Classical Poincare products for A, B, D and I2(m) from the permutation and signed-permutation models

Coordinate roots and the Gram isometry. In type $A_n$ take $E=H=\{x\in {\mathbb{R}}^{n+1}:{\sum }_i{x}_i=0\}$ and ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$; these roots have squared norm $2$, inner product $-1$ for adjacent indices and $0$ otherwise. In type $B_n$ take $E={\mathbb{R}}^n$, ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ for $i<n$ and ${\alpha }_n={\epsilon }_n$; their normalized adjacent inner products are $-\frac{1}{2}$ along the $3$-edges and $-\frac{1}{\sqrt{2}}=-\cos (\pi \mathrm{/}4)$ at the $4$-edge. In type $D_n$ take ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2=-({\epsilon }_1+{\epsilon }_2)$ and ${\alpha }_i={\epsilon }_{i-1}-{\epsilon }_i$ for $3\le i\le n$; the two fork inner products and chain inner products are $-1$, and the other pairs are orthogonal. For $I_2(m)$ take two normals of length $\sqrt{2}$ with angle $\pi -\pi \mathrm{/}m$. In each case the normalized roots ${\widehat{\alpha }}_s$ have Gram matrix $B(e_s,e_t)$. They form a basis: the type-$A$ differences form a basis of $H$, the type-$B$ roots are a coordinate-triangular basis, a relation among the type-$D$ roots has $c_n=0$ and successively $c_{n-1}=\cdots =c_3=0$, after which $c_1-c_2=-c_1-c_2=0$ forces $c_1=c_2=0$, and the two $I_2(m)$ normals are independent. Since $(e_s{)}_{s\in S}$ is a basis of $V$, $\varphi (e_s)={\widehat{\alpha }}_s$ extends to a linear isometry $\varphi :V\to E$.

Conjugating the canonical action. By [F7], $\varphi \rho (s){\varphi }^{-1}=r_{{\widehat{\alpha }}_s}$. The reflection formula gives the adjacent coordinate swaps in types $A_n,B_n,D_n$, the sign change of the last coordinate for $s_n$ in type $B_n$, the signed swap $(x_1,x_2)\mapsto (-x_2,-x_1)$ for $s_2$ in type $D_n$, and the two line reflections for $I_2(m)$. Let $G=⟨r_{{\widehat{\alpha }}_s}:s\in S⟩$. Its displayed generators are involutions, and [F2] transported by $\varphi$ shows that each pair product has the required Coxeter order, so [F3] gives a surjective homomorphism $\theta :W\to G$. The conjugate representation $\psi (w)={\varphi }^{-1}\theta (w)\varphi$ agrees with $\rho$ on every generator; uniqueness in [F3] gives $\psi =\rho$. Since $\rho$ is injective by [F15], $\theta$ is injective as well, hence $W\cong G$.

Identifying the model groups. In type $A_n$, the adjacent swaps generate $S_{n+1}$ by [F4]; if a permutation acts trivially on $H$, it fixes every ${\epsilon }_i-{\epsilon }_j$, which forces it to fix each index, so this action is faithful and $G\cong S_{n+1}$. In one-line notation, right multiplication by an adjacent transposition swaps two adjacent entries, changing only their mutual inversion and hence changing $\mathrm{inv}$ by exactly $1$ or $-1$. Therefore every word for $\pi$ has at least $\mathrm{inv}(\pi )$ letters. If $\pi \mathrm{\ne }1$, its one-line list has an adjacent descent; swapping that pair on the right reduces the inversion number by one. Repeating this operation reaches the increasing identity list after exactly $\mathrm{inv}(\pi )$ swaps, and reversing the swaps expresses $\pi$ by a word of that length. Thus $\mathrm{\ell }(\pi )=\mathrm{inv}(\pi )$. In type $B_n$, the reflections in roots ${\epsilon }_i$ negate coordinate $i$, those in ${\epsilon }_i-{\epsilon }_j$ swap coordinates $i,j$, and those in ${\epsilon }_i+{\epsilon }_j$ swap and negate coordinates $i,j$; these formulas show every root reflection is a signed permutation. Conversely the simple reflections $s_1,\dots ,s_{n-1}$ are adjacent swaps and $s_n$ is one coordinate sign change, so they generate all coordinate permutations and all individual sign changes, hence every signed permutation. As each is in the Weyl group and every root reflection is one of these signed permutations, $G=W(B_n)$. In type $D_n$, reflections in ${\epsilon }_i-{\epsilon }_j$ swap coordinates and those in ${\epsilon }_i+{\epsilon }_j$ swap and negate them; each has even sign parity, which is multiplicative under composition. The simple reflections $s_1,s_3,\dots ,s_n$ generate all coordinate permutations, and $s_1{s}_2$ negates coordinates $1,2$; conjugating this pair negation by permutations gives a negation of any pair of coordinates, and partitioning any even set into pairs gives every even sign pattern. Thus $G$ is the group of even signed permutations, and every root reflection lies in it, so $G=W(D_n)$. In type $I_2(m)$, $r=r_{{\widehat{\alpha }}_1}{r}_{{\widehat{\alpha }}_2}$ is a rotation by $2\pi \mathrm{/}m$ and has order $m$. Every word in two involutions reduces to an alternating word, hence to one of at most the $m$ powers $r^k$ or the $m$ elements $r^k{r}_{{\widehat{\alpha }}_1}$. The powers are distinct; the latter elements are distinct and have determinant $-1$, unlike the powers, so $\mathrm{\mid }G\mathrm{\mid }=2m$ and $G$ is dihedral.

Type $A_n$ ($n\ge 1$), $s_0=s_n$, $T=\{s_1,\dots ,s_{n-1}\}$. If $n=1$, then $T=\mathrm{\varnothing }$ and $W_T=\{1\}$ by [F9]; if $n\ge 2$, deleting $s_n$ from the displayed chain leaves type $A_{n-1}$. Let $y_j(x)=x_j{\mathrm{\mid }}_H$ and $z_j=-\sqrt{2}{y}_j$. Then $f_0\circ {\varphi }^{-1}=z_{n+1}$: $z_{n+1}$ vanishes on ${\widehat{\alpha }}_i=({\epsilon }_i-{\epsilon }_{i+1})\mathrm{/}\sqrt{2}$ for $i\le n-1$ and has value $1$ on ${\widehat{\alpha }}_n$. By step 3.1 the model group is $S_{n+1}$ acting by permuting the $z_j$, so $W\cdot f_0=\{z_1,\dots ,z_{n+1}\}$; each $s_i$ swaps $z_i,z_{i+1}$ and fixes the others, giving the path $z_1-z_2-\cdots -z_{n+1}$. For $j\le n$ the word $s_j{s}_{j+1}\cdots s_n$ carries $z_{n+1}$ to $z_j$ and has length $n+1-j$; for $j=n+1$ the empty word gives $z_{n+1}$. Conversely $d^{\mathrm{\prime }}(z_j):=n+1-j$ changes by at most one at every generator and vanishes at $z_{n+1}$, so $d^{\mathrm{\prime }}\le d$ by the graph-distance clause of [F6]. Hence $d=d^{\mathrm{\prime }}$ and ${\sum }_v{t}^{d(v)}={\sum }_{j=1}^{n+1}{t}^{n+1-j}=[n+1{]}_t$.

Type $B_n$ ($n\ge 2$), $s_0=s_1$, $T=\{s_2,\dots ,s_n\}$. Deleting $s_1$ leaves the type-$B_{n-1}$ chain; for $n=2$ the remaining one-generator Coxeter presentation is type $A_1$, also denoted $B_1$. Put $y_i(x)=x_i$ and $z_i=\sqrt{2}{y}_i$ on ${\mathbb{R}}^n$; then $f_0\circ {\varphi }^{-1}=z_1$, since it takes value $1$ on ${\widehat{\alpha }}_1$ and vanishes on every ${\widehat{\alpha }}_i$ for $i\ge 2$. By step 3.1 the model group is the signed permutation group, so $W\cdot f_0=\{\pm z_1,\dots ,\pm z_n\}$. The generators $s_i$ for $2\le i\le n-1$ swap $z_i,z_{i+1}$ together with their negatives, $s_1$ swaps $z_1,z_2$, and $s_n$ exchanges $z_n$ with $-z_n$; hence the Schreier graph is the path $z_1-z_2-\cdots -z_n-(-z_n)-\cdots -(-z_1)$. For $i=1$ the empty word carries $z_1$ to itself; for $2\le i\le n$, $s_{i-1}\cdots s_2{s}_1$ carries $z_1$ to $z_i$. For each $1\le i\le n$, $s_i{s}_{i+1}\cdots s_{n-1}{s}_n{s}_{n-1}\cdots s_1$ carries $z_1$ to $-z_i$. These words have lengths $i-1$ and $2n-i$. Thus $d(z_i)\le i-1$ and $d(-z_i)\le 2n-i$. The function $d^{\mathrm{\prime }}(z_i)=i-1$, $d^{\mathrm{\prime }}(-z_i)=2n-i$ vanishes at $z_1$ and changes by at most one along every edge, so $d^{\mathrm{\prime }}=d$ by [F6]. Hence ${\sum }_v{t}^{d(v)}={\sum }_{i=1}^n(t^{i-1}+t^{2n-i})=[2n{]}_t$.

Type $D_n$ ($n\ge 4$), $s_0=s_n$, $T=\{s_1,\dots ,s_{n-1}\}$. Deleting $s_n$ leaves the type-$D_{n-1}$ diagram; when $n=4$ the remaining three nodes form the path $s_1-s_3-s_2$, the type-$A_3$ presentation denoted $D_3$ in the low-rank convention. Put $y_i(x)=x_i$ and $z_i=\sqrt{2}{y}_i$ on ${\mathbb{R}}^n$; then $f_0\circ {\varphi }^{-1}=-z_n$, since $-\sqrt{2}{y}_n$ vanishes on ${\widehat{\alpha }}_i$ for $i\le n-2$, on ${\widehat{\alpha }}_{n-1}=({\epsilon }_{n-2}-{\epsilon }_{n-1})\mathrm{/}\sqrt{2}$ and on ${\widehat{\alpha }}_2=-({\epsilon }_1+{\epsilon }_2)\mathrm{/}\sqrt{2}$, while it takes value $1$ on ${\widehat{\alpha }}_n=({\epsilon }_{n-1}-{\epsilon }_n)\mathrm{/}\sqrt{2}$. By step 3.1 the model group is the even signed permutation group, so $W\cdot f_0=\{\pm z_1,\dots ,\pm z_n\}$. The generators $s_i$ for $i\mathrm{\ne }2$ swap adjacent coordinate functionals, while $s_2$ sends $z_1\mapsto -z_2$ and $z_2\mapsto -z_1$ and fixes $z_j$ for $j\ge 3$; thus the Schreier graph consists of the paths $z_1-\cdots -z_n$ and $(-z_1)-\cdots -(-z_n)$ joined by $z_1-(-z_2)$ and $(-z_1)-z_2$. The word $s_{j+1}{s}_{j+2}\cdots s_n$ carries $-z_n$ to $-z_j$ for $2\le j\le n$, so $d(-z_j)\le n-j$; the word $s_1{s}_3{s}_4\cdots s_n$ carries $-z_n$ to $-z_1$, giving $d(-z_1)\le n-1$. For $2\le j\le n$, $s_j{s}_{j-1}\cdots s_3{s}_2{s}_1{s}_3{s}_4\cdots s_n$ carries $-z_n$ to $z_j$ (the initial string is empty for $j=2$), and has length $n+j-2$. Also $s_2{s}_3\cdots s_n$ carries $-z_n$ to $z_1$, so $d(z_1)\le n-1$. The function $d^{\mathrm{\prime }}(-z_j)=n-j$ and $d^{\mathrm{\prime }}(z_j)=n+j-2$ vanishes at $-z_n$; each generator changes it by at most one along the displayed graph, so $d^{\mathrm{\prime }}=d$ by [F6]. Its values are $0,\dots ,n-2$ once each, $n-1$ twice, and $n,\dots ,2n-2$ once each. Therefore ${\sum }_v{t}^{d(v)}=(1+t+\cdots +t^{n-2})+2t^{n-1}+(t^n+\cdots +t^{2n-2})=[n{]}_t(1+t^{n-1})$.

Type $I_2(m)$, $s_0=s_1$, $T=\{s_2\}$. Here $T$ has one node and is type $A_1$. Let ${\eta }_1\in E^{\ast }$ be dual to ${\widehat{\alpha }}_1,{\widehat{\alpha }}_2$, so ${\eta }_1({\widehat{\alpha }}_1)=1$ and ${\eta }_1({\widehat{\alpha }}_2)=0$; then $f_0\circ {\varphi }^{-1}={\eta }_1$. By step 3.1 the group is dihedral of order $2m$, and the cosets $r^k{W}_T$ map bijectively to $v_k=r^k\cdot f_0$ by [F6]. Since $s_{2}r=r^{-1}{s}_2$ and $s_1=r{s}_2$, the left action is $s_2\cdot v_k=v_{-k}$ and $s_1\cdot v_k=v_{1-k}$. Omitting loops, these edges form the path $v_0-v_1-v_{-1}-v_2-v_{-2}-\cdots$ through all $m$ vertices. The vertex at position $2a$ is $(s_2{s}_1{)}^a\cdot f_0=v_{-a}$ and the vertex at position $2a+1$ is $s_1(s_2{s}_1{)}^a\cdot f_0=v_{a+1}$; these alternating words attain each path position. Thus [F6] gives distances $0,1,\dots ,m-1$ and ${\sum }_v{t}^{d(v)}=[m{]}_t$.

Parabolic recurrences and low-rank bases. By [F6], the orbit is in bijection with the cosets $w{W}_T$ and $d(v)$ is the length of the unique minimal representative of its coset. These are left cosets by [F14]. By [F16], each parabolic $W_T$ has the restricted Coxeter presentation and its intrinsic length equals the ambient length; deleting the selected node gives the smaller type named in each recurrence. The factorization in [F14] gives $P_W=P_{W^T}{P}_{W_T}$; the minimal representatives in $W^T$ therefore give $P_{W^T}={\sum }_v{t}^{d(v)}$, so each quotient polynomial in (2) yields the corresponding recurrence. The one-generator presentations for $A_1$ and $B_1$ coincide; the two-generator presentations for $A_2$ and $I_2(3)$ both have one edge labelled $3$, and those for $B_2$ and $I_2(4)$ both have one edge labelled $4$. The standard $D_3$ diagram is the three-node path $A_3$, so the generator bijection gives a length-preserving presentation isomorphism. Step 4.1 gives $P_{A_1}=[2{]}_t$ because $W_T=\{1\}$, and for $n\ge 2$ gives $P_{A_n}=P_{A_{n-1}}[n+1{]}_t$. For $n\ge 2$, $P_{B_n}=P_{B_{n-1}}[2n{]}_t$ with $P_{B_1}=P_{A_1}$. In type $D_2$, its two generators commute and are involutions. The presentation map $s_1\mapsto (1,0)$, $s_2\mapsto (0,1)$ into $\{0,1{\}}^2$ under componentwise addition modulo $2$, and the map $(a,b)\mapsto s_1^a{s}_2^b$ back are inverse homomorphisms: the first respects the Coxeter relators, and the second is a homomorphism because the generators commute and square to $1$. Hence its elements have lengths $0,1,1,2$, so $P_{D_2}=[2{]}_t^2$; this is the same two-generator presentation as $I_2(2)$ under the extended notation and the direct product $A_1\times A_1$. Also $P_{D_3}=P_{A_3}$ by the path-presentation isomorphism above, while for $n\ge 4$ step 4.3 gives $P_{D_n}=P_{D_{n-1}}[n{]}_t(1+t^{n-1})$. Step 4.4 gives $P_{I_2(m)}=P_{A_1}[m{]}_t$ for $m\ge 3$.

Telescoping the recurrences. The type-$A$ recurrence gives $P_{A_n}={\prod }_{k=2}^{n+1}[k{]}_t=[n+1{]}_t!$ and hence $P_{S_n}=P_{A_{n-1}}=[n{]}_t!$ for $n\ge 2$. The type-$B$ recurrence with $B_1=A_1$ gives $P_{B_n}={\prod }_{i=1}^n[2i{]}_t$ for $n\ge 1$. In type $D$, the identity $(1+t^{n-1})[n-1{]}_t=[2n-2{]}_t$ changes the recurrence for $n\ge 4$ into $P_{D_n}=[n{]}_t{\prod }_{i=1}^{n-1}[2i{]}_t$; this formula also holds for $D_3=A_3$ and for $D_2$ by step 5.1. Step 5.1 gives $P_{I_2(m)}=[2{]}_t[m{]}_t$.

Evaluation at one and group orders. By [F11] and [F13], $P_W(1)={\sum }_{w\in W}1=\mathrm{\mid }W\mathrm{\mid }$ (with cardinality as in [F10]); the polynomial evaluation is valid because the groups identified in step 3.1 are finite. The formulas give $P_{A_n}(1)=(n+1)!$, $P_{B_n}(1)=2^{n}n!$, $P_{D_n}(1)=2^{n-1}n!$ for $n\ge 2$, and $P_{I_2(m)}(1)=2m$. These are the group orders: [F12] gives $\mathrm{\mid }{S}_{n+1}\mathrm{\mid }=(n+1)!$ and $\mathrm{\mid }{S}_n\mathrm{\mid }=n!$; each permutation has $2^n$ sign assignments in type $B_n$ and $2^{n-1}$ even sign assignments in type $D_n$ (the first $n-1$ signs determine the last); the direct $D_2$ presentation in step 5.1 has four elements; and the dihedral group has order $2m$ by step 3.1. The low-rank coincidences are respected: $P_{A_1}=P_{B_1}$, $P_{A_2}=[2{]}_t[3{]}_t=P_{I_2(3)}$, $P_{B_2}=[2{]}_t[4{]}_t=P_{I_2(4)}$, $P_{D_2}=[2{]}_t^2$, and $P_{D_3}=[2{]}_t[3{]}_t[4{]}_t=P_{A_3}$. No invariant degrees and no exceptional diagrams are used.