Polynomiality for algebras of invariants associated with some In\"on\"u-Wigner contractions in type A

Abstract

This paper deals with polynomiality of algebras of symmetric invariants (or generated by symmetric semi-invariants) associated with some particular In\"on\"u-Wigner contractions in type A. More precisely we are interested with contractions of some parabolic subalgebras in type A with respect to their decomposition into their Levi factor and their nilpotent radical, the latter becoming an abelian ideal of the contraction. When the parabolic subalgebra $\mathfrak p$ has its Levi factor decomposing into three symmetric blocks, with respect to the antidiagonal, we show in this article that the algebra of symmetric invariants associated with the contraction of the canonical truncation of $\mathfrak p$ is a polynomial algebra, for which we can give the number of algebraically independent homogeneous generators, their weight and degree. The method relies on the construction of an adapted pair for such a contraction. As a by-product we obtain that the algebra generated by symmetric semi-invariants associated with the In\"on\"u-Wigner contraction of such a parabolic subalgebra $\mathfrak p$ in type A has a Weierstrass section and then is a polynomial algebra, when the central block is not of the same size as the two extremal blocks of the Levi factor.