Non-vanishing of Single, Double, and Triple Schubert Structure Constants

Abstract: The Schubert vanishing problem asks whether the single Schubert coefficients $c_{u,v}<sup>w$ are zero. In this paper, we consider the non-vanishing problems of double Schubert coefficients $c_{u,v}<sup>w(t)$ and triple Schubert coefficients $c_{u,v}<sup>w(t;y)$. We show that the non-vanishing of $c_{u,v}<sup>w(t;y)$ is completely determined by the non-vanishing of single Schubert coefficients. As a byproduct, we obtain the saturation property of the triple Littlewood--Richardson coefficients $c_{λ,μ}<sup>ν(t;y)$. Moreover, we pose a conjecture asserting that the non-vanishing of $c_{u,v}<sup>w(t)$ is also determined by the non-vanishing of single or triple Schubert coefficients. We prove a one-side inclusion of the conjecture. For the reverse inclusion, we show that the conjecture holds for the following three cases: the Pieri case, the separated descents case, and the inverse Grassmannian case.

• The paper completely determines triple Schubert supports from single supports, proves one direction of the double-support conjecture, and establishes the converse in Pieri, separated-descents, and inverse-Grassmannian cases.
• The authors prove saturation for triple Littlewood–Richardson coefficients and characterize their nonvanishing through Horn inequalities and Hermitian matrix majorization, extending classical and equivariant results.
• The results place triple Schubert vanishing in coNP unconditionally, while equivariant vanishing lies in coNP if the paper’s general double-support conjecture holds, leaving its full proof as the central open problem.

This paper, "Non-vanishing of Single, Double, and Triple Schubert Structure Constants" (2608.17378), by Chen, Fan, Xiong, and Yao, addresses the qualitative question of when the structure constants of (single, double, and triple) Schubert calculus are nonzero. Rather than computing the coefficients themselves, the authors define "supports" — the sets of permutations $w$ for which the corresponding coefficient is nonzero — and study the relations among the single support $\Omega^1(u,v)$, double (equivariant) support $\Omega^2(u,v)$, and triple support $\Omega^3(u,v)$. The main contributions are: (i) a complete determination of the triple support in terms of single supports; (ii) a general conjecture relating the double support to the single and triple supports, proved in one direction unconditionally and in the other direction for the Pieri, separated descents, and inverse Grassmannian cases; and (iii) a saturation theorem and Horn-type inequalities for triple Littlewood–Richardson coefficients, with consequences for the complexity status of equivariant Schubert vanishing.

Background and the support conjectures

Single Schubert polynomials $\mathfrak{S}_w(x)$, introduced by Lascoux and Schützenberger, represent Schubert classes on the flag variety, and their structure constants $c_{u,v}^w$ are the classical Schubert coefficients. Double Schubert polynomials $\mathfrak{S}_w(x;t)$ represent equivariant Schubert classes; the double coefficients $c_{u,v}^w(t)$ are Graham-positive, i.e., polynomials in the differences $t_{i+1}-t_i$ with nonnegative integer coefficients. The paper also employs triple Schubert calculus, in which two distinct secondary variable sets appear:

$$\mathfrak{S}_u(x;y)\cdot \mathfrak{S}_{v}(x;t)=\sum_{w} c_{u,v}^w(t;y)\cdot\mathfrak{S}_{w}(x;t).$$

A notable asymmetry is that $\Omega^1(u,v)$0 and $\Omega^1(u,v)$1 play different roles in the triple setting, so $\Omega^1(u,v)$2 in general.

The authors' central conjecture asserts that the double support is determined by the single support:

$\Omega^1(u,v)$3

They further conjecture the equivalent formulation

$\Omega^1(u,v)$4

inspired by their earlier work on inverse Grassmannian permutations. The authors state that they expect Conjecture (1) to hold in all Lie types.

Determination of the triple support

A key structural result is unconditional: the triple support is completely determined by the single (equivalently, double) support. Specifically, for $\Omega^1(u,v)$5,

$\Omega^1(u,v)$6

The "$\Omega^1(u,v)$7" direction uses left Demazure operators: applying $\Omega^1(u,v)$8 to both sides of the triple product shows that $\Omega^1(u,v)$9 whenever $\Omega^2(u,v)$0, and induction over a reduced decomposition of $\Omega^2(u,v)$1 gives the inclusion. The "$\Omega^2(u,v)$2" direction relies on Graham positivity of triple coefficients, $\Omega^2(u,v)$3 (a conjecture of Samuel proved by Gao and Xiong), which implies $\Omega^2(u,v)$4 if and only if $\Omega^2(u,v)$5. Setting $\Omega^2(u,v)$6, $\Omega^2(u,v)$7 and applying the Cauchy formula for Schubert polynomials then expresses the product as a nonnegative sum over single coefficients $\Omega^2(u,v)$8, forcing $\Omega^2(u,v)$9 for some reduced factorization.

Combining (3) with the basic inclusions $\Omega^3(u,v)$0 yields the equivalence of Conjectures (1) and (2), and proves unconditionally that $\Omega^3(u,v)$1 — the "$\Omega^3(u,v)$2" half of (1). The paper also records the length filtration $\Omega^3(u,v)$3 and the containment $\Omega^3(u,v)$4 in Bruhat order, the latter proved by a localization argument.

The reverse inclusion in three cases

The reverse inclusion of (2) is proved when explicit combinatorial formulas are available, in three cases summarized below.

Case	Single formula	Double formula	Triple formula
Pieri ($\Omega^3(u,v)$5 or $\Omega^3(u,v)$6)	Sottile	Robinson; Li–Ravikumar–Sottile–Yang	Samuel
Separated descents	Knutson–Zinn-Justin; Huang	Knutson–Zinn-Justin; Huang	Fan–Guo–Xiong; Samuel
Inverse Grassmannian	Pechenik–Weigandt; Wyser	Chen–Fan–Xiong–Yao	Chen–Fan–Xiong–Yao

Pieri case. When $\Omega^3(u,v)$7, the ordinary, double, and triple Pieri rules express the coefficients in terms of decreasing $\Omega^3(u,v)$8-paths in the $\Omega^3(u,v)$9-Bruhat order. The triple rule gives $\mathfrak{S}_w(x)$0, where $\mathfrak{S}_w(x)$1 is the path length and $\mathfrak{S}_w(x)$2 is the set of values fixed among the first $\mathfrak{S}_w(x)$3 positions. The double coefficient is the specialization at $\mathfrak{S}_w(x)$4, which vanishes precisely when the corresponding Grassmannian localization vanishes. The proof reduces to a Gale-order comparison: if $\mathfrak{S}_w(x)$5 in Bruhat order, then deleting the $\mathfrak{S}_w(x)$6 path labels from the Gale inequality for $\mathfrak{S}_w(x)$7 yields $\mathfrak{S}_w(x)$8, where $\mathfrak{S}_w(x)$9 is the Grassmannian permutation determined by $c_{u,v}^w$0, hence the localization is nonzero and $c_{u,v}^w$1. The case $c_{u,v}^w$2 with increasing $c_{u,v}^w$3-paths is analogous.

Separated descents case. Suppose $c_{u,v}^w$4. A lemma shows that $c_{u,v}^w$5 is contained in the set of $c_{u,v}^w$6 with $c_{u,v}^w$7 for all $c_{u,v}^w$8: since $c_{u,v}^w$9 is symmetric in $\mathfrak{S}_w(x;t)$0, it lies in the algebra generated by elementary symmetric functions and partial sums, and the triple Pieri rule shows that multiplication by each generator can only increase the first $\mathfrak{S}_w(x;t)$1 values in the Bruhat (Gale) sense. The remaining inclusion is proved via the bumpless pipe dream/puzzle model of Fan–Guo–Xiong, which gives $\mathfrak{S}_w(x;t)$2 over Schubert pipe puzzles. A puzzle has nonzero weight at $\mathfrak{S}_w(x;t)$3 exactly when no empty tile lies on the diagonal. The authors show constructively that if $\mathfrak{S}_w(x;t)$4 and $\mathfrak{S}_w(x;t)$5 for $\mathfrak{S}_w(x;t)$6, then the $\mathfrak{S}_w(x;t)$7-th pipe must pass through the $\mathfrak{S}_w(x;t)$8-th row, and a recursive "general droop" modification of any offending puzzle eliminates all empty diagonal tiles. This establishes $\mathfrak{S}_w(x;t)$9 and hence (2).

Inverse Grassmannian case. For inverse Grassmannian $c_{u,v}^w(t)$0, the authors use the preclan formalism from their companion work. Each pair $c_{u,v}^w(t)$1 arises from a Richardson preclan $c_{u,v}^w(t)$2, and the triple and double coefficients are given by affine Stanley symmetric functions evaluated at specialized variables, with support conditions expressed via a weak order action $c_{u,v}^w(t)$3 on preclans. Combining the criteria for $c_{u,v}^w(t)$4 and $c_{u,v}^w(t)$5 (via the sign-reversal involution $c_{u,v}^w(t)$6) with the double criterion — that $c_{u,v}^w(t)$7 be permutational, which by a lemma of the companion paper is equivalent to both $c_{u,v}^w(t)$8 and $c_{u,v}^w(t)$9 — gives (2) directly.

Saturation and Horn inequalities for triple LR-coefficients

Specializing to Grassmannian permutations, the Schubert coefficients become Littlewood–Richardson (LR) coefficients indexed by partitions. The separated descents result translates to

$t_{i+1}-t_i$0

and (3) gives $t_{i+1}-t_i$1.

The paper proves a saturation theorem for triple LR-coefficients: for all $t_{i+1}-t_i$2,

$t_{i+1}-t_i$3

generalizing the single saturation theorem of Knutson–Tao and the double version of Anderson–Richmond–Yong. The proof passes through a Horn-type characterization: assuming $t_{i+1}-t_i$4, the following are equivalent: (1) $t_{i+1}-t_i$5; (2) the Horn inequalities $t_{i+1}-t_i$6 hold for all admissible triples of subsets; (3) there exist $t_{i+1}-t_i$7 Hermitian matrices $t_{i+1}-t_i$8 with spectra $t_{i+1}-t_i$9 and $$\mathfrak{S}_u(x;y)\cdot \mathfrak{S}_{v}(x;t)=\sum_{w} c_{u,v}^w(t;y)\cdot\mathfrak{S}_{w}(x;t).$$0 in the Loewner order. The forward direction of (1)$$\mathfrak{S}_u(x;y)\cdot \mathfrak{S}_{v}(x;t)=\sum_{w} c_{u,v}^w(t;y)\cdot\mathfrak{S}_{w}(x;t).$$1(3) uses Horn's theorem on a partition $$\mathfrak{S}_u(x;y)\cdot \mathfrak{S}_{v}(x;t)=\sum_{w} c_{u,v}^w(t;y)\cdot\mathfrak{S}_{w}(x;t).$$2 with $$\mathfrak{S}_u(x;y)\cdot \mathfrak{S}_{v}(x;t)=\sum_{w} c_{u,v}^w(t;y)\cdot\mathfrak{S}_{w}(x;t).$$3, and the converse uses Fulton's majorization criterion. Since $$\mathfrak{S}_u(x;y)\cdot \mathfrak{S}_{v}(x;t)=\sum_{w} c_{u,v}^w(t;y)\cdot\mathfrak{S}_{w}(x;t).$$4 unless $$\mathfrak{S}_u(x;y)\cdot \mathfrak{S}_{v}(x;t)=\sum_{w} c_{u,v}^w(t;y)\cdot\mathfrak{S}_{w}(x;t).$$5 (a consequence of the localization lemma), this completely characterizes the triple support $$\mathfrak{S}_u(x;y)\cdot \mathfrak{S}_{v}(x;t)=\sum_{w} c_{u,v}^w(t;y)\cdot\mathfrak{S}_{w}(x;t).$$6. The double Horn characterization of Anderson–Richmond–Yong follows from the separated descents identity.

Complexity-theoretic consequences

The Schubert vanishing problem $$\mathfrak{S}_u(x;y)\cdot \mathfrak{S}_{v}(x;t)=\sum_{w} c_{u,v}^w(t;y)\cdot\mathfrak{S}_{w}(x;t).$$7 is a central object in geometric complexity theory; Mulmuley identified Schubert coefficients as fundamental constants whose vanishing must be understood. Recent work of Pak and Robichaux shows that classical Schubert vanishing lies in coRP unconditionally for all classical Lie types. The paper observes that, assuming Conjecture (1), the polynomial-size witness tuple $$\mathfrak{S}_u(x;y)\cdot \mathfrak{S}_{v}(x;t)=\sum_{w} c_{u,v}^w(t;y)\cdot\mathfrak{S}_{w}(x;t).$$8, together with the accepting random strings for the required classical nonvanishing tests (which exist since classical vanishing is in coRP), forms a coNP certificate. Consequently:

• Equivariant Schubert vanishing ($$\mathfrak{S}_u(x;y)\cdot \mathfrak{S}_{v}(x;t)=\sum_{w} c_{u,v}^w(t;y)\cdot\mathfrak{S}_{w}(x;t).$$9) lies in coNP, conditional on Conjecture (1).
• Triple Schubert vanishing ($\Omega^1(u,v)$00) lies in coNP unconditionally, since (3) is a theorem.

This is the first unconditional containment result for the triple vanishing problem in a standard complexity class, to the authors' knowledge, and it parallels the strongly polynomial-time result of Adve–Robichaux–Yong for double LR-coefficients on Grassmannians.

Limitations and open questions

The paper is explicit about the scope of its results. The reverse inclusion of Conjecture (1) — equivalently (2) — remains open in general; it is proved only in the Pieri, separated descents, and inverse Grassmannian cases, each of which depends on the availability of explicit positive combinatorial formulas for the relevant coefficients. The claim that equivariant Schubert vanishing lies in coNP is conditional on the full Conjecture (1). The authors also note that they expect Conjecture (1) to hold in Lie types other than $\Omega^1(u,v)$01, but no evidence beyond type $\Omega^1(u,v)$02 is given in the paper. Whether the coNP containment can be upgraded to unconditional status for equivariant vanishing, or to a deterministic polynomial-time algorithm in the spirit of Adve–Robichaux–Yong, is left open.

Conclusion

The paper establishes a clean framework in which the nonvanishing of triple Schubert structure constants is completely reduced to the classical single case, proves one direction of the analogous reduction for double (equivariant) constants unconditionally, and verifies the full reduction in three families where positive rules are known. The triple saturation theorem and Horn-type inequalities extend the eigenvalue-intersection picture of equivariant Schubert calculus, and the complexity corollary places the triple Schubert vanishing problem in coNP unconditionally. The principal open problem is the general validity of Conjecture (1), whose resolution would settle the complexity status of equivariant Schubert vanishing.