Schubert Vanishing Problem

• Schubert Vanishing Problem is defined as determining when coefficients or intersection numbers in Schubert calculus vanish, linking combinatorial and geometric formulations.
• It employs techniques such as perfect tableaux, polyhedral descriptions via Schubitopes, and probabilistic algorithms to achieve efficient vanishing tests.
• Geometric and algebraic approaches including perverse sheaves, LS paths, and valuation theory integrate to analyze vanishing orders and structure constants.

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 (Adve et al., 2021, Pak et al., 2024, Schürmann et al., 2023, Robles, 2012, Chirivì et al., 2022).

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 (Adve et al., 2021, Dizier et al., 2020, Robles, 2012, Chirivì et al., 2022).

Setting	Question	Representative result
Schubert polynomials	Decide whether $c_{\alpha,w}=0$	Polynomial-time algorithm via perfect tableaux and Schubitopes (Adve et al., 2018)
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 (Pak et al., 2024, Pak et al., 3 Apr 2025, Pak et al., 19 Sep 2025)
Schubert intersection numbers	Give sufficient vanishing criteria for $C_{w(1),\ldots,w(k)}$	Polynomial-time tableau and Schubitope tests (Dizier et al., 2020)
Generic Schubert-cell intersections	Control signed Euler characteristics	Nonnegativity from a generic vanishing theorem for perverse sheaves (Schürmann et al., 2023)
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 (Robles, 2012, Chirivì et al., 2022)

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 $c_{\alpha,w}=0$0 in $c_{\alpha,w}=0$1 is nonzero. Equivalently, the decision problem is whether $c_{\alpha,w}=0$2 (Adve et al., 2018).

The central combinatorial criterion is expressed in terms of the Rothe diagram $c_{\alpha,w}=0$3 and perfect tableaux. The key equivalence is

$c_{\alpha,w}=0$4

Here $c_{\alpha,w}=0$5 denotes perfect tableaux of shape $c_{\alpha,w}=0$6 and content $c_{\alpha,w}=0$7, where entries are column-injective, flagged by $c_{\alpha,w}=0$8, and every box is filled (Adve et al., 2018). The 2021 paper states the same criterion in the form

$c_{\alpha,w}=0$9

and uses it to derive the first polynomial-time algorithm for deciding vanishing of Schubert polynomial coefficients (Adve et al., 2021).

The polyhedral counterpart is the Schubitope. For a diagram $c_{u,v}^w=0$0,

$c_{u,v}^w=0$1

For Schubert polynomials, the Newton polytope of $c_{u,v}^w=0$2 is the Schubitope $c_{u,v}^w=0$3, and Schubert polynomials have saturated Newton polytopes, so

$c_{u,v}^w=0$4

This converts nonvanishing into polytope membership (Adve et al., 2018).

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 (Adve et al., 2018). 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 (Adve et al., 2021). By contrast, the exact counting problem is harder: computing the coefficient $c_{u,v}^w=0$5 explicitly is $c_{u,v}^w=0$6-complete (Adve et al., 2021).

3. Vanishing of Schubert structure constants and intersection numbers

A second major formulation concerns Schubert structure constants. In cohomological notation,

$c_{u,v}^w=0$7

and in Schubert-polynomial notation,

$c_{u,v}^w=0$8

The decision problem asks whether $c_{u,v}^w=0$9 (Pak et al., 2024, Pak et al., 3 Apr 2025).

Geometrically, these coefficients count generic intersections of Schubert varieties. One formulation is

${\sf coAM}$0

for generic flags ${\sf coAM}$1 (Pak et al., 3 Apr 2025). More generally, for a ${\sf coAM}$2-tuple of permutations,

${\sf coAM}$3

with ${\sf coAM}$4 generic (Dizier et al., 2020).

Before complete complexity classifications were available, generalized-permutahedral methods gave sufficient vanishing tests. For a Schubert problem ${\sf coAM}$5, the 2020 tableau test states: if a specified tableau set ${\sf coAM}$6 is empty, then ${\sf coAM}$7; the stronger asymmetric version gives: if ${\sf coAM}$8, then ${\sf coAM}$9 (Dizier et al., 2020). The same paper proves a Schubitope linear-inequality test: if ${\sf AM}\cap{\sf coAM}$0, then the vector ${\sf AM}\cap{\sf coAM}$1 must satisfy all defining inequalities of the Schubitope ${\sf AM}\cap{\sf coAM}$2 attached to the concatenated diagram ${\sf AM}\cap{\sf coAM}$3 (Dizier et al., 2020). 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

${\sf AM}\cap{\sf coAM}$4

for types ${\sf AM}\cap{\sf coAM}$5, assuming the Generalized Riemann Hypothesis, via polynomial-size lifted formulations and a reduction to the parametric Hilbert Nullstellensatz (Pak et al., 2024). The 2025 paper strengthens this to

${\sf AM}\cap{\sf coAM}$6

using lifted formulations, Mahajan–Vinay’s determinant construction, and Purbhoo’s algebraic criterion

${\sf AM}\cap{\sf coAM}$7

for generic unipotent elements ${\sf AM}\cap{\sf coAM}$8 (Pak et al., 3 Apr 2025).

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 ${\sf AM}\cap{\sf coAM}$9, with a randomized algorithm running in

$C_{w(1),\ldots,w(k)}$0

arithmetic operations and one-sided error; equivalently, the problem lies in $C_{w(1),\ldots,w(k)}$1 (Pak et al., 19 Sep 2025). Its criterion reduces vanishing to singularity of a determinant matrix $C_{w(1),\ldots,w(k)}$2, with

$C_{w(1),\ldots,w(k)}$3

and uses Schwarz–Zippel random evaluation to test whether the determinant polynomial vanishes identically (Pak et al., 19 Sep 2025).

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 $C_{w(1),\ldots,w(k)}$4 be a complete homogeneous variety with an action of a connected algebraic group $C_{w(1),\ldots,w(k)}$5, let $C_{w(1),\ldots,w(k)}$6 be locally closed affine subvarieties, assume that $C_{w(1),\ldots,w(k)}$7 is smooth and pure-dimensional, let $C_{w(1),\ldots,w(k)}$8 be a perverse sheaf on $C_{w(1),\ldots,w(k)}$9, and let $\mathfrak S_w$0 be a generic translate of $\mathfrak S_w$1. The generic vanishing theorem states: $\mathfrak S_w$2 where $\mathfrak S_w$3 and $\mathfrak S_w$4 are inclusions (Schürmann et al., 2023).

An immediate corollary is the signed Euler-characteristic inequality

$\mathfrak S_w$5

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 (Schürmann et al., 2023).

For partial flag varieties $\mathfrak S_w$6, the theorem yields positivity for generic triple intersections of Schubert cells. Writing

$\mathfrak S_w$7

with $\mathfrak S_w$8 and $\mathfrak S_w$9, the result is

$w$0

for generic $w$1 and all triples of Schubert cells (Schürmann et al., 2023).

These Euler characteristics are structure constants for Segre–Schwartz–MacPherson classes: $w$2 and the signed constants satisfy

$w$3

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 (Schürmann et al., 2023). 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 $w$4, the paper on Schubert varieties as variations of Hodge structure characterizes the Schubert varieties $w$5 that are variations of Hodge structure by

$w$6

equivalently,

$w$7

The invariant characteristic cohomology is then spanned by the dual Schubert classes indexed by these $w$8: $w$9 and the kernel of the projection is

$\alpha$0

Thus, the dual class $\alpha$1 vanishes in invariant characteristic cohomology unless $\alpha$2 (Robles, 2012). The same paper states that invariant characteristic cohomology is zero in odd degrees, and in even degree $\alpha$3 it is of Hodge type $\alpha$4.

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 $\alpha$5 (Chirivì et al., 2022). For an LS algebra $\alpha$6 of flag type and a maximal chain $\alpha$7 in the indexing poset, the paper constructs a positive valuation

$\alpha$8

with $\alpha$9 for LS paths $c_{\alpha,w}$0 supported on $c_{\alpha,w}$1. The associated quasi-valuation is

$c_{\alpha,w}$2

and the key identification is

$c_{\alpha,w}$3

for each LS path $c_{\alpha,w}$4 (Chirivì et al., 2022). In this sense, LS paths encode vanishing multiplicities with respect to the web of Schubert varieties.

The same framework produces a Newton–Okounkov body

$c_{\alpha,w}$5

and the paper states

$c_{\alpha,w}$6

the order complex of the Bruhat poset $c_{\alpha,w}$7 (Chirivì et al., 2022). 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}$8 (Adve et al., 2018, Adve et al., 2021). 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 (Dizier et al., 2020, Pak et al., 2024, Pak et al., 3 Apr 2025, Pak et al., 19 Sep 2025).

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 (Schürmann et al., 2023). 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 (Robles, 2012, Chirivì et al., 2022).

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.