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 [Formula: see text], whose action on the Gordan–Capelli basis of polynomial algebra [Formula: see text] 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 [Formula: see text]-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 [Formula: see text], called the Capelli immanants, that can be efficiently computed and provide a system of linear generators of [Formula: see text]. 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.

Generating Functions for Hecke Algebra Characters

Certain polynomials in n2 variables that serve as generating functions for symmetric group characters are sometimes called (Sn) character immanants. We point out a close connection between the identities of Littlewood–Merris–Watkins and Goulden–Jackson, which relate Sn character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra Hn(q), we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood–Merris–Watkins identities and selected Goulden–Jackson identities that relate Hn(q) character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis–Lê–Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.