Iwahori-Hecke algebra of symmetric group:S3

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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)  ?  ?  ?