Affine and cyclotomic type-A Hecke algebras

Let $F$ be a field and $q\in F^{\times }$ with finite quantum characteristic $e=e(F,q)$, $2\le e<\mathrm{\infty }$ (The quantum characteristic of an invertible Hecke parameter), let $I=\mathbb{Z}\mathrm{/}e\mathbb{Z}$ be the residue set with integer residue labels $0,1,\dots ,e-1$, and use the cyclic affine type-A Cartan datum with dominant weights $P^{+}$ and pairing $(\mathrm{\Lambda },{\alpha }_i)$ (The cyclic affine type A Cartan datum and its residues). In the nondegenerate branch the distinct spectral values of the labels are $q^i$, while in the degenerate branch they are $i\cdot 1_F$ (here $e=\mathrm{char}F$). Fix $\mathrm{\Lambda }\in P^{+}$ of level $\mathrm{\ell }={\sum }_{i\in I}(\mathrm{\Lambda },{\alpha }_i)$ and put ${\lambda }_i:=(\mathrm{\Lambda },{\alpha }_i)\ge 0$.

(NG) Nondegenerate case $q\mathrm{\ne }1$. The affine Hecke algebra of type $A_{d-1}$ at parameter $q$ is the unital $F$-algebra $H_d$ with generators $X_1^{\pm 1},\dots ,X_d^{\pm 1}$ and $T_1,\dots ,T_{d-1}$ subject to the relations $X_r{X}_s=X_s{X}_r,X_r{X}_r^{-1}=1,$ $T_r{X}_r{T}_r=q{X}_{r+1},T_r{X}_s=X_s{T}_r(s\mathrm{\notin }\{r,r+1\}),$ $T_r^2=(q-1)T_r+q,T_r{T}_{r+1}{T}_r=T_{r+1}{T}_r{T}_{r+1},T_r{T}_s=T_s{T}_r(\mathrm{\mid }r-s\mathrm{\mid }>1)\mathrm{.}$ The cyclotomic Hecke algebra is $H_d^{\mathrm{\Lambda }}:=H_d/⟨{\prod }_{i\in I}(X_1-q^i{)}^{{\lambda }_i}⟩\mathrm{.}$

(DG) Degenerate case $q=1$ (which forces $\mathrm{char}F=e>0$). The degenerate affine Hecke algebra $H_d$ is the unital $F$-algebra with generators $x_1,\dots ,x_d$ and $s_1,\dots ,s_{d-1}$ subject to the relations $x_r{x}_s=x_s{x}_r,s_r{x}_{r+1}=x_r{s}_r+1,s_r{x}_s=x_s{s}_r(s\mathrm{\notin }\{r,r+1\}),$ $s_r^2=1,s_r{s}_{r+1}{s}_r=s_{r+1}{s}_r{s}_{r+1},s_r{s}_t=s_t{s}_r(\mathrm{\mid }r-t\mathrm{\mid }>1);$ its ordered monomials $x_1^{m_1}\cdots x_d^{m_d}w$ form an $F$-basis by Degenerate affine Hecke ordered monomials form a basis. The degenerate cyclotomic Hecke algebra is $H_d^{\mathrm{\Lambda }}:=H_d/⟨{\prod }_{i\in I}(x_1-i{)}^{{\lambda }_i}⟩\mathrm{.}$

Conventions and boundary. Put $H_0^{\mathrm{\Lambda }}=F$; the cyclotomic polynomial is imposed on the FIRST polynomial generator, and $T_r$ acts on positions $r,r+1$. The Hecke parameter is the one of The generic type-A Hecke algebra ($T_i^2=(q-1)T_i+q$). At level one $\mathrm{\Lambda }={\mathrm{\Lambda }}_j$, the algebra $H_d^{\mathrm{\Lambda }}$ is the finite type-A Hecke algebra $H_{F,q}(S_d)$ if $q\mathrm{\ne }1$ and the group algebra $F{S}_d$ if $q=1$ (Cyclotomic Hecke algebras of level one are finite Hecke algebras). Only finite $e$ is treated: the source's $e=0$ branches ($q$ not a root of unity, and the characteristic-zero degeneration, whose quiver is $A_{\mathrm{\infty }}$) are outside this page's cyclic scope and are recorded as a deferred boundary in the batch notes. These presentations and the charged quotients are the ones of Affine type-A presentations and charged cyclotomic quotients; no embedding of the label set $I$ into $F$ is claimed in the nondegenerate branch, and no choice principle is used.