Quantum $W_{1+\infty }$ subalgebras of BCD type and symmetric polynomials

Abstract

The infinite affine Lie algebras of type ABCD, also called $\widehat{\mathfrak{gl}}(\infty )$, $\hat{\mathfrak{o}}(\infty )$, $\widehat{\mathfrak{sp}}(\infty )$, are equivalent to subalgebras of the quantum $W_{1+\infty }$ algebras. They have well-known representations on the Fock space of either a Dirac fermion (${\hat{A}}_{\infty }$), a Majorana fermion (${\hat{B}}_{\infty }$ and ${\hat{D}}_{\infty }$) or a symplectic boson (${\hat{C}}_{\infty }$). Explicit formulas for the action of the quantum $W_{1+\infty }$ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of the \textit{vertical presentation} of the quantum toroidal $\mathfrak{gl}(1)$ algebra Fock representation. They provide an alternative to the fermionic and bosonic expressions of the \textit{horizontal presentation}. Furthermore, these algebras are known to have a deep connection with symmetric polynomials. The action of the quantum $W_{1+\infty }$ generators leads to the derivation of Pieri-like rules and q-difference equations for these polynomials. In the specific case of ${\hat{B}}_{\infty }$, a q-difference equation is obtained for $Q$-Schur polynomials indexed by strict partitions.

46 pages, python sketch included (v2: minor changes, to appear in JMP)

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• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations