@enumeratio/hecke

Iwahori–Hecke algebra extensions for compute-engine: the $q$-deformation of the symmetric group algebra, with $T$-basis multiplication over $\mathbb{Z}[q]$. $H_n(q)$ keeps the symmetric-group basis exactly — one $T_w$ per permutation — and deforms only the product, by one rule applied a simple reflection at a time. See the guide.

Declaring

import { ComputeEngine } from "@cortex-js/compute-engine";
import { declareHecke } from "@enumeratio/hecke";

const ce = new ComputeEngine();
declareHecke(ce);

Heads

• HeckeAlgebra — the algebra itself, parameterised by $n$ (and $q$, symbolic unless specialized)
• HeckeT — the basis element $T_w$ for a permutation $w$
• HeckeIdentity — $T_e$, the identity of the algebra
• HeckeSpecialize — substitute a value for $q$

src/hecke.ts exports the underlying Element/Coefficients model directly — multiply, multiplyByBasis, reducedWord, length — for callers that want the $T$-basis arithmetic without going through compute-engine.

Usage

NonCommutativeMultiply(HeckeT([2, 1, 3], q), HeckeT([1, 3, 2], q))
  // HeckeT([2, 3, 1], q) — length goes up, so no q appears

NonCommutativeMultiply(HeckeT([2, 1, 3], q), HeckeT([2, 1, 3], q))
  // q * HeckeT([1, 2, 3], q) + (q - 1) * HeckeT([2, 1, 3], q) — the quadratic relation

AlgebraDimension(HeckeAlgebra(4, q))   // 24 — n!, same basis as Z[S_n]

Next

The guide has the length-up/length-down case split and works the braid relation $T_sT_tT_s = T_tT_sT_t$ through to the longest element of $S_3$.

Pages

• Hecke Algebras

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• Source: packages/symbols/algebras/hecke
• Depends on: boxes, engine, structures
• In development, also: config, manifest