Hall–Littlewood scalar product and monic functions

Let $s$ be an indeterminate and work in ${\mathrm{\Lambda }}_{\mathbb{Q}(s)}$, the scalar extension of The stable graded ring of symmetric functions. For a partition $\lambda$, let $m_r(\lambda )$ be the multiplicity of its part $r$, and put $z_{\lambda }={\prod }_{r\ge 1}{r}^{m_r(\lambda )}{m}_r(\lambda )!,b_{\lambda }(s)={\prod }_{r\ge 1}{\prod }_{j=1}^{m_r(\lambda )}(1-s^j),n(\lambda )={\sum }_{i\ge 1}(i-1){\lambda }_i\mathrm{.}$ The Hall–Littlewood scalar product is the symmetric bilinear form with $⟨p_{\lambda },p_{\mu }{⟩}_s={\delta }_{\lambda \mu }{z}_{\lambda }{\prod }_i(1-s^{{\lambda }_i}{)}^{-1}\mathrm{.}$ This defines a nondegenerate form, because the power sums are a rational basis (Power sums form a rational but not integral stable basis) and all displayed diagonal entries are nonzero. Different degrees are orthogonal; empty products are $1$, so $⟨1,1{⟩}_s=1$.

The monic Hall–Littlewood functions $P_{\lambda }(x;s)$ are characterized by $P_{\lambda }=m_{\lambda }+{\sum }_{\mu ⊲\lambda }{c}_{\lambda \mu }(s)m_{\mu },⟨P_{\lambda },P_{\mu }{⟩}_s=0(\lambda \mathrm{\ne }\mu ),$ where $⊲$ is strict dominance (Dominance order on partitions) and the monomials use The monomial symmetric functions form the integral stable basis. Existence, uniqueness, and the norm formula $⟨P_{\lambda },P_{\lambda }{⟩}_s=b_{\lambda }(s{)}^{-1}$ are proved in Jordan Hall classes are Hall–Littlewood functions with the transported Green pairing ↗, rather than assumed in this definition. Set $Q_{\lambda }=b_{\lambda }(s)P_{\lambda }$. For a real parameter $0<s<1$ the same formulas specialize wherever used below; the theorem proves their regularity there. This form differs from the ordinary symmetric-function Hall inner product, whose power-sum norms are $z_{\lambda }$.