Affine type-A presentations and charged cyclotomic quotients

Let $A$ be a commutative ring and let $z\in A$ be a unit. Fix an integer $n\ge 0$.

The nondegenerate affine presentation. The algebra $H_n^{\mathrm{a}\mathrm{f}\mathrm{f}}(A;z)$ is the $A$-algebra presented by commuting units $X_1,\dots ,X_n$ and generators $T_1,\dots ,T_{n-1}$ (The generic type-A Hecke algebra, The quantum characteristic of an invertible Hecke parameter) subject to $T_r^2=(z-1)T_r+z,T_r{T}_{r+1}{T}_r=T_{r+1}{T}_r{T}_{r+1},T_r{T}_s=T_s{T}_r(\mathrm{\mid }r-s\mathrm{\mid }>1),$ $T_r{X}_r{T}_r=z{X}_{r+1},T_r{X}_s=X_s{T}_r(s\mathrm{\notin }\{r,r+1\}),$ the last two relations being the Bernstein relation and the distant mixed commutation, and with $T_r$ invertible as a consequence of its quadratic relation. For $n=0$ the algebra is $A$.

The degenerate affine presentation. The algebra $H_n^{\mathrm{deg}}(A)$ is the $A$-algebra presented by commuting generators $x_1,\dots ,x_n$ and symmetric-group generators $s_1,\dots ,s_{n-1}$ subject to the Coxeter relations $s_r^2=1$, $s_r{s}_{r+1}{s}_r=s_{r+1}{s}_r{s}_{r+1}$, $s_r{s}_t=s_t{s}_r$ for $\mathrm{\mid }r-t\mathrm{\mid }>1$ together with the mixed relations $s_r{x}_r=x_{r+1}{s}_r-1,s_r{x}_{r+1}=x_r{s}_r+1,s_r{x}_t=x_t{s}_r(t\mathrm{\notin }\{r,r+1\})\mathrm{.}$

Auxiliary cyclotomic quotients. For a monic polynomial $f(u)=u^{\mathrm{\ell }}+a_{\mathrm{\ell }-1}{u}^{\mathrm{\ell }-1}+\cdots +a_0$ in $A[u]$ of degree $\mathrm{\ell }$ impose the single relation $f(X_1)=0$ (respectively $f(x_1)=0$) on the FIRST polynomial generator and call the resulting quotient the monic cyclotomic quotient; when $\mathrm{\ell }=0$, so that $f=1$, the quotient at positive rank is the zero algebra, while rank zero remains $A$. In the nondegenerate case the constant term $a_0$ is required to be a unit. The first-strand convention imposed here (cyclotomic relation on $X_1$, and later induction appending on the right) is the one used on this page.

The charged cyclic specialization. Let $A=F$ be a field, let $z=q_H\in F^{\times }$ have finite quantum characteristic $e\ge 2$ (The quantum characteristic of an invertible Hecke parameter), and let $I=\mathbb{Z}\mathrm{/}e\mathbb{Z}$ be the cyclic label set of residues, with a dominant cyclic type-A weight $({\lambda }_i{)}_{i\in I}$, ${\lambda }_i\ge 0$, and $\mathrm{\ell }={\sum }_i{\lambda }_i$ (The cyclic affine type A Cartan datum and its residues). If $q_H\mathrm{\ne }1$ use the nondegenerate presentation and put $c_i:=q_H^i$; if $q_H=1$ (which forces $\mathrm{char}F=e$) use the degenerate presentation and put $c_i:=i\cdot 1_F$. Impose $f(z_1):={\prod }_{i\in I}(z_1-c_i{)}^{{\lambda }_i},f(z_1)=0.$ on the first generator ($z_1=X_1$ in the nondegenerate branch and $z_1=x_1$ in the degenerate branch) and write $H_d^{\mathrm{\Lambda }}$ for the resulting charged cyclotomic quotient of rank $d$. This charged finite-$e$ branch is the scope of the present page; the arbitrary-base affine and monic definitions above supply its universal PBW and trace proof inputs. The definition itself asserts no basis, no spectrum, no primitive block decomposition and no Brundan–Kleshchev isomorphism. The grading variable $v$ of the KLR side is independent of $q_H$.

Well-definedness notes. The nondegenerate parameters $c_i=q_H^i$ are distinct because $q_H$ has multiplicative order exactly $e$; the degenerate parameters $c_i=i\cdot 1_F$ are distinct because $\mathrm{char}F=e$ makes $i\cdot 1_F\mathrm{\ne }j\cdot 1_F$ for $0\le i<j<e$. In the nondegenerate branch no embedding of the label set $\mathbb{Z}\mathrm{/}e\mathbb{Z}$ into $F$ is claimed or used: $i$ is only a label for the value $q_H^i$. All these definitions are given by generators, relations and a two-sided quotient ideal and use no choice principle.