For every complex simple Lie algebra mathfrakg, with rho the half-sum of its positive roots, and every dominant integral weight lambda∈ P_+, the irreducible representation V(lambda) occurs in V(rho)otimes V(rho) if and only if lambda≤ 2 rho, where lambda≤ 2 rho means that 2 rho-lambda=∑_i=1^r a_i alpha_i for some a_i∈Z_≥ 0 and simple roots alpha_1,…,alpha_r.

Kostant’s conjecture

Kostant’s conjecture asks whether $V(\lambda)$ occurs in $V(\rho)\otimes V(\rho)$ exactly when the dominant weight satisfies $\lambda\leq 2\rho$ . It is known in type $A$ and in several restricted forms, but no general proof or counterexample was found.

Known results

Berenstein–Zelevinsky proved the conjecture for $\mathfrak{sl}_{n+1}$ .
Chirivì–Kumar–Maffei (2016) proved the statement after applying a saturation factor; the factor is $1$ in type $A$ .
A 2023 paper proved the conjecture for vertices of the relevant polytope, and recorded affirmative computations for types $G_2$ and $F_4$ .
New component families and criteria provide further supporting evidence, not a complete proof.

May–August 2026 developments

A May 2026 computation verified the conjecture for all simple Dynkin types up to rank $7$ , but this is only a finite check. A journal paper by Arzu Boysal appeared online in August 2026 and addresses the conjecture; its Crossref record has no abstract, so the retrieved evidence does not establish what it proves. No sourced claim of a general proof or counterexample was found.

Current status (as of August 2026): Kostant’s conjecture is proved in type $A$ and under saturation, with additional special cases and rank- $7$ computations, but remains unresolved in general.

Kostant’s conjecture

Primary source

Additional references

Progress summary

Solutions 0