Commentarii Mathematici Helvetici

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Volume 72, Issue 1, 1997, pp. 84–100
DOI: 10.4171/CMH/72.1.7

Published online: 1997-03-31

Symmetric and non-symmetric quantum Capelli polynomials

F. Knop[1]

(1) Rutgers University, Piscataway, USA

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 = 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.

Keywords: Symmetric polynomials, Capelli identity,Macdonald polynomials, difference operators, Hecke algebras. Key words. Symmetric polynomials, Capelli identity, Macdonald polynomials, difference operators, Hecke algebras

Knop F.: Symmetric and non-symmetric quantum Capelli polynomials. Comment. Math. Helv. 72 (1997), 84-100. doi: 10.4171/CMH/72.1.7