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Kirillov's saturation conjecture for Schubert coefficients

For permutations u,v,w∈Snu,v,w\in S_n, let cu,vwc_{u,v}^w be the Schubert coefficients defined by

Su⋅Sv=∑w∈S∞cu,vwSw.\mathfrak{S}_u\cdot\mathfrak{S}_v=\sum_{w\in S_\infty}c_{u,v}^w\mathfrak{S}_w.

For w∈Snw\in S_n, write its Lehmer code as code(w)=(c1,…,cn){\sf{code}}(w)=(c_1,\ldots,c_n), and define code scaling by

N∗w=code−1(Nc1,Nc2,…,Ncn,0,…,0)∈SNn.N* w={\sf{code}}^{-1}(Nc_1,Nc_2,\ldots,Nc_n,0,\ldots,0)\in S_{Nn}.

Kirillov's conjecture. For every u,v,w∈Snu,v,w\in S_n and every integer N≥1N\geq 1,

cu,vw>0⟺cN∗u,N∗vN∗w>0.c_{u,v}^w>0\quad\Longleftrightarrow\quad c_{N*u,N*v}^{N*w}>0.

This conjecture proposes that positivity of Schubert coefficients is preserved exactly under code scaling, generalizing the saturation theorem for Littlewood–Richardson coefficients. The source states that it remained open until the paper in which this conjecture appears; no resolution is supplied in the provided material.

Sources & referencesView supporting material

Primary source

Igor Pak and Colleen Robichaux, “Saturation property fails for Schubert coefficients”, arXiv:2601.04182 (2026).

algebraic-combinatoricssaturationschubert-calculus

Progress summary

Refreshed
Solved

A 2026 paper gives explicit examples showing that Kirillov’s proposed preservation rule is false.

Kirillov formulated the conjecture in 2004: scaling the Lehmer codes of three permutations should preserve positivity of the corresponding Schubert coefficient. The conjecture is false, already in four-dimensional permutation examples.

2026 counterexamples

The paper constructs, for every n≥4n\geq 4, permutations with cu,vw=1c_{u,v}^{w}=1 but cN∗u,N∗vN∗w=0c_{N*u,N*v}^{N*w}=0 for every N>1N>1; its smallest example is u=2143u=2143, v=2134v=2134, w=4123w=4123. A later paper independently records these counterexamples and states that the stretched coefficients are eventually quasi-polynomial. The revised counterexample paper also reports that its first author and Slonim refuted the opposite implication, so both directions of the proposed equivalence fail. The analogous 22-step saturation question remains open.

Current status (as of August 2026): Kirillov’s conjecture is settled negatively; explicit counterexamples disprove the forward implication, and the revised paper reports that the reverse implication also fails.

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