MathDBThe divisibility conjecture for the -simplex
The divisibility conjecture for the -simplex
Arrange the entries in each layer of the -simplex as coefficients of a polynomial in according to the arrangement described in the paper. Divisibility conjecture. Each such polynomial is divisible by
This reformulates the observed 4-neighbor addition pattern through polynomial divisibility and is presented as a conjectural combinatorial property of the stable-envelope integral array.
Progress summary
No verified proof or counterexample to the divisibility conjecture has been found publicly.
The conjecture asserts that every polynomial layer associated with the -simplex is divisible by . Crawford, Kartik, and Lance present the underlying stable-envelope integral array in work posted in October 2025, but the retrieved material contains no verified resolution of this divisibility claim.
Current status (as of August 2026): The divisibility conjecture remains open; no publicly verified proof, counterexample, or claimed resolution was found in the retrieved sources.
Sources & referencesView supporting material
Primary source
Matthew Crawford, Pavan Kartik and Reese Lance, “Integrals of stable envelopes for cotangent bundles to Grassmannians”, arXiv:2510.21573 (2026).
Solutions 1
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Both the divisibility conjecture and its stronger positive-quotient form hold for every Grassmannian layer.
For nonnegative integers A,B,C, write d=A+B+C and define
q(A,B,C)=d!(d+1)!(B+1)/[A!C!(A+B+1)!(B+C+1)!],
with q=0 whenever any argument is negative. Set
Q_d(a,b,c)=∑_{A+B+C=d}q(A,B,C)a^A b^B c^C.
If H_n(a,b,c) is the unreduced Gr₂-simplex layer polynomial from the primary source, then the exact all-layer factorization is
H_n(a,b,c)=(a+2b+c)Q_{n−3}(a,b,c), n≥3.
To verify it directly from the stable-envelope formula in Example 3.2, set
A=i−1, B=j−i−1, C=n−j.
The source's explicit coefficient F_n(i,j) simplifies to
F_n(i,j) =(n−2)!(n−3)! T(A,B,C)/ [A!C!(A+B+1)!(B+C+1)!],
where direct expansion of its displayed bracket gives
T(A,B,C) =(B+1)A(A+B+1) +2B(A+B+1)(B+C+1) +(B+1)C(B+C+1).
Substituting the formula for q therefore yields the exact coefficient identity
F_n(i,j) =q(A−1,B,C)+2q(A,B−1,C)+q(A,B,C−1),
including all boundary positions under the negative-index convention. Comparing coefficients proves the factorization and thus divisibility by a+2b+c for every n.
Furthermore, the quotient has positive integer coefficients. Interpret q(A,B,C) as the number of nonnegative walks starting at height zero with four step types
U: height +1, weight b; D: height −1, weight ac/b; L_a: height 0, weight a; L_c: height 0, weight c.
For a walk contributing a^A b^B c^C and using s down-steps, the ballot/reflection principle gives
d!(B+1)/[(A−s)!(C−s)!s!(B+s+1)!]
possibilities. Summing over 0≤s≤min(A,C) and applying Chu–Vandermonde gives exactly the displayed q(A,B,C). Hence Q_d∈Z_{>0}[a,b,c] coefficientwise on its degree-d simplex.
In particular, the conjectured divisibility is strengthened to a completely explicit positive integral quotient, simultaneously yielding the reduced-simplex recurrence through the ballot-walk construction.