Hecke Algebras

The Iwahori–Hecke algebra is parameterised differently from every other family here. The hypercomplex units vary their generators; the diagram algebras vary which diagrams are admitted. keeps the basis of the symmetric group algebra exactly — one element per permutation, so the dimension is — and deforms the multiplication by a parameter .

The whole algebra is one rule, applied one simple reflection at a time:

The second line is the deformation, and it is the only place enters. Note what it does: a product of two basis elements is a linear combination, not one basis element times a scalar. That is new here — every earlier family's product landed back on a single basis element.

Length up, length down

When multiplying raises the Coxeter length, nothing happens but bookkeeping. When it lowers the length, appears.

#The two cases

First: ℓ goes up, so the product is a single T. Second: T_s² has ℓ going down, giving the quadratic relation q·T_e + (q−1)·T_s.

source

<notatio-cell value="CircleTimes(HeckeT([2, 1, 3]), HeckeT([1, 3, 2]))" />
<notatio-cell value="CircleTimes(HeckeT([2, 1, 3]), HeckeT([2, 1, 3]))" />

The quadratic relation — equivalently — is what replaces in the Coxeter group. At the two roots collide back onto and you recover an involution.

The braid relations survive the deformation

What does not change is the braid relations: for adjacent generators, and for distant ones. That is exactly why is well defined — it is the product over any reduced word for , and the tests check that for every in .

#Braid, and commutation

Both sides of the braid relation, then two distant generators in either order.

source

<notatio-cell value="CircleTimes(HeckeT([2, 1, 3]), HeckeT([1, 3, 2]), HeckeT([2, 1, 3]))" />
<notatio-cell value="CircleTimes(HeckeT([1, 3, 2]), HeckeT([2, 1, 3]), HeckeT([1, 3, 2]))" />

q = 1 is the symmetric group

This is the claim that makes "deformation" the right word, and it is the strongest test in the package: at the rule's two cases become one, and for every pair — checked exhaustively across , and .

#Specialising the parameter

q·T_e + (q−1)·T_s at q = 1 is just T_e — the group algebra. At q = 2 it stays spread.

source

<notatio-cell value="HeckeSpecialize(CircleTimes(HeckeT([2, 1, 3]), HeckeT([2, 1, 3])), 1)" />
<notatio-cell value="HeckeSpecialize(CircleTimes(HeckeT([2, 1, 3]), HeckeT([2, 1, 3])), 2)" />

So , and the symmetric group algebra reached as a diagram algebra is the same object by another route. Everything interesting — Kazhdan–Lusztig bases, the Jones polynomial via the Temperley–Lieb quotient, the representation theory at roots of unity — lives at .

Things worth knowing

Coefficients are exact polynomials in . They are compute-engine expressions, so they stay symbolic and simplify themselves; HeckeSpecialize(element, q) substitutes and drops whatever vanishes.

Sums compose. Because a product returns a linear combination, the parser reads sums back in — the result of one product is a valid operand for the next, which the tests check directly.

Dimension is , and Basis lists the up to ; past that the dimension still answers from the closed form.

Not built yet. The Kazhdan–Lusztig basis and the KL polynomials — the real reason anyone builds — and the Temperley–Lieb quotient, which would connect this page directly to the diagram algebras and to the Jones polynomial. Also only type A: the same rule works for any Coxeter system given its length function.