HeckeSpecialize

Substitute a value for the deformation parameter. At $q = 1$ the algebra collapses to the group algebra of $S_n$ — which is what makes 'deformation' the right word.

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

HeckeSpecialize(element, q)set $q$ and drop whatever vanishes

Typed signatures

Type	Package	Applies
(number, number) -> number	hecke	always

Domain: Hecke algebras

Details

• At $q = 1$ the two cases of the multiplication rule become one, and $T_u T_v = T_{uv}$ for every pair — checked exhaustively over $S_2$, $S_3$ and $S_4$
• Everything interesting about $H_n(q)$ — Kazhdan–Lusztig theory, the Jones polynomial, representations at roots of unity — lives at $q \ne 1$

Examples

Basic #

$q\,T_e + (q-1)T_s$ at $q = 1$ is just $T_e$

live ↗

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

Applications #

the ordinary product of permutations

live ↗

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

Properties #

at $q = 2$ it stays spread over two basis elements

live ↗

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

About

Examples	3
Crosswalk	WikipediaWikidatanLab

See also: HeckeT