Modules imitating the Iwahori–Hecke algebra

Soergel bimodules were introduced in 1992 by Wolfgang Soergel to study representations of Lie groups¹⁾. Computing tensor products between Soergel bimodules is formally equivalent to computations in the Iwahori–Hecke algebra. In fancy language: Soergel bimodules are a categorification of Iwahori–Hecke algebras. This phenomenon explains positivity properties of the latter.

Graphical calculus between Soergel bimodules²⁾

Here is a concrete example of this categorification: the Hecke algebra admits a basis $(b_w)$ called the Kazhdan-Lusztig basis. For a simple basis element $b_s$ we have $b_s^2=(v+v^{-1})b_s$ where $v$ is the formal parameter in the Hecke algebra. To each $b_w$ is associated a Soergel bimodule $B_w$. We have $$B_s⊗B_s≅B_s(1)+B_s(-1)$$ where $B_s(1)$ is a shift of $B_s$. This shift corresponds to the multiplication by $v$.

Elias–Williamson have developped a graphical calculus for Soergel bimodules (see the figure for an example).

A lovely introduction to Coxeter groups, the Hecke algebra and Soergel bimodules can be found here (Soergel bimodules in Section 4):

• 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

The advantage in the landscape of modules is that one can speak about an indecomposable module (those which can not be written as direct sum of two others). These are precisely the Soergel bimodules with shift $B_w(k)$ and can be used as a natural basis in the space of bimodules.

Let us get a feeling what a Soergel bimodule looks like. A Soergel bimodule is associated to each element of a Coxeter group and an integer (the “shift”). Take for example the permutation group $S_n$. The simplest Soergel bimodule, corresponding to the identity, is $R=R[x_1,\ldots,x_n]$. For the simple transposition $s=(12)$ and shift -1, the corresponding Soergel bimodule is given by $$B_s(-1)=R⊗_{R_s}R,$$ where $R_s=R[x_1+x_2,x_1x_2,x_3,\ldots,x_n]$, the polynomials invariant under $s$.

• Hecke algebra
• Categorification
• Coxeter group
• Weyl group