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hecke_algebra

Iwahori--Hecke algebra

Deformation of the permutation group

Iwahori–Hecke algebras appear in the representation theory of Lie groups. They are deformations of reflection groups.

Diagrammatics for multiplication in Hecke algebras

A Coxeter group (think either of a group generated by reflections, or the permutation group) is generated by simple reflections obeying $s^2=1$ and braid relations $(st)^m=1$. The Hecke algebra is generated by formal symbols $T_s$, obeying the braid relations and the quadratic relation $$T_s^2=(q-1)T_s+q.$$

This is a deformation of $s^2=1$ which is recovered for $q=1$.

Important features of the Hecke algebra (non-exhaustive):

  • Kazhdan–Lusztig basis with positivity properties
  • cellular structure
  • Satake correspondence

Material

A lovely introduction to Coxeter groups, the Hecke algebra and Soergel bimodules can be found here (Hecke algebras in Section 3):

  • Coxeter groups
  • Braid groups
  • Weyl groups
  • Kazhdan–Lusztig basis
  • Satake correspondence
hecke_algebra.txt · Last modified: 2021/09/13 12:17 by alex_th