Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary

Let $(S,m)$ be a finite Coxeter matrix, $W$ the presented Coxeter group with length $\mathrm{\ell }$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), and let $R$, $v_s$, $H$ and its generators $T_s$ be the generic Coxeter Hecke algebra of Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy (so $R=\mathbb{Z}[v_1^{\pm 1},\dots ,v_c^{\pm 1}]$ and $v_s=v_{[s]}$), with the standard basis elements $T_w$ of Reduced-word independence of T_w and the length-multiplication rules and The standard basis of the generic Hecke algebra and base change. Let $A^{+}$, $\gamma$, ${\sigma }_s$ be as in Artin monoid and Artin group presentations, and the canonical monoid-to-group map and $b_w$ as in The reduced positive section b_w, its length additivity, and the degree homomorphism. Finally let $V={\mathbb{R}}^S$, $B$ and $\rho :W\to \mathrm{G}\mathrm{L}(V)$ be the Coxeter form and canonical reflection representation of The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone, with root system $\mathrm{\Phi }$ (Descent of the reflection representation, unit root norms, and conjugation of reflections).

(1) Indexing and coefficient compatibility of the Artin and Hecke interfaces. Because the generators $T_s$ satisfy the braid relations of $H$, the universal property of the Artin monoid (Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares (1)) gives a unique monoid homomorphism

$\mathrm{\Theta }:A^{+}⟶(H,\cdot ,1),\mathrm{\Theta }({\sigma }_s)=T_s\mathrm{.}$

It satisfies $\mathrm{\Theta }(b_w)=T_w$ for every $w\in W$. Moreover the indexing is coefficient-compatible: $(T_w{)}_{w\in W}$ is an $R$-basis of $H$, and for every commutative ring $R^{\mathrm{\prime }}$ and ring homomorphism $R\to R^{\mathrm{\prime }}$ the specialised elements $(1\otimes T_w)$ form an $R^{\mathrm{\prime }}$-basis of $R^{\mathrm{\prime }}{\otimes }_{R}H$ (The standard basis of the generic Hecke algebra and base change (2),(4)). Thus the positive section $b$ and the Hecke standard basis use the same reduced-word indexing, and $\mathrm{\Theta }$ carries the monoid multiplication of $A^{+}$ to the multiplication of $H$ in that basis: $\mathrm{\Theta }(b_u{b}_v)=T_u{T}_v$ for all $u,v\in W$.

(2) Normalization conversions. In $H$: (i) $T_s^2=(v_s-v_s^{-1})T_s+1$ (the normalized or $T$-normalization); (ii) the multiplicative generators $S_s:=v_s{T}_s$ satisfy $(S_s-Q_s)(S_s+1)=0$, i.e. $S_s^2=(Q_s-1)S_s+Q_s$, where $Q_s:=v_s^2\in R$; for a single parameter $v_s=v$ and $Q=v^2$ this is the "$S=qT$, $Q=q^2$" convention with $q=v$; (iii) the opposite-sign generators $H_s:=-T_s$ satisfy $H_s^2=(v_s^{-1}-v_s)H_s+1$; (iv) the Soergel-calculus generators $T_s^{\mathrm{E}\mathrm{W}}:=-v_s^{-1}{T}_s$ satisfy $(T_s^{\mathrm{E}\mathrm{W}}+1)(T_s^{\mathrm{E}\mathrm{W}}-v_s^{-2})=0$, i.e. $(T_s^{\mathrm{E}\mathrm{W}}{)}^2=(v_s^{-2}-1)T_s^{\mathrm{E}\mathrm{W}}+v_s^{-2}$.

The assignments $T_s\leftrightarrow S_s=v_s{T}_s$, $T_s\leftrightarrow H_s=-T_s$ and $T_s\leftrightarrow T_s^{\mathrm{E}\mathrm{W}}=-v_s^{-1}{T}_s$, together with the identity on $T_s$, extend to $R$-algebra isomorphisms between the four corresponding presentations, with inverse displayed: $T_s=v_s^{-1}{S}_s=-H_s=-v_s{T}_s^{\mathrm{E}\mathrm{W}}$. In every case the braid relations are preserved: when $m$ is even the two alternating words contain each of $s,t$ exactly $m\mathrm{/}2$ times, and when $m$ is odd the generators $s,t$ are joined by an odd-labelled edge, so their parameters satisfy $v_s=v_t$ (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy); in both cases the product of the scaling constants attached to the letters is the same on the two sides (with a single parameter simply $v_s^m$, $(-1{)}^m$ or $(-v_s^{-1}{)}^m$). For the single-parameter specialization, put $H_w:=(-1{)}^{\mathrm{\ell }(w)}{T}_w$ and $T_w^{\mathrm{K}\mathrm{L}}:=v^{-\mathrm{\ell }(w)}{H}_w$, where each basis element is the product along a reduced expression. Thus $H_s=-T_s$ as in (iii), $T_s^{\mathrm{K}\mathrm{L}}=T_s^{\mathrm{E}\mathrm{W}}$ as in (iv), and $H_w=v^{\mathrm{\ell }(w)}{T}_w^{\mathrm{K}\mathrm{L}}$. Here $T^{\mathrm{K}\mathrm{L}}$ denotes the original Kazhdan–Lusztig normalization with parameter $q=v^{-2}$; it is distinct from the normalized $T$-basis in (i). For any finite family of polynomials $P_{y,x}(q)$ and an element written in that normalization as

$C_x=v^{\mathrm{\ell }(x)}{\sum }_y{P}_{y,x}(v^{-2})T_y^{\mathrm{K}\mathrm{L}},$

its coefficients in $C_x={\sum }_y{h}_{y,x}{H}_y$ are exactly $h_{y,x}=v^{\mathrm{\ell }(x)-\mathrm{\ell }(y)}{P}_{y,x}(v^{-2})$. This proves the coefficient conversion used in Remark 3.2 of the Hodge-theory source whenever such an expansion is supplied, without constructing a canonical basis or asserting that the supplied polynomials are Kazhdan–Lusztig polynomials.

(3) Root-length matching for a supplied root system. Let $\beta$ be a symmetric bilinear form on a real vector space $E$ and let ${\alpha }_s\in E$ ($s\in S$) be vectors with $\mathrm{\mid }{\alpha }_s\mathrm{\mid }:=\beta ({\alpha }_s,{\alpha }_s{)}^{1\mathrm{/}2}>0$ such that

$\frac{\beta ({\alpha }_s,{\alpha }_t)}{\mathrm{\mid }{\alpha }_s\mathrm{\mid }\mathrm{\mid }{\alpha }_t\mathrm{\mid }}=-c(s,t)(s\mathrm{\ne }t),$

where $c(s,t):=\cos (\pi \mathrm{/}m(s,t))$ for finite $m(s,t)$ and $c(s,t):=1$ when $m(s,t)=\mathrm{\infty }$ (the convention of the Coxeter form The real Coxeter form, its radical, reflections, and form-preserving maps, so that the right-hand side is the number $B(e_s,e_t)$), that $\{{\alpha }_s:s\in S\}$ is linearly independent, and that the assignment $v\mapsto v-\frac{2\beta (v,{\alpha }_s)}{\beta ({\alpha }_s,{\alpha }_s)}{\alpha }_s$ defines a representation of $W$ on $E$. Then the linear map

$\phi :V⟶E,\phi (e_s):=\frac{{\alpha }_s}{\mathrm{\mid }{\alpha }_s\mathrm{\mid }},$

is an isomorphism of $V$ onto $\mathrm{span}\{{\alpha }_s:s\in S\}$, it satisfies $\beta (\phi u,\phi w)=B(u,w)$ for all $u,w\in V$, and it intertwines the canonical representation with the reflection representation generated by the ${\alpha }_s$: $\phi (\rho (s)v)=r_{{\alpha }_s}(\phi (v))$ for all $s\in S$, $v\in V$. Consequently the labelled Coxeter diagram fixes $B$ and the normalized products $\beta ({\alpha }_s,{\alpha }_t)\mathrm{/}(\mathrm{\mid }{\alpha }_s\mathrm{\mid }\mathrm{\mid }{\alpha }_t\mathrm{\mid })$, but neither the root lengths $\mathrm{\mid }{\alpha }_s\mathrm{\mid }$ nor the individual numbers $\beta ({\alpha }_s,{\alpha }_t)$; a supplied root-system treatment with its own root-length normalization, coroot system and lattice data matches this canonical form exactly under the displayed normalization, and such a match is a hypothesis on the supplier, not a consequence of the Coxeter matrix. No root system, coroot system, Cartan matrix or lattice is constructed or identified here.

(4) The reflection-faithfulness boundary for Soergel applications. (a) For a field $k$, a realization of $(W,S)$ in the sense of Elias–Williamson is a free finite-rank $k$-module $\mathfrak{h}$ with subsets $\{{\alpha }_s^{\vee }\}\subset \mathfrak{h}$, $\{{\alpha }_s\}\subset {\mathfrak{h}}^{\ast }$ satisfying $⟨{\alpha }_s^{\vee },{\alpha }_s⟩=2$, the reflection assignment $v\mapsto v-⟨v,{\alpha }_s⟩{\alpha }_s^{\vee }$ defining a representation of $W$, and the following condition on each distinct pair with finite $m=m(s,t)$. Put $x:=⟨{\alpha }_s^{\vee },{\alpha }_t⟩$ and $y:=⟨{\alpha }_t^{\vee },{\alpha }_s⟩$, and define $p_0=q_0=0$, $p_1=q_1=1$, $p_{j+1}=x{q}_j-p_{j-1}$ and $q_{j+1}=y{p}_j-q_{j-1}$ for $j\ge 1$. Require $p_m=q_m=0$ (the two-colored quantum-number condition (3.3), with the recursion of Definition 3.5 of the source). An infinite label imposes no such condition. Such a realization is faithful when the action of $W$ on $\mathfrak{h}$ is faithful, and reflection faithful when it is faithful and the assignment sending each reflection of $W$ (each conjugate of a simple reflection) to its full fixed space $\mathrm{Fix}(w):=\{v:w(v)=v\}$ is a bijection onto $\{\mathrm{Fix}(w):w\in W,\mathrm{codim}\mathrm{Fix}(w)=1\}$ (the source's Definition 3.8). In the canonical case $k=\mathbb{R}$, $\mathfrak{h}={⨁}_s\mathbb{R}{\alpha }_s^{\vee }$ and $⟨{\alpha }_t^{\vee },{\alpha }_s⟩=-2\cos (\pi \mathrm{/}m(s,t))$, with the source's convention $\pi \mathrm{/}\mathrm{\infty }:=0$ under which the value for $m(s,t)=\mathrm{\infty }$ is $-2$. (b) Every element of $W$ fixes the radical $\mathrm{rad}(B)=\{v\in V:B(v,e_s)=0\mathrm{\forall }s\in S\}$ pointwise: $\rho$ preserves $B$ (Descent of the reflection representation, unit root norms, and conjugation of reflections (2)) and $r_a(v)=v$ whenever $B(v,a)=0$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (2)), so each generator $r_{e_s}$ and hence $\rho (W)$ fixes $\mathrm{rad}(B)$ pointwise. (c) Suppose $\dim \mathrm{rad}(B)=\dim V-1\ge 1$. Then $\mathrm{rad}(B)$ is a hyperplane, and for every root $\alpha \in \mathrm{\Phi }$ the reflection $r_{\alpha }$ has fixed hyperplane $\ker B(-,\alpha )$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (2), Descent of the reflection representation, unit root norms, and conjugation of reflections (4)), which contains $\mathrm{rad}(B)$ and therefore equals it; moreover the simple reflections $\rho (s)\mathrm{\ne }\rho (t)$ ($s\mathrm{\ne }t$) are distinct reflections with the same fixed hyperplane. Hence the correspondence "reflection $\mapsto$ its fixed hyperplane" fails to be injective, and the canonical realization is not reflection faithful in the sense of Elias–Williamson even though $\rho$ is faithful (The root-length criterion and faithfulness of the canonical reflection representation (3)). (d) Therefore a faithful canonical representation may not be substituted for a reflection-faithful realization in an argument that assumes reflection faithfulness. The source records Soergel's construction of a reflection-faithful representation for every Coxeter group over $\mathbb{R}$ (Example 3.2(2) and the discussion after Definition 3.8). Reflection faithfulness is a sufficient hypothesis for the classical Soergel theory over an infinite field of characteristic different from $2$, but is not necessary for every Soergel application: the same discussion records Libedinsky's extension to the geometric realization and defines a Soergel realization as a faithful realization to which Soergel's techniques apply. That property is not asserted to be equivalent to reflection faithfulness. A concrete rank-two instance is computed on the companion page. (e) No Kazhdan–Lusztig canonical basis, bar-invariant basis, cell theory, Soergel bimodule or positivity statement is constructed here; those use the standard basis and bar involution of The standard basis of the generic Hecke algebra and base change and The reversal anti-involution, generator invertibility, the bar operator and the multiplicative normalization together with the conversions of (2), and remain in their designated proof homes.