The normalized type-A Hecke algebra and its bar involution

Let $n\ge 1$ and $A:=\mathbb{Z}[v^{\pm 1}]$. For $n\ge 2$, define the normalized type-A Hecke algebra $H_v(n)$ to be the associative unital $A$-algebra presented by generators $H_{s_1},\dots ,H_{s_{n-1}}$ with relations $H_{s_i}^2=1+(v^{-1}-v)H_{s_i},H_{s_i}{H}_{s_{i+1}}{H}_{s_i}=H_{s_{i+1}}{H}_{s_i}{H}_{s_{i+1}},H_{s_i}{H}_{s_j}=H_{s_j}{H}_{s_i}(\mathrm{\mid }i-j\mathrm{\mid }>1)\mathrm{.}$ For $n=1$, set $H_v(1):=A$.

Here $S_n=\mathrm{Sym}(\{1,\dots ,n\})$, $s_i=(ii+1)$, and $\mathrm{\ell }$ is inversion length as in The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition and Permutation Weyl group and inversion length. For $w\in S_n$, let $T_w$ be the standard basis element of the generic Hecke algebra of The generic type-A Hecke algebra, formed from any reduced expression of $w$, and after the coefficient specialization $q\mapsto v^{-2}$ put $H_w:=v^{\mathrm{\ell }(w)}{T}_w$. Equivalently, $H_w=H_{s_{i_1}}\cdots H_{s_{i_k}}$ for a reduced expression $w=s_{i_1}\cdots s_{i_k}$. The standard-basis theorem The standard basis of the generic type-A Hecke algebra and rescaling by units show that $\{H_w:w\in S_n\}$ is an $A$-basis of $H_v(n)$.

Right multiplication by a generator is $H_w{H}_{s_i}=H_{w{s}_i}\text{if }\mathrm{\ell }(w{s}_i)=\mathrm{\ell }(w)+1,H_w{H}_{s_i}=H_{w{s}_i}+(v^{-1}-v)H_w\text{if }\mathrm{\ell }(w{s}_i)=\mathrm{\ell }(w)-1.$

Dictionary with the generic normalization. If the generic parameter in The generic type-A Hecke algebra is denoted by $q$, its relation is $T_i^2=(q-1)T_i+q$. The coefficient map $q\mapsto v^{-2}$ is the stated specialization, and $H_{s_i}=v{T}_i$. This gives $H_{s_i}^2=1+(v^{-1}-v)H_{s_i}$; the braid and commutation relations are unchanged. The two multiplication cases above are the standard-basis rule after the same rescaling.

Bar assignment. Let $F$ be the free associative $\mathbb{Z}[v^{\pm 1}]$-algebra on the generator symbols. Define the semilinear algebra map ${\iota }_0:F\to F$ by ${\iota }_0(v)=v^{-1}$ and ${\iota }_0(H_{s_i})=H_{s_i}-(v^{-1}-v)$. In the quotient $H_v(n)$, the quadratic relation makes this latter element the two-sided inverse $H_{s_i}^{-1}$. The next lemma proves that ${\iota }_0$ preserves the defining ideal and descends to a well-defined involution $\iota$ on $H_v(n)$.

All items on this page use this normalization. It agrees with Elias–Williamson §3.2 under $H_x=v^{\mathrm{\ell }(x)}{T}_x$ and $q=v^{-2}$.