# Iwahori-Hecke algebra of symmetric group:S3

The Iwahori-Hecke algebra of symmetric group:S3 over a commutative unital ring $R$ is defined as the $R[q]$-algebra:

$R[q] = \langle T_1,T_2 \rangle/ \langle (T_1 - q)(T_1 + 1), (T_2 - q)(T_2 + 1), T_1T_2T_1 - T_2T_1T_2 \rangle$

Specializing to $q = 1$ gives the group algebra over $R$ of symmetric group:S3:

$R[S_3] = R\langle T_1,T_2,T_3 \rangle/\langle T_1^2 - 1, T_2^2 - 1, T_1T_2T_1 - T_2T_1T_2 \rangle$

The thing below needs to be fixed

Element $()$ $\! T_1 \sim (1, 2)$ $\! T_2 \sim (2, 3)$ $\! T_1T_2T_1 \sim (1,3)$ $\! T_1T_2 \sim (1, 2, 3)$ $\! T_2T_1 \sim (1, 3, 2)$
$()$ $()$ $(1, 2)$ $(2, 3)$ $(1,3)$ $(1, 2, 3)$ $(1, 3, 2)$
$\! T_1 \sim(1, 2)$ $(1,2)$ $q() + (q - 1)(1,2)$ $q(1, 2, 3) + (q - 1)(1,3)$ $q(1, 3, 2) + (q - 1)(1,2,3)$ $q(2, 3) + (q - 1)(1,2,3)$ $(1, 3)$
$\! T_2 \sim (2, 3)$ $(2,3)$ $(1,3,2)$ $q() + (q - 1)(2,3)$ $q(1, 2, 3) + (q - 1)(1,3)$ $(1, 3)$ $q(1, 2) + (q - 1)(1,3,2)$
$\! T_1T_2T_1 \sim (1,3)$ $(1,3)$ $q(1, 2, 3) + (q - 1)(1,3)$ $q(1, 3, 2) + (q - 1)(1,3)$  ?  ?  ?
$\! T_1T_2 = (1, 2, 3)$ $(1, 2, 3)$ $(1,3)$ $q(2, 3) + (q - 1)(1,2,3)$  ?  ?
$\! T_2T_1 = (1, 3, 2)$ $(1, 3, 2)$ $q(2, 3) + (q - 1)(1,3,2)$ $(1, 3)$  ?  ?  ?