# Iwahori-Hecke algebra of symmetric group:S2

Note that although the symmetric group 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 is defined as the -algebra:

In other words, as a -algebra, it is generated by a single element which satisfies the relation , or .

Specializing to a particular numerical value means considering a -algebra obtained by quotienting the base ring by the ideal , i.e., setting downstairs.

Specializing to gives the group algebra over of the symmetric group, i.e.:

The multiplication table in general is given as follows:

Identity element, , or | or | |
---|---|---|

or | ||

or |

## Interpretation over fields

Specializing to 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 elements. The basis represents the set of possible relative positions of pairs of complete flags in a two-dimensional vector space over . Two flags have relative position if they are identical, and otherwise. We can now interpret the multiplication as follows:

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