Length generating series, descent-class series, spherical subsets, and the multivariate descent polynomial

Let $(W,S)$ be a Coxeter system with $S$ finite, presented group $W$, length function $\mathrm{\ell }$ and standard parabolic subgroups $W_I=⟨s:s\in I⟩$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), and let $D_L(w)=\{s\in S:\mathrm{\ell }(sw)<\mathrm{\ell }(w)\}$ and $D_R(w)=\{s\in S:\mathrm{\ell }(ws)<\mathrm{\ell }(w)\}$ be the descent sets with the conventions fixed in Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups (2).

(1) Length generating series. Let $A\subseteq W$. The length generating series (also Poincare series) of $A$ is $P_A(t):={\sum }_{w\in A}{t}^{\mathrm{\ell }(w)}\in \mathbb{Z}\mathrm{⟦}t\mathrm{⟧}$ (Formal power series over a commutative ring and the coefficient-extraction functional $[x^n]$). It is well defined: for every $n\in \mathbb{N}$ the fiber $\{w\in W:\mathrm{\ell }(w)=n\}$ is finite. Evaluation of the $n$-letter words gives a map $S^n\to W$, and each element of the fiber is the value of one of its reduced words; $S^n$ is finite by The product rule: $\mid A\times B\mid =\mid A\mid \mid B\mid$, and $\mid {\prod }_{i<m}{A}_i\mid ={\prod }_{i<m}\mid A_i\mid$. Hence $[t^n]P_A=\mathrm{\mid }\{w\in A:\mathrm{\ell }(w)=n\}\mathrm{\mid }$ is the cardinality (The cardinality $\mid A\mid$ of a finite set) of a finite set, viewed as an integer. This includes $S=\mathrm{\varnothing }$: the length-zero fiber is $\{1\}$ and every positive-length fiber is empty. If $1\in A$ then $P_A(0)=1$ and $P_A(t)$ is a unit of $\mathbb{Z}\mathrm{⟦}t\mathrm{⟧}$ with a recursively determined formal inverse (A formal power series is a unit exactly when its constant coefficient is a unit); if $1\mathrm{\notin }A$ then $P_A(0)=0$ and $P_A(t)$ is not a unit. Thus $P_A(t)$ is a unit exactly when $1\in A$. No convergence, radius of convergence or evaluation at a real number is asserted.

(2) Spherical subsets and descent-class series. A subset $I\subseteq S$ is spherical when the standard parabolic $W_I$ is finite. For $I\subseteq J\subseteq S$ the descent-class series is $D_I^J(t):={\sum }_{\begin{array}{c}w\in W\\ I\subseteq D_R(w)\subseteq J\end{array}}{t}^{\mathrm{\ell }(w)}\in \mathbb{Z}\mathrm{⟦}t\mathrm{⟧}\mathrm{.}$

This series is well defined coefficientwise because its length-$n$ summation set is a subset of the finite length-$n$ fiber in (1).

(3) Multivariate descent polynomial (finite $W$ only). For finite $W$ define the marked multivariate descent polynomial $\widehat{W}(\mathbf{x},\mathbf{y},t):={\sum }_{w\in W}{t}^{\mathrm{\ell }(w)}{\prod }_{s\in D_R(w)}{x}_s{\prod }_{s\in S\setminus D_R(w)}{y}_s\in \mathbb{Z}[x_s,y_s:s\in S][t]$ (Polynomial rings in finitely many commuting indeterminates by iteration). This is a polynomial because the sum is finite. Setting every $y_s=1$ recovers the usual descent polynomial ${\sum }_{w\in W}{t}^{\mathrm{\ell }(w)}{\prod }_{s\in D_R(w)}{x}_s$. For $I\subseteq J\subseteq S$, substitute $x_s=1,y_s=0$ for $s\in I$, $x_s=y_s=1$ for $s\in J\setminus I$, and $x_s=0,y_s=1$ for $s\mathrm{\notin }J$. A term survives exactly when $I\subseteq D_R(w)\subseteq J$, and then its descent/non-descent factors all equal $1$; hence this evaluation is $D_I^J(t)$, also a polynomial. This includes $I=\mathrm{\varnothing },J=S$ (the full series $P_W$) and $I=J$ (the exact descent set $I$).

(4) Conventions and limits. $P_{\mathrm{\varnothing }}=0$ and $P_{\{1\}}=1$. For infinite $S$ no scalar Poincare series is defined here; finite $S$ is the standing hypothesis that ensures the finite-coefficient argument in (1). This definition makes no assertion that $P_A$ is rational, that $W^J$ has a length-additive interpretation, that $D_I^J(t)$ has the inclusion-exclusion expansion, or that $W_{D_R(w)}$ is finite; those are separate results, not part of the definitions above. No choice principle is used.