Invariant differential operators and the generalized symmetric group

2021/11/10 by Ibrahim Nonkané, Nonkané, Ibrahim, Latévi M. Lawson +1
Chemistry · Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Axial and Atropisomeric Chirality Synthesis #FOS: Mathematics #Molecular spectroscopy and chirality #Primary 13N10 #Representation Theory (math.RT) #Secondary 20C30

paper · pdf · doi:10.48550/arxiv.2111.05655

openalex publication_date 2021/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the decomposition of the direct image of π₊(\Oc_X) the polynomial ring \Oc_X as a \D-module, under the map π: \spec \Oc_X → \spec \Oc_X^G(r,n), where \Oc_X^G(r,n) is the ring of invariant polynomial under the action of the wreath product G(r,p):= \ZZ / r \ZZ \wr \Sc_n . We first describe the generators of the simple components of π₊(\Oc_X) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a \D-module decomposition of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants, differential operators, on the higher Specht polynomials.

Related

• Higher Specht polynomials and modules over the Weyl algebra
• Invariant rings and representations of the symmetric groups
• Decomposition of the polynomials over the spherical subalgebra
• Some implications of a conjecture of Zabrocki to the action of S_n on polynomial differential forms
• Decomposition of modules over invariant differential operators