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  "body": "\n## Definition\n\nThis is the Iwahori–Hecke algebra.\nLet $(W, S)$ be a [[Coxeter system|G6FND7X69J]], with Coxeter matrix $(m_{st})_{s,t \\in S}$, so that $m_{st}$ is the order of $st$ in $W$.\nLet $R$ be a [[commutative ring|0310REJJY8]] with identity and let $q \\in R$.\nThe *Hecke algebra* $H_q(W)$ is the [[associative algebra|3CTW1P4HBP]] over $R$ generated by elements $T_s$ for $s \\in S$, subject to the relations:\n\n1. $(T_s - q)(T_s + 1) = 0$ for every $s \\in S$.\n2. $T_s T_t T_s \\cdots = T_t T_s T_t \\cdots$, with $m_{st}$ factors on each side, for all $s \\ne t$ in $S$ with $m_{st} \u003c \\infty$.\n\n## Remarks\n\nFor $w \\in W$ with a reduced expression $w = s_1 \\cdots s_k$, put $T_w = T_{s_1} \\cdots T_{s_k}$.\nBy the braid relations and Matsumoto's theorem $T_w$ does not depend on the reduced expression, and the $T_w$ for $w \\in W$ form a basis of $H_q(W)$ as a [[free|0DZ9DKAKZR]] $R$-module.\nThe multiplication is $T_s T_w = T_{sw}$ if $\\ell(sw) \u003e \\ell(w)$ and $T_s T_w = (q - 1) T_w + q T_{sw}$ if $\\ell(sw) \u003c \\ell(w)$, where $\\ell$ is the length function.\nAt $q = 1$ the relations become $T_s^2 = 1$ and the braid relations, which present the group algebra $R[W]$.\nExample: $W = \\set{1, s}$ of order $2$ gives $H_q(W) = R[T]/((T - q)(T + 1))$, free of rank $2$ with basis $1, T$.\nNon-example: for $W = S_3$ with $S = \\set{s, t}$ and $m_{st} = 3$, dropping relation 2 changes the algebra.\nAt $q = 1$ the result is the group algebra of the infinite dihedral group $\\langle s, t \\mid s^2 = t^2 = 1 \\rangle$, which has infinite rank, while $H_1(S_3) = R[S_3]$ has rank $6$.\nMany authors use the parameter $q^{1/2}$ or $v$ with $q = v^2$ and rescale the generators.\n",
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