Decomposition of modules over invariant differential operators

2015/06/20 by Rikard Bøgvad, Bögvad, Rikard, Rolf Källström +1
Mathematics · Physics and Astronomy · #14F10 20C30 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1506.06229

openalex publication_date 2015/06/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let G be a finite subgroup of the linear group of a finite-dimensional complex vector V, B=\operatorname S(V) be the symmetric algebra, \mathcal D=\mathcal D^G_B the ring of G-invariant differential operators, and \mathcal D^- its subring of negative degree operators. We prove that M↦ M^ann= \operatorname Ann_{\mathcal D^-}(M) defines an isomorphism between the category of \mathcal D-submodules of B and a category of modules formed as lowest weight spaces. This is applied to a construction of simple \mathcal D-submodules of B when G is a generalized symmetric group, to show that B^ann is a so-called Gelfand model. Using differential algebra and lowest weight methods we also prove branching rules, entailing the main results in the representation theory of the symmetric group, such as a differential construction of the Young basis.

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