Canonical truncation and semi-invariants for general contractions with nonzero central truncation

Determine the canonical truncation of an Inönü–Wigner contraction when the central component of the canonical truncation of the original parabolic subalgebra is nonzero, and characterize the resulting algebra of symmetric semi-invariants in terms of symmetric invariant algebras.

Background

The paper studies Inönü–Wigner contractions of parabolic subalgebras, obtained by making the nilpotent radical abelian, and investigates polynomiality of the associated algebras of symmetric invariants and symmetric semi-invariants. For a parabolic subalgebra whose Levi factor has three symmetric blocks, the canonical truncation contains a nonzero one-dimensional central component.

In this setting, the relationship between the canonical truncation of the original algebra and that of its contraction is not generally understood. Resolving this issue would clarify how symmetric semi-invariants of the contraction relate to symmetric invariants of an appropriate truncated contraction.

References

When ${\Lambda}\neq{0}$ it is not clear in general what should be the canonical truncation $\widetilde{\Lambda}$ of the contraction $\widetilde$ and then what should be the algebra $Sy\bigl(\widetilde\bigr)$ generated by symmetric semi-invariants in terms of algebra of symmetric invariants (in view of Eq. \ref{semitrunc}).

— Polynomiality for algebras of invariants associated with some Inönü-Wigner contractions in type A  (2609.01173 - Fauquant-Millet, 1 Sep 2026) in Section 1, subsection labeled “The context” (paragraph beginning “When $_{\Lambda}\neq\{0\}$”)

We cannot conclude for the moment whether $Sy\bigl(\widetilde\bigr)=Y\bigl(\widetilde'\bigr)$ is or not polynomial in this case.

— Polynomiality for algebras of invariants associated with some Inönü-Wigner contractions in type A  (2609.01173 - Fauquant-Millet, 1 Sep 2026) in Section 1, subsection labeled “The context”; Section “The case when the three blocks are of the same size,” Remark following Lemma \ref{samesize}

When $3s=n$ that is, when the three blocks of the standard Levi factor $\mathfrak r$ of $$ are of the same size $s$, it is not clear whether the canonical truncation $\widetilde_{\Lambda}$ of the contraction $\widetilde$ is equal to the derived subalgebra $\widetilde'$ of $\widetilde$ or to the contraction $\widetilde{{\Lambda}$ of the canonical truncation ${\Lambda}$ of $$.

— Polynomiality for algebras of invariants associated with some Inönü-Wigner contractions in type A  (2609.01173 - Fauquant-Millet, 1 Sep 2026) in Section “The case when the three blocks are of the same size”