Quantum W1+∞ subalgebras of BCD type and symmetric polynomials

Abstract

The infinite affine Lie algebras of type ABCD, also called gl̂(∞), ô(∞), and sp̂(∞), are equivalent to subalgebras of the quantum W1+∞ algebras. They have well-known representations on the Fock space of a Dirac fermion (Â∞), a Majorana fermion (B̂∞ and D̂∞), or a symplectic boson (Ĉ∞). Explicit formulas for the action of the quantum W1+∞ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of the vertical presentation of the quantum toroidal gl(1) algebra Fock representation. They provide an alternative to the fermionic and bosonic expressions of the horizontal presentation. Furthermore, these algebras are known to have a deep connection with symmetric polynomials. The action of the quantum W1+∞ generators leads to the derivation of Pieri-like rules and q-difference equations for these polynomials. In the specific case of B̂∞, a q-difference equation is obtained for Q-Schur polynomials indexed by strict partitions.