B-tilde versus C-tilde: the n=2 coincidence and the duality behind the difference

At $n=2$, [F1] specifies the common $(4,4)$ path; [F3] realizes it from each dual root-length assignment. The finite systems coincide by [F9], and their two scalings are exchanged by [F4]. For $n\ge 3$, the $B$ graph has a degree-three vertex and one $4$-edge, whereas the $C$ graph has degrees at most two and two $4$-edges. A graph isomorphism would biject each vertex's neighbors and preserve degree, which is impossible for these degree sequences.

By [F12], if $c_4:=\cos (\pi \mathrm{/}4)$ and $c_3:=\cos (\pi \mathrm{/}3)$, then $c_4=\sqrt{2}\mathrm{/}2$ and $c_3=1\mathrm{/}2$: positivity follows from $0<\pi \mathrm{/}4,\pi \mathrm{/}3<\pi \mathrm{/}2$ and strict decrease to $\cos (\pi \mathrm{/}2)=0$; for $c_4$ the double-angle identity gives $2c_4^2=1$, and for $c_3$ it gives $2c_3^2-1=\cos (2\pi \mathrm{/}3)=-c_3$, so $(2c_3-1)(c_3+1)=0$ and $c_3=1\mathrm{/}2$. Put $c=c_4$. For ${\tilde{C}}_n$, take endpoint coordinates $1$ and all $n-1$ interior coordinates $\sqrt{2}$. Each endpoint equation is $1-c\sqrt{2}=0$; at $n=2$ the central equation is $\sqrt{2}-2c=0$; for $n\ge 3$, a vertex next to an end has equation $\sqrt{2}-c-\sqrt{2}\mathrm{/}2=0$, and any other interior vertex has equation $\sqrt{2}-(\sqrt{2}+\sqrt{2})\mathrm{/}2=0$. For ${\tilde{B}}_n$, $n\ge 3$, use $x_0=x_1=1$, $x_2=\cdots =x_{n-1}=2$, $x_n=\sqrt{2}$. At each branch leaf the equation is $1-x_2\mathrm{/}2=0$. At $v_2$, the equation is $2-(1+1)\mathrm{/}2-c\sqrt{2}=0$ when $n=3$, and $2-(1+1+2)\mathrm{/}2=0$ when $n\ge 4$. Every intervening chain vertex (if any) has equation $2-(2+2)\mathrm{/}2=0$; the vertex $v_{n-1}$ for $n\ge 4$ has equation $2-1-c\sqrt{2}=0$, and the final vertex has equation $\sqrt{2}-2c=0$. Thus the displayed vectors are positive kernel vectors; [F2] independently gives the semidefinite corank-one assertion. At $n=2$ use the $C$ vector $(1,\sqrt{2},1)$ for both names.

In the coordinate models [F6], for $k\ge 2$ the finite Coxeter groups of type $B_k$ or $C_k$ are identified by [F5] with the Weyl groups on the displayed root sets. Their simple reflections interchange neighboring coordinates or change the last coordinate's sign; conjugating by permutations permits each sign change. Every root reflection is a signed permutation, and these generators give all signed permutations, so the group has order $2^{k}k!$. For $D_k$, $k\ge 4$, [F5,F6] identify the finite Coxeter group with the Weyl group of roots $\pm e_i\pm e_j$. Its adjacent swaps and the reflection in $e_{k-1}+e_k$ generate exactly the permutations with an even number of sign changes: composing that reflection with the swap gives a double sign change, and conjugates give all pairs. Every root reflection has even sign parity, so the group has order $2^{k-1}k!$. For $k=3$, the remaining all-$3$ path is $A_3=D_3$ by [F9]; its coordinate roots $e_i-e_j$ on the sum-zero subspace of ${\mathbb{R}}^4$ give all transpositions of four coordinates, so its group is $S_4$ of order $24$. A one-vertex $A_1$ factor has the sign reflection group of order $2$, and the rank-zero group is trivial.

Every finite subgroup $H$ of either affine group fixes a point: average the finite orbit $Hx$ for any $x$; affine linearity makes its barycentre fixed by $H$. By [F8]'s strict closed fundamental domain, conjugate this point into $\bar{A}$. For $p\in \bar{A}$, let $T(p)$ be the types of facets containing $p$. The affine relation in [F8] gives $T(p)⊊S$. The intersection formula in [F8] and support criterion [F10] show $\mathrm{Stab}(p)=W_{T(p)}$: if $wp=p$, then $p\in w\bar{A}\cap \bar{A}$, so every type in $S(w)$ is in $T(p)$, hence $w\in W_{T(p)}$; conversely every generator indexed by $T(p)$ fixes $p$. This stabilizer is finite by [F8]'s proper-parabolic clause. The face containing $p$ has a vertex $v$; all its containing facet reflections fix $v$, so $H\le W_{T(p)}\le \mathrm{Stab}(v)$. Applying the same stabilizer identity to $v$ shows $\mathrm{Stab}(v)=W_{T(v)}$, a finite proper parabolic. At a vertex $v$, the $n$ incident facets have a unique intersection, so their normals span the ambient direction space and their reflections have common fixed-point set exactly $\{v\}$. If a finite group contained $\mathrm{Stab}(v)$, its barycentre would be fixed by this stabilizer, hence would equal $v$; the group would then be contained in $\mathrm{Stab}(v)$. Thus every vertex stabilizer is maximal finite, and every maximal finite subgroup is conjugate to one.

Set $N=2^{n}n!$ and let $n\ge 3$. Deleting the endpoint $v_n$ of the $4$-edge from ${\tilde{B}}_n$ leaves the finite all-$3$ $D_n$ graph (at $n=3$, the path $A_3=D_3$). Its parabolic is the stabilizer of the opposite vertex, hence maximal finite by Step 1.4, and has order $N\mathrm{/}2$ by Step 1.3. Deleting vertex $v_i$, $0\le i\le n$, from ${\tilde{C}}_n$ leaves two finite terminal-$4$ paths of ranks $i$ and $n-i$, interpreted as $B_1=A_1$ or rank-zero trivial blocks at the ends. Its vertex stabilizer has order $2^{n}i!(n-i)!$ by the restricted presentation [F10] and Step 1.3; the two blocks commute and their presented group is the direct product. At $i=0,n$ this is $N$. For $1\le i\le n-1$, $\left(\genfrac{}{}{0px}{}{n}{i}\right)\ge n$: the ratio $\left(\genfrac{}{}{0px}{}{n}{i+1}\right)\mathrm{/}\left(\genfrac{}{}{0px}{}{n}{i}\right)=(n-i)\mathrm{/}(i+1)$ shows the binomial coefficients increase to the middle and then decrease symmetrically, so their minimum on these indices is $\left(\genfrac{}{}{0px}{}{n}{1}\right)=\left(\genfrac{}{}{0px}{}{n}{n-1}\right)=n$. Hence the order is at most $N\mathrm{/}n<N\mathrm{/}2$. By Step 1.4 these are all maximal finite subgroup orders in $W_a(C_n)$. The two groups therefore cannot be abstractly isomorphic, since an isomorphism preserves finiteness, maximality and subgroup order.

The different coroot lattices can also be seen directly. For $B_n$ with short roots $\pm e_i$, its coroots are $\pm 2e_i$ and $\pm e_i\pm e_j$, whose integer span is $\{z\in {\mathbb{Z}}^n:{\sum }_i{z}_i\text{ is even}\}$: the generators have even sum, and subtracting $z_i(e_i-e_n)$ for $i<n$ leaves an even multiple of $e_n$. For $C_n$ with long roots $\pm 2e_i$, their coroots include all $\pm e_i$, so the lattice is ${\mathbb{Z}}^n$. Both finite Weyl groups are the same signed permutation group by [F6], and [F3] identifies their affine facet diagrams with the standard ${\tilde{B}}_n,{\tilde{C}}_n$ diagrams; Step 1.1 shows these differ in degree data. Step 2.1 proves the stronger abstract distinction for $n\ge 3$. Their chambers have dimension $n$; when $n=2$ the common affine diagram yields isomorphic Coxeter groups by [F3,F11] despite the dual coordinate normalizations. The decomposition [F7] identifies the translation lattices in these semidirect products with the two coroot lattices just computed. Finally, label $4$ is the actual product order by [F3], not a bond multiplicity, and the extended weight-lattice group is a distinct comparison object by [F11]. All comparisons use finite coordinates and averaging, without Choice.