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The Iwahori-Hecke algebra of symmetric group:S3 over a commutative unital ring R {\displaystyle R} is defined as the R [ q ] {\displaystyle R[q]} -algebra:
R [ q ] = ⟨ T 1 , T 2 ⟩ / ⟨ ( T 1 − q ) ( T 1 + 1 ) , ( T 2 − q ) ( T 2 + 1 ) , T 1 T 2 T 1 − T 2 T 1 T 2 ⟩ {\displaystyle R[q]=\langle T_{1},T_{2}\rangle /\langle (T_{1}-q)(T_{1}+1),(T_{2}-q)(T_{2}+1),T_{1}T_{2}T_{1}-T_{2}T_{1}T_{2}\rangle }
Specializing to q = 1 {\displaystyle q=1} gives the group algebra over R {\displaystyle R} of symmetric group:S3:
R [ S 3 ] = R ⟨ T 1 , T 2 , T 3 ⟩ / ⟨ T 1 2 − 1 , T 2 2 − 1 , T 1 T 2 T 1 − T 2 T 1 T 2 ⟩ {\displaystyle R[S_{3}]=R\langle T_{1},T_{2},T_{3}\rangle /\langle T_{1}^{2}-1,T_{2}^{2}-1,T_{1}T_{2}T_{1}-T_{2}T_{1}T_{2}\rangle }
The thing below needs to be fixed
![{\displaystyle R[q]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f30d2f9081f9455412df93790fbc441005131110)
![{\displaystyle R[q]=\langle T_{1},T_{2}\rangle /\langle (T_{1}-q)(T_{1}+1),(T_{2}-q)(T_{2}+1),T_{1}T_{2}T_{1}-T_{2}T_{1}T_{2}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/498d177294e9f18738d73abd2693e8e7205e8283)
![{\displaystyle R[S_{3}]=R\langle T_{1},T_{2},T_{3}\rangle /\langle T_{1}^{2}-1,T_{2}^{2}-1,T_{1}T_{2}T_{1}-T_{2}T_{1}T_{2}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/027977f41d4955b30ca2402b76527cd9d869d816)