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

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.