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  "body": "\n## Summary\n\nThe name Hecke algebra covers several related algebras; the most common is the Iwahori–Hecke algebra of a [[Coxeter group|G6FND7X69J]].\nThe Iwahori–Hecke algebra is a deformation of the group algebra of the Coxeter group, depending on a parameter $q$.\nAt $q = 1$ it is the group algebra.\nLet $G$ be a finite group of Lie type over a field with $q$ elements.\nIwahori showed that the Hecke algebra is the endomorphism algebra of the permutation module of $G$ on the cosets of a Borel subgroup.\nOther Hecke algebras are built from double cosets of a subgroup, as for the [[Hecke operators|S4SV3QZ8B1]] of number theory, or from locally compact groups.\n\n## Explanations\n\n### L1 human\n\nA Coxeter group, such as the [[symmetric group|0515A6D2Y1]], is generated by reflections $s$ with $s^2 = 1$ and some braid relations.\nThe Hecke algebra keeps the braid relations but replaces $s^2 = 1$ by a rule with a parameter $q$, namely $T_s^2 = (q - 1) T_s + q$.\nWhen $q = 1$ this is again $T_s^2 = 1$, so the Hecke algebra is a one-parameter family of algebras through the group algebra.\nFor the group with two elements, the Hecke algebra is spanned by $1$ and $T$, with $T^2 = (q-1)T + q$.\nThese algebras appear in the representation theory of finite and $p$-adic groups, in knot invariants and in the combinatorics of Kazhdan–Lusztig polynomials.\n",
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