The set of standard Capelli bitableaux and the set of standard Young-Capelli bitableaux are bases of U(gl(n)), whose action on the Gordan-Capelli basis of polynomial algebra ℂ[Mn,n] have remarkable properties (see, e.g. [A. Brini, A. Palareti and A. Teolis, Gordan-Capelli series in superalgebras, Proc. Natl. Acad. Sci. USA 85 (1988) 1330-1333; A. Brini and A. Teolis, Young-Capelli symmetrizers in superalgebras, Proc. Natl. Acad. Sci. USA 86 (1989) 775-778; A. Brini and A. Teolis, Capelli bitableaux and &-forms of general linear Lie superalgebras, Proc. Natl. Acad. Sci. USA 87 (1990) 56-60; A. Brini and A. Teolis, Capelli's theory, Koszul maps, and superalgebras, Proc. Natl. Acad. Sci. USA 90 (1993) 10245-10249.]). We introduce a new class of elements of U(gl(n)), called the Capelli immanants, that can be efficiently computed and provide a system of linear generators of U(gl(n)). The Okounkov quantum immanants [A. Okounkov, Quantum immanants and higher Capelli identities, Transform Groups 1 (1996) 99-126; A. Okounkov, Young basis, Wick formula, and higher Capelli identities, Int. Math. Res. Not. 1996(17) (1996) 817-839.] - quantum immanants, for short - are proved to be simple linear combinations of diagonal Capelli immanants, with explicit coefficients. Quantum immanants can also be expressed as sums of double Young-Capelli bitableaux. Since double Young-Capelli bitableaux uniquely expands into linear combinations of standard Young-Capelli bitableaux, this leads to canonical presentations of quantum immanants, and, furthermore, it does not involve the computation of the irreducible characters of symmetric groups.

Young–Capelli bitableaux, Capelli immanants in U(gl(n)) and the Okounkov quantum immanants / A. Brini; A. Teolis. - In: JOURNAL OF ALGEBRA AND ITS APPLICATIONS. - ISSN 0219-4988. - STAMPA. - 20:7(2021), pp. 2150123.1-2150123.44. [10.1142/S0219498821501231]

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

A. Brini;
2021

Abstract

The set of standard Capelli bitableaux and the set of standard Young-Capelli bitableaux are bases of U(gl(n)), whose action on the Gordan-Capelli basis of polynomial algebra ℂ[Mn,n] have remarkable properties (see, e.g. [A. Brini, A. Palareti and A. Teolis, Gordan-Capelli series in superalgebras, Proc. Natl. Acad. Sci. USA 85 (1988) 1330-1333; A. Brini and A. Teolis, Young-Capelli symmetrizers in superalgebras, Proc. Natl. Acad. Sci. USA 86 (1989) 775-778; A. Brini and A. Teolis, Capelli bitableaux and &-forms of general linear Lie superalgebras, Proc. Natl. Acad. Sci. USA 87 (1990) 56-60; A. Brini and A. Teolis, Capelli's theory, Koszul maps, and superalgebras, Proc. Natl. Acad. Sci. USA 90 (1993) 10245-10249.]). We introduce a new class of elements of U(gl(n)), called the Capelli immanants, that can be efficiently computed and provide a system of linear generators of U(gl(n)). The Okounkov quantum immanants [A. Okounkov, Quantum immanants and higher Capelli identities, Transform Groups 1 (1996) 99-126; A. Okounkov, Young basis, Wick formula, and higher Capelli identities, Int. Math. Res. Not. 1996(17) (1996) 817-839.] - quantum immanants, for short - are proved to be simple linear combinations of diagonal Capelli immanants, with explicit coefficients. Quantum immanants can also be expressed as sums of double Young-Capelli bitableaux. Since double Young-Capelli bitableaux uniquely expands into linear combinations of standard Young-Capelli bitableaux, this leads to canonical presentations of quantum immanants, and, furthermore, it does not involve the computation of the irreducible characters of symmetric groups.
2021
Young–Capelli bitableaux, Capelli immanants in U(gl(n)) and the Okounkov quantum immanants / A. Brini; A. Teolis. - In: JOURNAL OF ALGEBRA AND ITS APPLICATIONS. - ISSN 0219-4988. - STAMPA. - 20:7(2021), pp. 2150123.1-2150123.44. [10.1142/S0219498821501231]
A. Brini; A. Teolis
File in questo prodotto:
File Dimensione Formato  
24T122850.333.pdf

accesso aperto

Tipo: Postprint
Licenza: Licenza per Accesso Aperto. Creative Commons Attribuzione (CCBY)
Dimensione 1.3 MB
Formato Adobe PDF
1.3 MB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/781109
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 2
  • ???jsp.display-item.citation.isi??? 2
social impact