Hecke Algebras

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

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

$$T_s \cdot T_w = \begin{cases} T_{sw} & ℓ(sw) > ℓ(w) \\ q\,T_{sw} + (q-1)\,T_w & ℓ(sw) < ℓ(w)\end{cases}$$

The second line is the deformation, and it is the only place $q$ 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, $q$ 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.

In$\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, q)⊗\mathrm{HeckeT}(\bigl\lbrack1, 3, 2\bigr\rbrack, q)$Out$\mathrm{HeckeT}(\bigl\lbrack2, 3, 1\bigr\rbrack, q)$In$\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, q)⊗\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, q)$Out$q\mathrm{HeckeT}(\bigl\lbrack1, 2, 3\bigr\rbrack, q)+(q-1)\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, q)$

source · editable

<notatio-cell value="CircleTimes(HeckeT([2, 1, 3], q), HeckeT([1, 3, 2], q))" data-allow-mismatch="children" /> <notatio-cell value="CircleTimes(HeckeT([2, 1, 3], q), HeckeT([2, 1, 3], q))" data-allow-mismatch="children" />

The quadratic relation $T_s^2 = q + (q-1)T_s$ — equivalently $(T_s - q)(T_s + 1) = 0$ — is what replaces $s^2 = 1$ in the Coxeter group. At $q = 1$ the two roots collide back onto $± 1$ and you recover an involution.

The braid relations survive the deformation

What does not change is the braid relations: $T_sT_tT_s = T_tT_sT_t$ for adjacent generators, and $T_sT_t = T_tT_s$ for distant ones. That is exactly why $T_w$ is well defined — it is the product over any reduced word for $w$, and the tests check that for every $w$ in $S_4$.

#Braid, and commutation

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

In$\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, q)⊗\mathrm{HeckeT}(\bigl\lbrack1, 3, 2\bigr\rbrack, q)⊗\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, q)$Out$\mathrm{HeckeT}(\bigl\lbrack3, 2, 1\bigr\rbrack, q)$In$\mathrm{HeckeT}(\bigl\lbrack1, 3, 2\bigr\rbrack, q)⊗\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, q)⊗\mathrm{HeckeT}(\bigl\lbrack1, 3, 2\bigr\rbrack, q)$Out$\mathrm{HeckeT}(\bigl\lbrack3, 2, 1\bigr\rbrack, q)$

source · editable

<notatio-cell value="CircleTimes(HeckeT([2, 1, 3], q), HeckeT([1, 3, 2], q), HeckeT([2, 1, 3], q))" data-allow-mismatch="children" /> <notatio-cell value="CircleTimes(HeckeT([1, 3, 2], q), HeckeT([2, 1, 3], q), HeckeT([1, 3, 2], q))" data-allow-mismatch="children" />

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 $q = 1$ the rule's two cases become one, and $T_u \cdot T_v = T_{uv}$ for every pair — checked exhaustively across $S_2$, $S_3$ and $S_4$.

#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.

In$\mathrm{HeckeSpecialize}(\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, q)⊗\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, q), 1)$Out$\mathrm{HeckeT}(\bigl\lbrack1, 2, 3\bigr\rbrack, 1)$In$\mathrm{HeckeSpecialize}(\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, q)⊗\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, q), 2)$Out$2\mathrm{HeckeT}(\bigl\lbrack1, 2, 3\bigr\rbrack, 2)+\mathrm{HeckeT}(\bigl\lbrack2, 1, 3\bigr\rbrack, 2)$

source · editable

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

So $H_n(1) = \mathbb{Z}S_n$, 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 $q ≠ 1$.

Things worth knowing

Coefficients are exact polynomials in $q$. They are compute-engine expressions, so they stay symbolic and simplify themselves; HeckeSpecialize(element, q) substitutes and drops whatever vanishes. The parameter is an argument of every head (HeckeT(w, q), HeckeAlgebra(n, q)), a symbol or a number of your choosing, and elements with different parameters do not multiply.

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 $n!$, and Basis lists the $T_w$ up to $n = 6$; 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 $H_n(q)$ — 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.