Finite Hecke specialization and its regular-trace determinant

Let $k$ be a field of characteristic zero, let $S$ be a finite Coxeter generator set, and assume that the presented Coxeter group $W$ is finite. Let $\mathcal{C}$ be the set of odd-edge connected classes of $S$ from Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy: one class joins generators connected by a chain of finite odd braid labels. Introduce one invertible parameter $v_C$ for each $C\in \mathcal{C}$ and put $R_k:=k[v_C^{\pm 1}:C\in \mathcal{C}]$. Let $H_{R_k}(W)$ be the scalar extension of the universal Hecke algebra to $R_k$; its relations are $(T_s-v_{[s]})(T_s+v_{[s]}^{-1})=0$ and the Coxeter braid relations. By The standard basis of the generic Hecke algebra and base change, $H_{R_k}(W)$ is a free $R_k$-module with basis $(T_w{)}_{w\in W}$.

For $z\in H_{R_k}(W)$, let $L_z(a):=za$. Define the trace of an $R_k$-linear endomorphism of $H_{R_k}(W)$ to be the sum of the diagonal entries of its matrix in this specified basis. Thus the regular-trace pairing over $R_k$ is $B_W(x,y):={\mathrm{tr}}_{R_k}(L_{xy})$, with Gram matrix $G_W=({\mathrm{tr}}_{R_k}(L_{T_x{T}_y}){)}_{x,y\in W}$ and determinant $D_W\in R_k$.

For a homomorphism of commutative rings $\varphi :R_k\to K$ to a field $K$, define the specialized algebra $H_{\varphi }:=K{\otimes }_{R_k}{H}_{R_k}(W)$ and use the same symbol $T_w$ for the specialized basis element $1\otimes T_w$. For $z\in H_{\varphi }$, let $L_z:H_{\varphi }\to H_{\varphi }$ be left multiplication, $L_z(x)=zx$. The regular-trace pairing on $H_{\varphi }$ is

$B_{\varphi }(x,y):={\mathrm{Tr}}_K(L_{xy})\mathrm{.}$

Its Gram matrix in the standard basis is $G_{\varphi }=({\mathrm{Tr}}_K(L_{T_x{T}_y}){)}_{x,y\in W}$, and its regular-trace determinant is $D_{\varphi }:=\det (G_{\varphi })$.

The matrix of $L_z$ in the standard basis has entries in $R_k$, so applying $\varphi$ to that matrix gives the left-multiplication matrix on $H_{\varphi }$. Since trace is the sum of diagonal entries, $D_{\varphi }=\varphi (D_W)$. If another $K$-basis has coordinate matrix $P\in {\mathrm{GL}}_{\mathrm{\mid }W\mathrm{\mid }}(K)$ relative to $(T_w)$, its Gram matrix is $P^{\mathsf{T}}{G}_{\varphi }P$, so its determinant is $\det (P{)}^2{D}_{\varphi }$. In particular, nonvanishing of the regular-trace determinant is independent of the chosen field basis. The regular-trace pairing is distinct from the canonical symmetrizing trace pairing used on the preceding trace page.