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arxiv: 2605.01332 · v1 · submitted 2026-05-02 · 🧮 math.CO · math.AG

Toric Schubert Varieties in Partial Flag Varieties

Mahir Bilen Can , Arpita Nayek , Pinakinath Saha This is my paper

Pith reviewed 2026-05-09 14:39 UTC · model grok-4.3

classification 🧮 math.CO math.AG
keywords toric Schubert varietiespartial flag varietiesDeodhar decompositionsmoothness criteriaCartan integersCoxeter-type elementsjoin-distributive latticesRichardson varieties
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The pith

Toric Schubert varieties in partial flag varieties admit an explicit combinatorial fan description that yields necessary and sufficient smoothness conditions via Cartan integers from reduced expressions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an explicit combinatorial description of the fan associated to a toric Schubert variety in a partial flag variety. By leveraging Deodhar's decomposition of Richardson varieties and Pasquier's earlier results, the authors produce a model for the cones of this fan that depends only on data from the Weyl group. This description yields necessary and sufficient conditions for smoothness expressed through Cartan integers of reduced expressions. It also proves a lattice-theoretic property for intervals in the quotient poset when the top element is of Coxeter type. A sympathetic reader would care because these results convert questions about the geometry of singularities into verifiable combinatorial statements that can be checked from group-theoretic data alone.

Core claim

Using Deodhar's decomposition of Richardson varieties and the work of Pasquier, we give an explicit description of the fan of a toric Schubert variety, leading to a combinatorial model for its cones. As an application, we obtain necessary and sufficient conditions for smoothness of toric Schubert varieties in terms of the Cartan integers associated to a reduced expression. Furthermore, we prove that for a Coxeter-type element w in W^P, the interval [e,w] in W^P is a supersolvable join-distributive lattice. Finally, we apply these results to the study of spherical and horospherical Schubert varieties, providing a combinatorial method for checking the smoothness via the associated toric Schott

What carries the argument

The fan of the toric Schubert variety, constructed via an explicit combinatorial model from Deodhar's decomposition of Richardson varieties combined with Pasquier's results.

Load-bearing premise

That Deodhar's decomposition of Richardson varieties and Pasquier's results apply directly to toric Schubert varieties in partial flag varieties to produce a fan description whose combinatorial conditions on Cartan integers are necessary and sufficient for smoothness.

What would settle it

A specific reduced expression for a Weyl group element where the associated Cartan integers satisfy the paper's conditions for smoothness, yet geometric computation of the toric Schubert variety reveals a singularity.

read the original abstract

In this article, we investigate the toric Schubert varieties in partial flag varieties $G/P$ for a connected semisimple algebraic group $G$. Using Deodhar's decomposition of Richardson varieties and the work of Pasquier, we give an explicit description of the fan of a toric Schubert variety, leading to a combinatorial model for its cones. As an application, we obtain necessary and sufficient conditions for smoothness of toric Schubert varieties in terms of the Cartan integers associated to a reduced expression. Furthermore, we prove that for a Coxeter-type element $w \in W^P$, the interval $[e,w]_{W^P}$ is a supersolvable join-distributive lattice. Finally, we apply these results to the study of spherical and horospherical Schubert varieties, providing a combinatorial method for checking the smoothness via the associated toric Schubert varieties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper investigates toric Schubert varieties inside partial flag varieties G/P for a connected semisimple algebraic group G. Using Deodhar's decomposition of Richardson varieties together with Pasquier's prior results, it supplies an explicit combinatorial description of the fan of such a toric Schubert variety and a model for its cones. As an application it derives necessary-and-sufficient smoothness criteria phrased in terms of the Cartan integers attached to a reduced expression. It further proves that when w is a Coxeter-type element of W^P the Bruhat interval [e,w]_{W^P} is a supersolvable join-distributive lattice, and it applies the same machinery to obtain combinatorial smoothness tests for spherical and horospherical Schubert varieties.

Significance. If the claimed fan description and the ensuing smoothness criteria are valid, the work supplies a concrete combinatorial handle on a geometrically natural class of toric varieties inside flag varieties. The lattice-theoretic statement adds a new structural result about Bruhat intervals for Coxeter-type elements, while the applications to spherical and horospherical cases give a practical test that could be used in explicit computations. The paper therefore strengthens the bridge between the geometry of G/P and the combinatorics of Coxeter groups and root systems.

major comments (2)
  1. [§3 (fan construction)] The central claim that Deodhar's decomposition together with Pasquier's results directly yields an explicit fan description for the toric locus inside G/P (rather than inside a Richardson variety in G/B) is load-bearing for both the smoothness criterion and the lattice statement. The manuscript must verify that the torus action and the parabolic embedding do not alter the cone generators or their lattice indices; without an explicit check that the reduced-word data survive the quotient by P, the necessity and sufficiency of the Cartan-integer conditions remain conditional.
  2. [§5 (lattice result)] The proof that [e,w]_{W^P} is supersolvable and join-distributive for Coxeter-type w likewise rests on the same combinatorial model being faithful to the geometry. If the fan description in §3 contains an implicit normalization on the choice of reduced expression or on the parabolic subgroup, the lattice claim may hold only under additional hypotheses not stated in the abstract.
minor comments (2)
  1. [Introduction] The introduction should include a short table or diagram illustrating the passage from a reduced word to the associated Cartan integers and then to the smoothness criterion, to make the combinatorial model easier to follow.
  2. [§2] Notation for the poset W^P and the interval [e,w]_{W^P} is introduced without an explicit reminder of the Bruhat order restricted to the quotient; a one-sentence clarification would prevent confusion with the full Weyl group.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments identify points where additional explicit verification would strengthen the exposition. We address each below and have revised the manuscript to incorporate the requested clarifications while preserving the original statements and proofs.

read point-by-point responses

  1. Referee: [§3 (fan construction)] The central claim that Deodhar's decomposition together with Pasquier's results directly yields an explicit fan description for the toric locus inside G/P (rather than inside a Richardson variety in G/B) is load-bearing for both the smoothness criterion and the lattice statement. The manuscript must verify that the torus action and the parabolic embedding do not alter the cone generators or their lattice indices; without an explicit check that the reduced-word data survive the quotient by P, the necessity and sufficiency of the Cartan-integer conditions remain conditional.

    Authors: We agree that an explicit verification of invariance under the parabolic quotient strengthens the argument. The construction begins with the Deodhar decomposition of the Richardson variety in G/B and then descends via the projection π: G/B → G/P. Because the torus T is the same and the parabolic subgroup P is generated by the Levi factor together with the unipotent radical corresponding to the omitted simple roots, the weights determining the cone generators (the negative roots appearing in the reduced expression) remain unchanged; only directions orthogonal to the parabolic are quotiented out. We have inserted a new paragraph immediately after the statement of the fan in §3.2 that records this lattice-index preservation, citing the standard fact that minimal-length representatives in W^P give the same Cartan integers as their lifts to W. With this addition the necessity and sufficiency of the Cartan-integer smoothness criteria hold unconditionally for the toric Schubert variety in G/P. revision: yes

  2. Referee: [§5 (lattice result)] The proof that [e,w]_{W^P} is supersolvable and join-distributive for Coxeter-type w likewise rests on the same combinatorial model being faithful to the geometry. If the fan description in §3 contains an implicit normalization on the choice of reduced expression or on the parabolic subgroup, the lattice claim may hold only under additional hypotheses not stated in the abstract.

    Authors: The combinatorial model used for the lattice statement is the poset [e,w]_{W^P} equipped with the covering relations coming from the same reduced expressions that label the rays of the fan. Deodhar’s decomposition supplies a canonical collection of subexpressions independent of any particular choice of reduced word once the parabolic quotient is fixed; the join-distributive and supersolvable properties are then verified directly on this poset by induction on length, using only the root-system data encoded in the Cartan matrix. No further normalization on the reduced expression or on P is imposed beyond the standard definition of W^P. Consequently the lattice-theoretic claim holds for every Coxeter-type element w ∈ W^P exactly as stated, without additional hypotheses. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external citations of Deodhar and Pasquier

full rationale

The paper attributes its central fan description explicitly to Deodhar's decomposition of Richardson varieties together with Pasquier's prior results, which are independent external contributions. Smoothness criteria via Cartan integers and the supersolvable join-distributive lattice property for Coxeter-type elements are derived as applications of that model. No equations, definitions, or steps in the provided text reduce the claimed results to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation chain is self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Central claims rest on two cited combinatorial decompositions and standard facts about Weyl groups and root systems; no free parameters or new entities are introduced in the abstract.

axioms (3)
  • domain assumption Deodhar's decomposition of Richardson varieties applies to the toric Schubert setting
    Invoked to obtain the explicit fan description.
  • domain assumption Pasquier's prior results on toric varieties or fans are valid and combinable with Deodhar decomposition
    Used to produce the combinatorial cone model.
  • domain assumption Cartan integers associated to reduced expressions determine smoothness via the fan
    Basis for the necessary-and-sufficient smoothness criterion.

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