Stable Hilbert series of \mathcal S(\mathfrak g)^K for classical groups

2005/10/29 by Jeb F. Willenbring, Willenbring, Jeb F.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.math/0510649

openalex publication_date 2005/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a classical symmetric pair, (G,K), with \mathfrak g = Lie(G), we provide descriptions of the Hilbert series of the algebra of K-invariant vectors in the associated graded algebra of \mathcal U(\mathfrak g) viewed as a K-representation under restriction of the adjoint representation. The description illuminates a certain stable behavior of the Hilbert series, which is investigated in a case-by-case basis. We note that the stable Hilbert series of one symmetric pair often coincides with others. Also, for the case of the real form U(p,q) we derive a closed expression for the Hilbert series when min(p,q) → ∞.

Related

• Stability theorems for multiplicities in graded S_n-modules
• On the graded algebras associated with Hecke symmetries, II. The Hilbert series
• Invariant rings and representations of the symmetric groups
• Hilbert Functions of \mathfrak S_n-Stable Artinian Gorenstein Algebras
• Hilbert series of symmetric ideals in infinite polynomial rings via\n formal languages