Young-Capelli bitableaux, Capelli immanants in U(gl(n)) and the Okounkov\n quantum immanants

2018/07/26 by Andrea Brini, Brini, Andrea, Antonio Teolis +1
Mathematics · Physics and Astronomy · #17B10 05E05 05E15 17B35 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1807.10045

openalex publication_date 2018/07/26 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28

Abstract

We propose a new approach to a unified study of determinants, permanents,\nimmanants, (determinantal) bitableaux and symmetrized bitableaux in the\npolynomial algebra C[M_{n, n}] as well as of their Lie analogues in the\nenveloping algebra U(gl(n)). This leads to new relevant classes of elements\nin U(gl(n)): Capelli bitableaux, right Young-Capelli bitableaux and Capelli\nimmanants. The set of standard Capelli bitableaux and the set of standard right\nYoung-Capelli bitableaux are bases of U(gl(n)), whose action on the\nGordan-Capelli basis of C[M_{n, n}] have remarkable properties. Capelli\nimmanants can be efficiently computed and provide a system of generators of\nU(gl(n)). The Okounkov quantum immanants are proved to be simple linear\ncombinations of Capelli immanants. Several examples are provided throughout the\npaper.\n

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