Iwahori-Hecke algebra of the symmetric group

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Definition

Let R be a unital ring and n be a natural number. The Iwahori-Hecke algebra of the symmetric group of degree n over the ring R is defined as the R[q]-algebra given by the presentation:

\langle T_1, T_2, \dots T_{n-1} \mid (T_i - q)(T_i + 1) = 0, T_iT_j = T_jT_i \ \forall |i - j| > 1, T_iT_{i+1}T_i = T_{i+1}T_iT_{i+1} \rangle.

In other words, it is the Iwahori-Hecke algebra corresponding to the symmetric group of degree n, viewed as a Coxeter group, in the usual way: Symmetric group on a finite set is a Coxeter group.

For a field with q elements, specializing to the value q gives the Hecke algebra of the general linear group GL_n(\mathbb{F}_q).

Particular cases

n Iwahori-Hecke algebra of the symmetric group of degree n
2 Iwahori-Hecke algebra of symmetric group:S2
3 Iwahori-Hecke algebra of symmetric group:S3
4 Iwahori-Hecke algebra of symmetric group:S4