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:

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:

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