---
title: Schubert Vanishing Problem
url: https://www.emergentmind.com/topics/schubert-vanishing-problem
type: topic
---

# Schubert Vanishing Problem

The Schubert Vanishing Problem appears in several distinct but related settings in Schubert calculus. In one formulation, it asks whether a coefficient of a Schubert polynomial vanishes; in another, whether a Schubert structure constant \(c_{u,v}^w\) or a generic Schubert intersection number is zero; in further geometric formulations, it concerns vanishing of characteristic classes or orders of vanishing along webs of Schubert subvarieties. Across these settings, the problem links Schubert polynomials, Newton polytopes, tableaux, intersection theory on flag varieties, perverse sheaves, valuations, and complexity-theoretic classifications [2103.05195][2412.02064][2303.13833][1208.5453][2203.12992].

## 1. Terminological scope

In current usage, the expression refers to several questions that share Schubert-theoretic input but differ in the precise object whose vanishing is being tested. The literature includes coefficient-vanishing questions for Schubert polynomials, vanishing of Schubert intersection numbers in flag varieties, vanishing or nonvanishing of structure constants in cohomology, vanishing of Schubert classes in invariant characteristic cohomology, and vanishing multiplicities along Schubert stratifications [2103.05195][2010.14332][1208.5453][2203.12992].

| Setting | Question | Representative result |
|---|---|---|
| Schubert polynomials | Decide whether \(c_{\alpha,w}=0\) | Polynomial-time algorithm via perfect tableaux and Schubitopes [1810.10361] |
| Flag-variety structure constants | Decide whether \(c_{u,v}^w=0\) | \({\sf coAM}\) under GRH, then \({\sf AM}\cap{\sf coAM}\) under GRH, then probabilistic polynomial time [2412.02064][2504.03004][2509.16467] |
| Schubert intersection numbers | Give sufficient vanishing criteria for \(C_{w(1),\ldots,w(k)}\) | Polynomial-time tableau and Schubitope tests [2010.14332] |
| Generic Schubert-cell intersections | Control signed Euler characteristics | Nonnegativity from a generic vanishing theorem for perverse sheaves [2303.13833] |
| Characteristic cohomology / vanishing orders | Determine which Schubert classes survive, or encode vanishing multiplicities | Schubert VHS survive in invariant characteristic cohomology; LS paths encode vanishing multiplicities [1208.5453][2203.12992] |

This multiplicity of meanings is substantive rather than merely terminological. The coefficient problem is a support question for Schubert polynomials, whereas the structure-constant problem is an incidence question for generic Schubert-variety intersections. The geometric variants replace binary vanishing by finer data such as signed Euler characteristics, survival in characteristic cohomology, or valuations encoding vanishing orders.

## 2. Vanishing of coefficients of Schubert polynomials

A standard formulation asks: given a Schubert polynomial \(\mathfrak S_w\) indexed by a permutation \(w\), and a monomial exponent vector \(\alpha\), decide whether the coefficient \(c_{\alpha,w}\) of \(x^\alpha\) in \(\mathfrak S_w\) is nonzero. Equivalently, the decision problem is whether \(c_{\alpha,w}\neq 0\) [1810.10361].

The central combinatorial criterion is expressed in terms of the Rothe diagram \(D(w)\) and perfect tableaux. The key equivalence is
\[
c_{\alpha, w} > 0 \iff {\sf PerfectTab}(D(w), \alpha)\neq \emptyset \iff {\sf PerfectTab}_{<}(D(w), \alpha)\neq \emptyset .
\]
Here \({\sf PerfectTab}(D(w),\alpha)\) denotes perfect tableaux of shape \(D(w)\) and content \(\alpha\), where entries are column-injective, flagged by \(\tau(r,c)\le r\), and every box is filled [1810.10361]. The 2021 paper states the same criterion in the form
\[
c_{a, w} > 0 \quad \iff \quad \text{PerfectTab}(D(w), a) \neq \emptyset ,
\]
and uses it to derive the first polynomial-time algorithm for deciding vanishing of Schubert polynomial coefficients [2103.05195].

The polyhedral counterpart is the Schubitope. For a diagram \(D\subseteq [n]^2\),
\[
\mathcal{S}_D = \left\{ \alpha \in \mathbb{R}_{\geq 0}^n : \sum_{i=1}^n \alpha_i = \# D,\ \forall S \subseteq [n],\ \sum_{i\in S} \alpha_i \leq \theta_D(S) \right\}.
\]
For Schubert polynomials, the Newton polytope of \(\mathfrak S_w\) is the Schubitope \(\mathcal S_{D(w)}\), and Schubert polynomials have saturated Newton polytopes, so
\[
\alpha \in \mathcal{S}_{D(w)} \cap \mathbb{Z}^n \iff c_{\alpha, w} \neq 0 .
\]
This converts nonvanishing into polytope membership [1810.10361].

Algorithmically, the tableau criterion can be encoded as feasibility of a polytope defined by linear inequalities, and total unimodularity implies that integer feasibility can be checked in polynomial time [1810.10361]. The 2021 account likewise formulates the problem via an indicator polytope whose integer points correspond to perfect tableaux and shows that the relevant constraint matrix is totally unimodular [2103.05195]. By contrast, the exact counting problem is harder: computing the coefficient \(c_{a,w}\) explicitly is \(\#P\)-complete [2103.05195].

## 3. Vanishing of Schubert structure constants and intersection numbers

A second major formulation concerns Schubert structure constants. In cohomological notation,
\[
\sigma_u \smallsmile \sigma_v = \sum_{w} c_{u,v}^w \sigma_w ,
\]
and in Schubert-polynomial notation,
\[
S_u \cdot S_v = \sum_w c_{u, v}^w S_w .
\]
The decision problem asks whether \(c_{u,v}^w=0\) [2412.02064][2504.03004].

Geometrically, these coefficients count generic intersections of Schubert varieties. One formulation is
\[
c_{u,v}^w = \#\bigl( X_u(F_\bullet) \cap X_v(G_\bullet) \cap X_{w_\circ w}(E_\bullet) \bigr),
\]
for generic flags \(F_\bullet,G_\bullet,E_\bullet\) [2504.03004]. More generally, for a \(k\)-tuple of permutations,
\[
C_{w(1), \ldots, w(k)} := \text{number of points in } \bigcap_{i=1}^k g_i X_{w(i)},
\]
with \(g_i\) generic [2010.14332].

Before complete complexity classifications were available, generalized-permutahedral methods gave sufficient vanishing tests. For a Schubert problem \((w(1),\ldots,w(k))\), the 2020 tableau test states: if a specified tableau set \(\mathrm{Tab}\) is empty, then \(C_{w(1),\ldots,w(k)}=0\); the stronger asymmetric version gives: if \(\mathrm{Tab}'=\emptyset\), then \(C^{w(k)}_{w(1),\ldots,w(k-1)}=0\) [2010.14332]. The same paper proves a Schubitope linear-inequality test: if \(C_{w(1),\ldots,w(k)}>0\), then the vector \((n-1,n-2,\ldots,1,0)\) must satisfy all defining inequalities of the Schubitope \(\mathcal S_D\) attached to the concatenated diagram \(D\) [2010.14332]. These tests are polynomial-time sufficient criteria, not necessary-and-sufficient characterizations.

Subsequent work placed the general structure-constant vanishing problem into progressively lower complexity classes. The 2024 paper proves that
\[
\{c_{u,v}^w =^? 0\}\ \text{is in}\ \coAM
\]
for types \(A,B,C\), assuming the Generalized Riemann Hypothesis, via polynomial-size lifted formulations and a reduction to the parametric Hilbert Nullstellensatz [2412.02064]. The 2025 paper strengthens this to
\[
\text{Schubert vanishing} \in \mathbf{AM} \cap \mathbf{coAM} \quad \text{assuming GRH},
\]
using lifted formulations, Mahajan–Vinay’s determinant construction, and Purbhoo’s algebraic criterion
\[
c_{u,v}^{w} \neq 0 \quad \Longleftrightarrow \quad
\rho R_u \rho^{-1} + \omega R_v \omega^{-1} + \tau R_{w_0 w} \tau^{-1} = \mathfrak{n}
\]
for generic unipotent elements \(\rho,\omega,\tau\) [2504.03004].

The 2025 probabilistic result gives an algorithmic resolution for classical types. It states that Schubert vanishing can be decided in probabilistic polynomial time for types \(A,B,C,D\), with a randomized algorithm running in
\[
O\left( k n^{8.75} \log \frac{1}{\varepsilon} \right)
\]
arithmetic operations and one-sided error; equivalently, the problem lies in \(\mathsf{coRP}\) [2509.16467]. Its criterion reduces vanishing to singularity of a determinant matrix \(M\), with
\[
c(u_1,\ldots,u_k)=0 \iff \det M \equiv 0 ,
\]
and uses Schwarz–Zippel random evaluation to test whether the determinant polynomial vanishes identically [2509.16467].

## 4. Generic vanishing, perverse sheaves, and positivity of Schubert-cell intersections

A different but closely related line of work studies generic vanishing and signed Euler characteristics on homogeneous varieties. Let \(X\) be a complete homogeneous variety with an action of a connected algebraic group \(G'\), let \(A,B_0\subset X\) be locally closed affine subvarieties, assume that \(B_0\) is smooth and pure-dimensional, let \(\mathcal P\) be a perverse sheaf on \(A\), and let \(B=gB_0\) be a generic translate of \(B_0\). The generic vanishing theorem states:
\[
H^i(B, j_{A*} (\mathcal{P}|_{A \cap B})) \cong H^i (A, j_{B!} (\mathcal{P}|_{A \cap B})) = 0
\qquad \text{for all } i \ne -\mathrm{codim}_X B,
\]
where \(j_A: A\cap B\to A\) and \(j_B: A\cap B\to B\) are inclusions [2303.13833].

An immediate corollary is the signed Euler-characteristic inequality
\[
(-1)^{\operatorname{codim} B} \chi(A \cap B,\mathcal{P}|_{A\cap B}) \ge 0 .
\]
The theorem applies to affine locally closed subvarieties and generic translates, and its proof uses perverse sheaves, affineness, Artin’s vanishing theorem, Kleiman’s transversality theorem, and a generic base-change isomorphism for sheaves; it does not use rank-one local system twisting [2303.13833].

For partial flag varieties \(X=G/P\), the theorem yields positivity for generic triple intersections of Schubert cells. Writing
\[
E_{u,v,w'} := (-1)^d \chi \left( X_u^\circ \cap g X_v^\circ \cap h X_{w'}^\circ \right),
\]
with \(d=\dim\left(X_u^\circ \cap g X_v^\circ \cap h X_{w'}^\circ\right)\) and \(w'.W_P = w_0 w.W_P\), the result is
\[
E_{u,v,w'} = (-1)^d \chi\left(X_u^\circ \cap gX_v^\circ \cap hX_{w'}^\circ\right) \ge 0
\]
for generic \(g,h\) and all triples of Schubert cells [2303.13833].

These Euler characteristics are structure constants for Segre–Schwartz–MacPherson classes:
\[
\operatorname{SSM}(X_u^\circ) \cdot \operatorname{SSM}(X_v^\circ)
= \sum_{w'} a_{u,v}^{w'} \operatorname{SSM}(X_{w'}^\circ),
\qquad
a_{u,v}^{w'} = \chi(X_u^\circ \cap gX_v^\circ \cap hX_{w'}^\circ),
\]
and the signed constants satisfy
\[
E_{u,v,w'} = (-1)^d a_{u,v}^{w'} .
\]
This identifies the theorem as a positivity result for the signs of generic Schubert-cell intersection invariants rather than a decision procedure for zero versus nonzero [2303.13833]. The paper also states that the theorem applies to iterated intersections of more than three Schubert cells.

## 5. Cohomological and valuation-theoretic variants

In Hodge-theoretic language, the Schubert vanishing problem concerns which Schubert classes survive in invariant characteristic cohomology. For a compact dual \(\check D\), the paper on Schubert varieties as variations of Hodge structure characterizes the Schubert varieties \(X_w\subset \check D\) that are variations of Hodge structure by
\[
X_w \text{ is a VHS if and only if } w \in W_\sI^\varphi,
\]
equivalently,
\[
W_\sI^\varphi = \{ w \in W^\varphi: \ \varrho_w(\ttT_\varphi) = |w| \}.
\]
The invariant characteristic cohomology is then spanned by the dual Schubert classes indexed by these \(w\):
\[
H^\bullet_\sI(D)^{G_\mathbb{R}} = \operatorname{span}_\mathbb{C} \{ x_w : w \in W_\sI^\varphi \},
\]
and the kernel of the projection is
\[
\ker p_\sI = \bigoplus_{w \in W^\varphi \setminus W^\varphi_\sI} \mathscr{H}_w .
\]
Thus, the dual class \(x_w\) vanishes in invariant characteristic cohomology unless \(w\in W_\sI^\varphi\) [1208.5453]. The same paper states that invariant characteristic cohomology is zero in odd degrees, and in even degree \(2\ell\) it is of Hodge type \((\ell,\ell)\).

A different geometric meaning of vanishing arises from LS algebras, standard monomial theory, and valuations on Schubert varieties. In this formulation, the question is to describe the order of vanishing of sections with respect to a prescribed web of Schubert subvarieties in a partial flag variety \(G/P\) [2203.12992]. For an LS algebra \(A\) of flag type and a maximal chain \(C\) in the indexing poset, the paper constructs a positive valuation
\[
v_C : A \setminus \{0\} \to \mathbb{Q}^C
\]
with \(v_C(p_T)=T\) for LS paths \(T\) supported on \(C\). The associated quasi-valuation is
\[
v(x) = \min_C v_C(x),
\]
and the key identification is
\[
v(p_T)=T
\]
for each LS path \(T\) [2203.12992]. In this sense, LS paths encode vanishing multiplicities with respect to the web of Schubert varieties.

The same framework produces a Newton–Okounkov body
\[
\Delta_v(A) = \overline{ \bigcup_{r \geq 1} \left\{ \frac{1}{r} v(x) \mid x \in A_r \setminus \{0\} \right\} },
\]
and the paper states
\[
\Delta_v(A) = \mathcal{A}(W^w),
\]
the order complex of the Bruhat poset \(W^w\) [2203.12992]. It also proves compatibility with Seshadri stratification constructions and a semi-toric degeneration of Schubert varieties. Here “vanishing” is not a binary emptiness question but a valuation-theoretic datum recording multiplicity along a stratified Bruhat web.

## 6. Conceptual synthesis

Taken together, these results show that the Schubert Vanishing Problem is not a single theorem but a family of vanishing phenomena organized around Schubert combinatorics. At the support level of Schubert polynomials, vanishing is governed by perfect tableaux, Schubitopes, and saturated Newton polytopes, with deterministic polynomial-time algorithms for deciding \(c_{\alpha,w}=0\) [1810.10361][2103.05195]. At the level of cohomological structure constants, the problem becomes a question about generic intersections of Schubert varieties and admits polyhedral sufficient tests, Arthur–Merlin upper bounds under GRH, and probabilistic polynomial-time algorithms for classical types [2010.14332][2412.02064][2504.03004][2509.16467].

The perverse-sheaf approach adds a sign-sensitive refinement: for generic Schubert-cell intersections, the relevant invariant is not merely nonemptiness but the signed Euler characteristic, and a generic vanishing theorem forces the expected sign for the structure constants of Segre–Schwartz–MacPherson classes [2303.13833]. The Hodge-theoretic and LS-algebraic variants show that “vanishing” can also mean disappearance of Schubert classes in characteristic cohomology or the order of vanishing of sections along nested Schubert strata [1208.5453][2203.12992].

A plausible implication is that the phrase functions as an umbrella term for support, incidence, and multiplicity problems that all become tractable once Schubert geometry is translated into the appropriate combinatorial or algebraic language: tableaux and Schubitopes for coefficient support, lifted polynomial formulations and determinant criteria for structure constants, perverse sheaves for signed intersection invariants, and valuations for vanishing orders.

Source: https://www.emergentmind.com/topics/schubert-vanishing-problem