The generic type-A Hecke algebra

Let $n\ge 0$ be an integer, with $S_0=\{1\}$ as in Partitions, English diagrams, and conjugation. Let $A:=\mathbb{Z}[v^{\pm 1}]$ be the ring of Laurent polynomials in one indeterminate $v$ over $\mathbb{Z}$. For $n\ge 2$ the generic type-A Hecke algebra $H_v(n)$ is the associative unital $A$-algebra presented by generators $T_1,\dots ,T_{n-1}$ and the relations $T_i^2=(v-1)T_i+v(1\le i<n),T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1}(1\le i\le n-2),T_i{T}_j=T_j{T}_i(1\le i,j<n,\mathrm{\mid }i-j\mathrm{\mid }>1),$ that is, the quotient of the free unital associative $A$-algebra on $T_1,\dots ,T_{n-1}$ by the two-sided ideal generated by these relators. The first relation is written in the RG-13 quadratic normalization $(T_i-v)(T_i+1)=0$, equivalent to the displayed form over $A$. For $n\le 1$ there are no generators and we set $H_v(n):=A$.

Standard basis elements. Let $S_n=\mathrm{Sym}(\{1,\dots ,n\})$ with simple transpositions $s_i=(ii+1)$ (The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition), and let $w\in S_n$ have inversion length $\mathrm{\ell }(w)=\mathrm{\ell }$ (Permutation Weyl group and inversion length). Choose a reduced expression $w=s_{i_1}\cdots s_{i_{\mathrm{\ell }}}$, that is, a word of the minimal length $\mathrm{\ell }(w)$ representing $w$, and put $T_w:=T_{i_1}\cdots T_{i_{\mathrm{\ell }}}\in H_v(n),$ the empty product when $w$ is the identity of $S_n$, so that $T_w=1$ for that element. That $T_w$ is independent of the chosen reduced expression, and that the elements $T_w$ form an $A$-basis of $H_v(n)$, is proved in the standard-basis theorem below (The standard basis of the generic type-A Hecke algebra ↗); the notation distinguishes the generator $T_i$ with $1\le i\le n-1$ from $T_w$ for the group element $w$.

Specializations. Let $R$ be a commutative $A$-algebra with structure map $\mathbb{Z}[v^{\pm 1}]\to R$, $v\mapsto v_0$, and suppose $v_0\in R^{\times }$. The specialization of $H_v(n)$ at $v_0$ is the $R$-algebra $R{\otimes }_A{H}_v(n)$; it is presented over $R$ by the images of $T_1,\dots ,T_{n-1}$ subject to the same relations with $v$ replaced by $v_0$, and we write $H_{v_0}(n)$ for it. Two specializations are used on this page: $v\mapsto 1$, where the quadratic relation becomes $T_i^2=1$ together with the braid and commutation relations, and $v\mapsto q$, where $q$ is the prime power defining ${\mathrm{GL}}_n({\mathbb{F}}_q)$; the latter specializes the generic algebra to the Hecke algebra attached to the finite general linear group. The condition $v_0\in R^{\times }$ is part of the definition of a specialization and, for a nonzero coefficient ring $R$, excludes $v_0=0$. The zero ring is allowed: its unique element is a unit and its specialization is the zero algebra.

Dictionary to the Soergel normalization. The type-A Soergel item def-type-a-hecke-algebra-in-soergel-normalization, homed later in the reading order, uses $A_S=\mathbb{Z}[v_S^{\pm 1}]$ and $q=v_S^{-2}$. Its algebra is the base change $A_S{\otimes }_{A,v\mapsto v_S^{-2}}{H}_v(n)$: substitution makes its quadratic, braid and commutation relations identical to the ones here. This coefficient map is not a Laurent-ring isomorphism; its image is $\mathbb{Z}[v_S^{\pm 2}]$, and $A_S$ is free of rank two over that image with basis $1,v_S$. Numerical specializations must therefore satisfy $v_{\text{here}}=v_S^{-2}$. This comparison is orientation only and is not a premise of the definition or its standard-basis proof.