if $w$ avoids a set of ten patterns. This set includes the seven from Case 1 plus three additional patterns of length 5:
     • $25314_h, 24315_h, 14325_h, 15324_h$.
   • The authors prove that avoiding the length-4 pattern $2413_h$in$G_h$ is equivalent to avoiding the set of length-5 patterns in the general case.

C. Smoothness Implications

• Sufficient Condition: By combining the regularity results with the work of Insko, Precup, and Woo (who linked regularity to smoothness for connected intersections), the authors show that avoiding these patterns is a sufficient condition for the smoothness of the Hessenberg Schubert variety $\Omega_{w,h}$.
• Necessary Condition: They also provide a necessary condition for the smoothness of the intersection $\Omega_w \cap Hess(S, h)$.

4. Key Results

• Generalization of Classical Results: When $h = (n, n, \dots, n)$, the Hessenberg variety is the full flag variety. In this case, the associated patterns reduce to the classical $2143$and$1324$, recovering the known smoothness criteria for Schubert varieties (Theorem 1.1).
• Permutahedral Variety: When $h = (2, 3, \dots, n, n)$, the variety is the permutohedral variety (smooth). The authors show that in this specific case, all permutations avoid the relevant patterns, consistent with the known smoothness of the variety.
• Algorithmic Check: The results provide a concrete, combinatorial algorithm to check the smoothness of these varieties by inspecting the permutation $w$ and the function $h$ for specific subword patterns.

5. Significance

• Bridging Geometry and Combinatorics: The paper successfully translates a deep geometric property (smoothness) into a purely combinatorial condition (pattern avoidance) dependent on the Hessenberg function.
• Resolution of Open Problems: It extends the classical theory of Schubert variety smoothness to the more complex setting of Hessenberg Schubert varieties, which are central to the Stanley-Stembridge conjecture and the study of cohomology modules.
• Foundation for Future Work: While the paper establishes a sufficient condition for smoothness, it explicitly leaves open the question of whether this condition is also necessary for the smoothness of the variety itself (as opposed to the intersection). This sets the stage for future research into the precise geometric characterization of singularities in these varieties.

In summary, the paper provides a rigorous combinatorial framework using GKM graphs and pattern avoidance to determine the regularity of Hessenberg Schubert varieties, offering a powerful tool for researchers in algebraic combinatorics and algebraic geometry.