Quantum W1+∞ subalgebras of BCD type and symmetric polynomials

Abstract

The infinite affine Lie algebras of type ABCD, also called $\widehat{\mathfrak{g}\mathfrak{l}}(\infty )$ , $\widehat{\mathfrak{o}}(\infty )$ , and $\widehat{\mathfrak{s}\mathfrak{p}}(\infty )$ , are equivalent to subalgebras of the quantum W_1+∞ algebras. They have well-known representations on the Fock space of a Dirac fermion (${\widehat{A}}_{\infty }$ ), a Majorana fermion (${\widehat{B}}_{\infty }$ and ${\widehat{D}}_{\infty }$ ), or a symplectic boson (${\widehat{C}}_{\infty }$ ). Explicit formulas for the action of the quantum W_1+∞ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of the vertical presentation of the quantum toroidal $\mathfrak{g}\mathfrak{l}(1)$ 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 W_1+∞ generators leads to the derivation of Pieri-like rules and q-difference equations for these polynomials. In the specific case of ${\widehat{B}}_{\infty }$ , a q-difference equation is obtained for Q-Schur polynomials indexed by strict partitions.