Iwahori-Hecke algebra of symmetric group:S2

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Note that although the symmetric group S_2 is isomorphic as an abstract group to cyclic group:Z2, it is important that we are thinking of it as a symmetric group in this case since the definition of Iwahori-Hecke algebra of the symmetric group relies on thinking of the group as a symmetric group.

The Iwahori-Hecke algebra of symmetric group:S2 over a commutative unital ring R is defined as the R[q]-algebra:

R[q]\langle T \rangle/ \langle (T - q)(T + 1) \rangle

In other words, as a R[q]-algebra, it is generated by a single element T which satisfies the relation (T - q)(T + 1) = 0, or T^2 = q + (q - 1)T.

Specializing q to a particular numerical value q_0 \in R means considering a R-algebra obtained by quotienting the base ring by the ideal \langle q - q_0 \rangle, i.e., setting q = q_0 downstairs.

Specializing to q = 1 gives the group algebra over R of the symmetric group, i.e.:

R[S_2] \cong R[T]/(T^2 - 1)

The multiplication table in general is given as follows:

Identity element, 1, or () T or (1,2)
1 or () 1 T
T or (1,2) T q + (q - 1)(T)

Interpretation over fields

Specializing to q a prime power gives the Hecke algebra of the general linear group for the general linear group of degree two over the finite field with q elements. The basis \{ 1, T \} represents the set of possible relative positions of pairs of complete flags in a two-dimensional vector space over \mathbb{F}_q. Two flags have relative position 1 if they are identical, and T otherwise. We can now interpret the multiplication as follows: