JournalscmhVol. 72, No. 1pp. 84–100

Symmetric and non-symmetric quantum Capelli polynomials

  • F. Knop

    Rutgers University, Piscataway, USA
Symmetric and non-symmetric quantum Capelli polynomials cover
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Abstract

We introduce families of symmetric and non-symmetric polynomials (the quantum Capelli polynomials) which depend on two parameters q and t. They are defined in terms of vanishing conditions. In the differential limit (q=tαandt1)(q = t^\alpha\, {\rm and}\, t \to 1) they are related to Capelli identities. It is shown that the quantum Capelli polynomials form an eigenbasis for certain q-difference operators. As a corollary, we obtain that the top homogeneous part is a symmetric/non-symmetric Macdonald polynomial. Furthermore, we study the vanishing and integrality properties of the quantum Capelli polynomials.

Cite this article

F. Knop, Symmetric and non-symmetric quantum Capelli polynomials. Comment. Math. Helv. 72 (1997), no. 1, pp. 84–100

DOI 10.4171/CMH/72.1.7