Iwahori-Hecke algebra of symmetric group:S2

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:

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: