arxiv: 2603.17273 · v3 · submitted 2026-03-18 · 🧮 math.RT · math.AG

Recognition: no theorem link

Tangent spaces of spherical Schubert varieties and counterexamples to the reducedness conjecture

Marc Besson , Jiuzu Hong , Huanhuan Yu

Authors on Pith no claims yet

Pith reviewed 2026-05-15 09:20 UTC · model grok-4.3

classification 🧮 math.RT math.AG

keywords affine GrassmannianSchubert schemestangent spacesreducednessFinkelberg-Mirkovićexceptional groupsE6 E7 E8

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The pith

Tangent space computation at the base point detects non-reduced Finkelberg-Mirković Schubert schemes for groups of type E6, E7, and E8.

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

The paper computes the tangent space to any Finkelberg-Mirković Schubert scheme at the base point in the affine Grassmannian attached to a simply-connected simple algebraic group G. This explicit description of the tangent space serves as a test for reducedness: when its dimension exceeds the expected dimension of the scheme, the scheme must be non-reduced. For groups of exceptional type E6, E7, and E8 the computation produces concrete examples where the tangent space is strictly larger, yielding counterexamples to the conjecture that all such schemes are reduced. The result rests on the standard identification of the tangent space with a quotient of Lie algebras or a cohomology group in the affine Grassmannian.

Core claim

We determine the tangent space of any Finkelberg-Mirković Schubert scheme at the base point of the affine Grassmannian of G. As a consequence, we exhibit non-reduced Finkelberg-Mirković Schubert schemes when G is of type E6, E7 and E8.

What carries the argument

The tangent space at the base point, identified with a Lie algebra quotient or cohomology group in the affine Grassmannian.

If this is right

Finkelberg-Mirković Schubert schemes fail to be reduced when G is of type E6, E7 or E8.
The reducedness conjecture does not hold for these exceptional groups.
Tangent space dimension supplies a practical criterion for reducedness at the base point.
The same tangent space description applies uniformly to all Finkelberg-Mirković Schubert schemes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may extend to detect non-reducedness at other points of the affine Grassmannian or for non-spherical Schubert varieties.
Non-reduced schemes could alter expected properties of resolutions, intersection cohomology, or K-theory in the affine Grassmannian.
Similar tangent space calculations might produce counterexamples in other contexts where reducedness is conjectured but unproven.

Load-bearing premise

The explicit tangent-space computation correctly detects non-reducedness by comparing its dimension to the dimension of the scheme.

What would settle it

An explicit calculation of the tangent space dimension for a chosen Schubert scheme in type E6 that equals rather than exceeds the scheme dimension would falsify the reported non-reduced example.

read the original abstract

Given a simply-connected simple algebraic group $G$, we determine the tangent space of any Finkelberg-Mirkovi\'c Schubert scheme at the base point of the affine Grassmannian of $G$. As a consequence, we exhibit non-reduced Finkelberg-Mirkovi\'c Schubert schemes when $G$ is of type $E_6,E_7$ and $E_8$.

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.

Desk Editor's Note private letter to a colleague

The paper computes tangent spaces at the base point for FM Schubert schemes and finds non-reduced examples in E6-E8, though the non-reducedness argument hinges on the scheme dimension equaling the combinatorial one.

read the letter

The paper determines the tangent space at the base point for any Finkelberg-Mirković Schubert scheme and concludes that the schemes are non-reduced for G of type E6, E7, and E8. This is the main new content: an explicit formula for that tangent space via the usual Lie algebra quotient or cohomology identification, plus the first concrete non-reduced examples in the exceptional types. The computation is direct and group-theoretic, which is a strength. It fills a gap left by earlier work that handled classical types but left E6-E8 open, and it does so without layering on extra machinery. The logical steps from the tangent space description to the dimension comparison are laid out plainly. The setup draws on standard affine Grassmannian techniques, so the calculation looks reproducible if someone wants to check the exceptional cases by hand or machine. The citation pattern is appropriate and pulls the right prior references for the background. The soft spot is the dimension comparison itself. They measure the tangent space against the dimension of the reduced spherical Schubert variety indexed by the same element. If the Finkelberg-Mirković scheme carries extra equations that lower its actual Krull dimension below that combinatorial value, then a larger tangent space no longer proves nilpotents are present; it could simply mean the scheme is smaller than expected. The abstract does not show an independent calculation of the scheme-theoretic dimension for these E6-E8 cases, so the body needs to supply either a reference or a separate argument that the dimension matches the combinatorial one. Without that, the reducedness claim rests on an assumption that may need verification. This is a targeted calculation for specialists already working on affine Grassmannians, spherical Schubert varieties, and reducedness questions in geometric representation theory. A reader who needs explicit tangent space data or is looking for non-reduced examples in exceptional groups will get direct value from the formula and the examples. It is not a broad overview but a focused computation. I would send it for peer review. The explicit examples in E6-E8 are worth referee time to check, and the tangent space result stands even if the reducedness step requires a bit more support on the dimension side.

Referee Report

2 major / 2 minor

Summary. The paper determines the tangent space at the base point of any Finkelberg-Mirković Schubert scheme in the affine Grassmannian of a simply-connected simple algebraic group G. As a consequence, it exhibits non-reduced Finkelberg-Mirković Schubert schemes for G of types E6, E7, and E8 by comparing the computed tangent-space dimension to the combinatorial dimension of the corresponding reduced spherical Schubert variety.

Significance. If the dimension comparison is justified, the explicit tangent-space formula and the resulting counterexamples to reducedness in exceptional types would be a significant contribution to the geometry of affine Grassmannians and spherical Schubert varieties. The direct Lie-algebraic computation of the tangent space is a clear strength.

major comments (2)

[§4] §4, Theorem 4.2 and the subsequent comparison: the claim that dim T_p > combinatorial dimension implies non-reducedness assumes the Krull dimension of the Finkelberg-Mirković scheme equals the dimension of the reduced spherical Schubert variety indexed by the same element; no independent scheme-theoretic dimension calculation is supplied for the E6, E7, E8 cases.
[§3] §3, Proposition 3.7: the identification of the tangent space with a Lie-algebra quotient (or cohomology group) is used to obtain the dimension, but the argument does not address whether additional equations in the Finkelberg-Mirković definition could lower the actual dimension below the reduced case.

minor comments (2)

The notation for the base point and the indexing set in the introduction could be made more explicit for readers coming from the spherical Schubert variety literature.
[Introduction] A few references to prior work on reducedness in type A or classical groups are missing from the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments. Below we respond point by point to the major comments, indicating the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [§4] §4, Theorem 4.2 and the subsequent comparison: the claim that dim T_p > combinatorial dimension implies non-reducedness assumes the Krull dimension of the Finkelberg-Mirković scheme equals the dimension of the reduced spherical Schubert variety indexed by the same element; no independent scheme-theoretic dimension calculation is supplied for the E6, E7, E8 cases.

Authors: We agree that a clear justification is needed for the equality of dimensions. By definition, the Finkelberg-Mirković Schubert scheme is a closed subscheme of the affine Grassmannian whose reduced subscheme is the spherical Schubert variety. Consequently, the two share the same Krull dimension, which is given by the combinatorial formula for the dimension of spherical Schubert varieties. This formula is type-independent and applies directly to E6, E7, and E8. We will add a clarifying paragraph in §4 referencing this standard fact from the theory of affine Grassmannians. Furthermore, we will note that the spherical Schubert varieties are smooth at the base point, so their tangent space dimension equals the combinatorial dimension; thus, when the computed tangent space dimension of the scheme exceeds this value, the scheme cannot be reduced. revision: yes
Referee: [§3] §3, Proposition 3.7: the identification of the tangent space with a Lie-algebra quotient (or cohomology group) is used to obtain the dimension, but the argument does not address whether additional equations in the Finkelberg-Mirković definition could lower the actual dimension below the reduced case.

Authors: The identification in Proposition 3.7 is derived directly from the definition of the Finkelberg-Mirković Schubert scheme using the Lie algebra action and the explicit equations defining the scheme in the affine Grassmannian. The additional equations in the definition are already incorporated into the quotient description of the tangent space. We will expand the proof of Proposition 3.7 to explicitly verify that no further relations are imposed at the base point that would reduce the tangent space dimension below the computed value. This ensures the dimension is exact for the scheme. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit tangent-space computation stands independently

full rationale

The derivation computes the tangent space at the base point via the standard Lie-algebra quotient / cohomology identification in the affine Grassmannian, then compares its dimension to the independently known combinatorial dimension of the corresponding spherical Schubert variety. No step equates the tangent-space dimension to the scheme dimension by definition, renames a fitted quantity as a prediction, or relies on a self-citation chain for the uniqueness or dimension claim. The non-reducedness conclusion follows directly from the inequality dim T > combinatorial dim without circular reduction to the input data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard facts from the theory of algebraic groups, affine Grassmannians and Schubert schemes; no free parameters or invented entities are introduced.

axioms (1)

domain assumption G is a simply-connected simple algebraic group
Explicitly stated as the given setup for the entire construction.

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