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.

中文翻译：

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BCD型和对称多项式的QuantumW1+∞子代数

ABCD 类型的无限仿射李代数，也称为 $\widehat{\mathfrak{G}\mathfrak{升}}(\infty )$, $\widehat{\mathfrak{○}}(\infty )$， 和 $\widehat{\mathfrak{秒}\mathfrak{磷}}(\infty )$, 等价于量子W _1+∞代数的子代数。他们对狄拉克费米子的福克空间有众所周知的表示（${\widehat{\mathrm{一种}}}_{\infty }$), 马约拉纳费米子 (${\widehat{乙}}_{\infty }$ 和 ${\widehat{D}}_{\infty }$)，或辛玻色子 (${\widehat{C}}_{\infty }$）。为每个表示都提出了量子W _1+∞子代数对 Fock 状态的作用的显式公式。这些公式等价于量子环形的垂直表示$\mathfrak{G}\mathfrak{升}(1)$代数福克表示。它们提供了水平表示的费米子和玻色子表达式的替代方案。此外，众所周知，这些代数与对称多项式有很深的联系。量子W _1+∞发生器的作用导致这些多项式的类皮耶里规则和 q 差分方程的推导。在特定情况下${\widehat{乙}}_{\infty }$，对于由严格分区索引的Q- Schur 多项式获得 q 差分方程。