The Hall inner product on symmetric functions

The Hall inner product is the graded $\mathbb{Z}$-bilinear form on the stable ring $\mathrm{\Lambda }$ (The stable graded ring of symmetric functions) characterized by $⟨h_{\lambda },m_{\mu }{⟩}_H={\delta }_{\lambda \mu }$ for all partitions $\lambda ,\mu$ (Partitions, English diagrams, and conjugation, Elementary and complete families freely generate the stable ring, The monomial symmetric functions form the integral stable basis), where ${\delta }_{\lambda \mu }=1$ if $\lambda =\mu$ and $0$ otherwise. In particular, homogeneous components of unequal degrees are orthogonal. For the empty partition, $h_{\mathrm{\varnothing }}=m_{\mathrm{\varnothing }}=1$, so $⟨1,1{⟩}_H=1$.

For each $d\ge 0$, the proven integral bases give unique finite expansions $f_d={\sum }_{\lambda ⊢d}{a}_{\lambda }{h}_{\lambda },g_d={\sum }_{\mu ⊢d}{b}_{\mu }{m}_{\mu }$ for $f_d,g_d\in {\mathrm{\Lambda }}^d$. Define their degree-$d$ pairing by $⟨f_d,g_d{⟩}_H:={\sum }_{\lambda ,\mu ⊢d}{a}_{\lambda }{b}_{\mu }{\delta }_{\lambda \mu }\mathrm{.}$ For $f={\sum }_d{f}_d$ and $g={\sum }_d{g}_d$ in the algebraic direct sum $\mathrm{\Lambda }$, set $⟨f,g{⟩}_H:={\sum }_d⟨f_d,g_d{⟩}_H\mathrm{.}$ Only finitely many $d$ contribute because $f$ and $g$ have finite degree support. The unique expansions in the two integral bases make this a well-defined $\mathbb{Z}$-bilinear form; the displayed basis rule determines it uniquely on all of $\mathrm{\Lambda }$.