The type-A Hecke algebra in Soergel normalization

Coefficients. Let $A:=\mathbb{Z}[v,v^{-1}]$ be the ring of Laurent polynomials in one indeterminate with integer coefficients, a commutative ring containing $\mathbb{Z}$ as the constant Laurent polynomials, and put $q:=v^{-2}\in A$. Both $v$ and $q$ are units of $A$, indeed $A^{\times }=\{\pm v^m:m\in \mathbb{Z}\}$, while $q-1=v^{-2}-1$ is a non-zero, non-invertible element of the domain $A$ and is never inverted below; the only element of $A$ inverted anywhere on this page is $v$ itself, through $q=v^{-2}$.

The algebra. For $n\ge 2$ let $H_n=H_{S_n}$ be the unital associative $A$-algebra presented by generators $T_1,\dots ,T_{n-1}$ and relations $T_i^2=(q-1)T_i+q,T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1},T_i{T}_j=T_j{T}_i(\mathrm{\mid }i-j\mathrm{\mid }>1)\mathrm{.}$ The braid and commutation relations are those of the Coxeter presentation of $S_n$ (Type-A reduced words and the Coxeter presentation), while its involution relations are replaced by the displayed Hecke quadratic relations; $H_n$ is the quotient of the free associative $A$-algebra on $T_1,\dots ,T_{n-1}$ by these relations. It is not the quotient of the group algebra $A[S_n]$ by the ideal generated by the quadratic relations. Indeed, in $A[S_n]$ the involutions satisfy $T_i^2=1$, so the ideal generated there by the relations $T_i^2=(q-1)T_i+q$ is generated by $(T_i-q)(T_i+1)=(T_i^2-1)+(1-q)(T_i+1)=(1-q)(T_i+1)\mathrm{.}$ Already for $n=2$, the generic algebra $H_2=A[T_1]\mathrm{/}((T_1-q)(T_1+1))$ is free over $A$ on $1,T_1$, since its defining polynomial is monic of degree two. Thus $(1-q)(T_1+1)\mathrm{\ne }0$ in $H_2$, whereas that element vanishes in the indicated quotient of $A[S_2]$; the generator-preserving presentation through the group algebra is impossible. Since $T_i^2=(q-1)T_i+q$ is equivalent over $A$ to $(T_i-q)(T_i+1)=0$, we may use either form.

Normalized generators. Put $H_i:=v(T_i+1)=v{T}_i+v\in H_n\mathrm{.}$ Then $H_i$ is a unit-free normalization: it is triangular, with leading term $v{T}_i$. Expanding the quadratic relation gives $T_i^2=(q-1)T_i+q$, so $H_i^2=v^2(T_i+1{)}^2=v^2(T_i^2+2T_i+1)=v^2((q-1)T_i+q+2T_i+1)=v^2((q+1)T_i+(q+1))=(v^2(q+1))(T_i+1)\mathrm{.}$ Since $v^{2}q=1$, the scalar is $v^{2}q+v^2=1+v^2$, so $H_i^2=(1+v^2)(T_i+1)\cdot v\cdot v^{-1}=v^{-1}(1+v^2)H_i=(v^{-1}+v)H_i,$ that is $H_i^2=(v+v^{-1})H_i$. Conversely $T_i=v^{-1}{H}_i-1$, so the two generating sets determine each other over $A$. The element $H_i$ is the bar-invariant generator of Elias–Williamson, §2.1, where it is written with the same scalar $v+v^{-1}$.

Small $n$ and scope. For $n\le 1$ there are no generators and we set $H_n:=A$; the statements about $T_i,H_i$ are vacuous. This presentation is the internal normalization used by this page: it records no specialization to a finite field, no Tits deformation theorem and no positivity or Kazhdan–Lusztig statement. The standard basis $\{T_w\}$ indexed by reduced words, its multiplication rule by simple generators, and the comparison with products of the $H_i$ are supplied by The standard basis of the type-A Hecke algebra and its multiplication rule, which also makes the notation $T_w$ for $w\in S_n$ well defined.