Specializations of symmetric functions and the series $\sigma$

Work in ${\mathrm{\Lambda }}_{\mathbb{Q}}:=\mathbb{Q}{\otimes }_{\mathbb{Z}}\mathrm{\Lambda }$, the $\mathbb{Q}$-scalar extension of the stable graded ring (The stable graded ring of symmetric functions). Its power sums $p_1,p_2,\dots$ (Power sums $p_k$ and complete homogeneous symmetric polynomials $h_k$) are algebraically independent generators: ${\mathrm{\Lambda }}_{\mathbb{Q}}=\mathbb{Q}[p_1,p_2,\dots ]$, and for every $d$ the products $p_{\lambda }$, $\lambda ⊢d$, form a $\mathbb{Q}$-basis of ${\mathrm{\Lambda }}_{\mathbb{Q}}^d$ (Power sums form a rational but not integral stable basis).

A specialization is a $\mathbb{Q}$-algebra homomorphism $f\mapsto f[A]$ from ${\mathrm{\Lambda }}_{\mathbb{Q}}$ to a commutative $\mathbb{Q}$-algebra. By the algebraic independence of the power sums, a specialization is determined by the sequence of values $(p_k[A]{)}_{k\ge 1}$, and conversely every sequence of elements of a commutative $\mathbb{Q}$-algebra extends uniquely to a specialization. For specializations $A,B$, a scalar $c\in \mathbb{Q}$ and a one-letter alphabet $t$ one defines the specializations $A+B$, $-A$, $AB$, $c\cdot A$ and $tA$ by their power-sum values

$p_k[A+B]:=p_k[A]+p_k[B],p_k[-A]:=-p_k[A],p_k[AB]:=p_k[A]p_k[B],$

$p_k[c\cdot A]:=c{p}_k[A],p_k[t]:=t^k\text{ and hence }{p}_k[tA]=t^k{p}_k[A]\mathrm{.}$

Here $c\cdot A$ is a coefficient multiplicity, to be distinguished from the one-letter alphabet $t$, for which $p_k[t]=t^k$ is a power and $p_k[tA]=t^k{p}_k[A]$; all occurrences of $tX$, $t^{-1}$, $tA$ and $\sigma [t]$ on this page use this one-letter convention. The symbol $1$ denotes the one-letter alphabet with $p_k[1]=1$ for all $k$, and $-1$ denotes its plethystic negative with $p_k[-1]=-1$ for all $k$ (Power sums $p_k$ and complete homogeneous symmetric polynomials $h_k$); it is not the ordinary one-letter alphabet whose letter is minus one, for which $p_k$ would equal $(-1{)}^k$. One writes $f[A+B]$, $f[-A]$, $f[AB]$ and $f[c\cdot A]$ for the images of $f$ under these specializations.

Define $X^{\perp }$ to be the specialization with $f[X^{\perp }]:=f^{\perp }$ for all $f$, where $f^{\perp }$ is the adjoint of multiplication by $f$ with respect to the Hall inner product $⟨,{⟩}_H$ (The Hall inner product on symmetric functions), characterized by $⟨f^{\perp }g,h{⟩}_H=⟨g,fh{⟩}_H$. It exists and is unique because the Hall form is nondegenerate on each finite-dimensional graded piece ${\mathrm{\Lambda }}_{\mathbb{Q}}^d$; the adjoints of commuting multiplication operators commute, so the operators $f^{\perp }$ span a commutative $\mathbb{Q}$-algebra and $X^{\perp }$ is a specialization in the sense above.

Define the series $\sigma [X]:={\sum }_{n\ge 0}{h}_n[X],$ $h_n$ the complete homogeneous symmetric function (Power sums $p_k$ and complete homogeneous symmetric polynomials $h_k$). Here $\sigma$ is interpreted in the graded completion of ${\mathrm{\Lambda }}_{\mathbb{Q}}$ (or with a formal degree variable): it is a formal series, not an element of ${\mathrm{\Lambda }}_{\mathbb{Q}}$, and every occurrence of $\sigma$ below is to be read degree by degree. Every Hall pairing of a series with a polynomial is the finite sum obtained by truncating to the degrees of the polynomial. No convergence of a scalar infinite sum is asserted, and no value of $\sigma$ at a specialization with infinitely many letters is interpreted analytically.