Polynomial ring representations of endomorphisms of exterior powers

Behzad, Ommolbanin ^[3] ; Contiero, André ^[1] ; Martins, Renato Vidal ^[1] ; Gatto, Letterio ^[2]
1. [1] Universidade Federal de Minas Gerais
  
  Universidade Federal de Minas Gerais
  
  Brasil
2. [2] Polytechnic University of Turin
  
  Polytechnic University of Turin
  
  Torino, Italia
3. [3] Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
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Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 73, Fasc. 1, 2022, págs. 107-133
Idioma: inglés
DOI: 10.1007/s13348-020-00310-5
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Resumen
- An explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.