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

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

• Nicolas Libedinsky: Gentle introduction to Soergel bimodules I: The basics, https://nicolaslibedinsky.cl/gentle-introduction-to-soergel-bimodules-i-the-basics/, arXiv https://arxiv.org/abs/1702.00039

• Coxeter groups
• Braid groups
• Weyl groups
• Kazhdan–Lusztig basis
• Soergel bimodules
• Satake correspondence