A new modular plethystic SL₂(\mathbbF)-isomorphism Sym^N-1E ⊗ \bigwedge^N+1 Sym^d+1E ≅ Δ^(2,1^N-1) Sym^d E

2024/05/07 by Alvaro L. Martinez, Mark Wildon, Martinez, Alvaro L. +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2405.04631

Abstract

Let \mathbbF be a field and let E be the natural representation of SL₂(\mathbbF). Given a vector space V, let Δ^(2,1^N-1)V be the kernel of the multiplication map \bigwedge^N V ⊗ V → \bigwedge^N+1V. We construct an explicit SL₂(\mathbbF)-isomorphism Sym^N-1E ⊗ \bigwedge^N+1 Sym^d+1E ≅ Δ^(2,1^N-1) Sym^d E. This SL₂(\mathbbF)-isomorphism is a modular lift of the q-binomial identity q^(N(N-1))/(2)[N]_q \binomd+1N+1_q = s_(2,1^N-1)(1,q,…, q^d), where s_(2,1^N-1) is the Schur function for the partition (2,1^N-1). This identity, which follows from our main theorem, implies the existence of an isomorphism when \mathbbF is the field of complex numbers but it is notable, and not typical of the general case, that there is an explicit isomorphism defined in a uniform way for any field.

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