The Demazure Conjecture

Definitions and mathematical statement

The stored question is: "Can representations of semisimple algebraic groups be characterized over the integers?" For a field $k$, a finite-dimensional rational representation of an algebraic group $G/k$ is a morphism of algebraic groups $\rho:G\to\mathrm{GL}(V)$, where $V$ is a finite-dimensional $k$-vector space. This explains one standard meaning of representation but does not specify what the imported word "characterized" asks. To ask for a representation over ${\mathbb{Z}}$ one would need, for example, a specified group scheme over ${\mathbb{Z}}$, an appropriate integral module, a group action, and a precise base-change comparison. These choices are absent, and this draft does not impose them.

The original background attributes the problem to Michel Demazure in the 1970s and describes integral structures and deformation of positive-characteristic irreducible representations to characteristic zero. The supplied corrective review, checked 2026-08-17, explicitly says that this wording does not accurately formulate the historical Demazure conjecture. It instead identifies that historical conjecture with arbitrary-characteristic cohomology vanishing and character formulas for Schubert varieties. The record gives no exact line bundle, Schubert variety, vanishing range, or character identity, so it does not contain enough data to substitute a precise historical theorem here.

The same review reports the historical vanishing and character results as established in the literature: Lakshmibai--Musili--Seshadri supplied classical-group standard monomial theory, with characteristic-free general semisimple/reductive results through Kempf vanishing, Frobenius splitting, and uniform standard monomial/path theory. It cites Littelmann for uniform standard monomial theory for symmetrizable Kac--Moody algebras. Its status is solved / SOLVED-IN-LITERATURE for that named historical target, while the registry status field remains open. Crucially, the review assigns medium confidence to the mapping of the malformed record to the historical conjecture, not to the historical resolution. The broad integral-characterization or modular-lifting question in the stored wording is not thereby asserted solved.

Short English statement

The record asks broadly about integral forms of representations, but its corrective review identifies a different, historically solved Demazure conjecture about Schubert-variety vanishing and characters. The two descriptions cannot be merged into one precise theorem without repairing the source formulation.

Sources

• Local UnsolvedMath registry, permanent ID 1200, source code catalogue entry, "The Demazure Conjecture"; supplied statement, background attribution to Michel Demazure in the 1970s, and corrective literature triage.
• A. Ramanathan, "Equations defining Schubert varieties and Frobenius splittings of diagonals," Publications Mathématiques de l'IHÉS 65 (1987), 61--90: https://www.numdam.org/article/PMIHES_1987__65__61_0.pdf. Supplied primary reference for the arbitrary-characteristic vanishing and character formulation.
• V. Lakshmibai and C. S. Seshadri, "Geometry of G/P-V," Journal of Algebra 100 (1986), 462--557: https://citeseerx.ist.psu.edu/document?doi=d2008112efe3875dc57e923d89a97fb6dd7113bc&repid=rep1&type=pdf. Supplied primary reference for the classical-scope standard monomial result.
• Peter Littelmann, "Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras," Journal of the American Mathematical Society 11 (1998), 551--567: https://doi.org/10.1090/S0894-0347-98-00268-9. Supplied primary reference for uniform standard monomial bases.
• Venkatramani Lakshmibai et al., "C. S. Seshadri (1932--2020)," Notices of the AMS 68 (2021): https://jointmathematicsmeetings.org/journals/notices/202111/noti2383/noti2383.html. Supplied secondary participant history concerning the gap in Demazure's proof, corrected classical work, and later completion.
• Supplied literature triage checked 2026-08-17, SOLVED-IN-LITERATURE, with medium-confidence source mapping and an explicit requirement to repair the dataset statement; dated triage does not exclude later or unindexed results.