Invariants of symplectic and orthogonal groups acting on GL(n,\mathbb C)-modules

2017/07/18 by Vesselin Drensky, Elitza Hristova, Drensky, Vesselin +1
Mathematics · #05E05 #13A50 #15A72 #15A75 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1707.05893

openalex publication_date 2017/07/18 · openalex created_date 2017/07/31 · openalex updated_date 2026/07/28

Abstract

Let GL(n) = GL(n, \mathbb C) denote the complex general linear group and let G ⊂ GL(n) be one of the classical complex subgroups O(n), SO(n), and Sp(2k) (in the case n = 2k). We take a polynomial GL(n)-module W and consider the symmetric algebra S(W). Extending previous results for G=SL(n), we develop a method for determining the Hilbert series H(S(W)^G, t) of the algebra of invariants S(W)^G. Then we give explicit examples for computing H(S(W)^G, t). As a further application, we extend our method to compute also the Hilbert series of the algebras of invariants Λ(S² V)^G and Λ(Λ² V)^G, where V = \mathbb Cⁿ denotes the standard GL(n)-module.

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