HeckeT

The basis element $T_w$ of the Iwahori–Hecke algebra $H_n(q)$, indexed by a permutation in one-line notation. Multiplication is the $q$-deformation of the symmetric group's.

WikipediaIwahori–Hecke algebra↗WikidataQ1012612↗nLabIwahori–Hecke algebra↗

HeckeT([2,1,3], q)$T_w$ for the permutation $w$

Typed signatures

Type	Package	Applies
(list<integer>, number) -> number	hecke	always

Domain: Hecke algebras

Details

• $T_s \cdot T_w = T_{sw}$ when the length goes up, and $q\,T_{sw} + (q-1)\,T_w$ when it goes down — the second case is the entire deformation
• So a product of two basis elements is a LINEAR COMBINATION, unlike the other algebra families here; sums read back in, so products compose
• The quadratic relation is $T_s^2 = q + (q-1)T_s$, i.e. $(T_s-q)(T_s+1) = 0$, replacing $s^2 = 1$
• The braid relations survive the deformation, which is why $T_w$ is the product over ANY reduced word for $w$
• Coefficients stay exact polynomials in $q$; use HeckeSpecialize to pick a value
• Use the ordered product (NonCommutativeMultiply or $\otimes$) — $H_n(q)$ is not commutative

Examples

Basic #

length goes up, so no $q$ appears

live ↗

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

$n!$ — the same basis as $\mathbb{Z}S_n$

live ↗

In$\mathrm{AlgebraDimension}(\mathrm{HeckeAlgebra}(4, q))$Out$24$

Properties #

the quadratic relation $T_s^2 = q + (q-1)T_s$

live ↗

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

$T_sT_tT_s$ — the longest element of $S_3$

live ↗

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

…and $T_tT_sT_t$ gives the same, which is the braid relation

live ↗

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

About

Examples	5
Crosswalk	WikipediaWikidatanLab

See also: HeckeSpecialize, NonCommutativeMultiply, Basis