Corpus ID: 233481226

Modular plethystic isomorphisms for two-dimensional linear groups

@inproceedings{McDowell2021ModularPI,
  title={Modular plethystic isomorphisms for two-dimensional linear groups},
  author={Eoghan McDowell and Mark Wildon},
  year={2021}
}

• Eoghan McDowell, M. Wildon
• Published 2 May 2021
• Mathematics

Let E be the natural representation of the special linear group SL2(K) over an arbitrary field K. We use the two dual constructions of the symmetric power when K has prime characteristic to construct an explicit isomorphism SymmSym E ∼= Sym`SymmE. This generalises Hermite reciprocity to arbitrary fields. We prove a similar explicit generalisation of the classical Wronskian isomorphism, namely SymmSym E ∼= ∧m Sym`+m−1E. We also generalise a result first proved by King, by showing that if ∇ is the Schur functor for the partition λ and λ◦ is the complement of λ in a rectangle with `+1 rows, then ∇SymE ∼= ∇ ◦ Sym`E. To illustrate that the existence of such ‘plethystic isomorphisms’ is far from obvious, we end by proving that the generalisation ∇SymE ∼= ∇ ′ Sym )−`(λ)E of the Wronskian isomorphism, known to hold for a large class of partitions over the complex field, does not generalise to fields of prime characteristic, even after considering all possible dualities.