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2 Group Algebras and Coxeter Groups

For a finite group \(G\) and a field \(k\), the group algebra \( k[G] \) is the algebra over \(k\) with dimension \( \vert G\vert \) and a basis indexed by the elements of \(G\). The multiplication in \( k[G] \) is given by the multiplication in \(G\). As an example, the group algebra \( \mathbb {C}[S_n]\) has dimension \(n!\) with basis denoted by permutations \(\sigma \in S_n\). The multiplication of two basis vectors equals another basis vector, induced from the multiplication in the symmetric group \(S_n\).

The symmetric group \(S_n\) is an example of a Coxeter group. We can present \(S_n\) with generators \(s_1,\ldots ,s_{n-1}\) where each \(s_i\) is the permutation \( (i i+1)\). Then the relations are \[ s_i^2=1, \quad (s_is_{i+1})^3=1, \quad (s_is_j)^2=1 \text { if } \vert i-j\vert \neq 1. \] The condition \( (s_is_j)^2=1 \) is equivalent to \( s_is_j = s_j^{-1}s_i^{-1} = s_js_i\), which means that \(s_i\) and \(s_j\) commute.

A Coxeter group is a group \(W\) generated by a finite set \(S = \{s_1, \ldots , s_n\}\) of involutions, subject only to relations of the form \((s_i s_j)^{m_{ij}} = 1\), where \(m_{ii} = 1\) for all \(i\), \(m_{ij} = m_{ji} \geq 2\) for \(i \neq j\), and \(m_{ij} \in \mathbb {N} \cup \{\infty \}\) (with the convention that no relation is imposed when \(m_{ij} = \infty \)). The integers \(m_{ij}\) can be collected in a Coxeter matrix \(M = (m_{ij})_{1 \leq i, j \leq n}\). For example, the symmetric group \(S_{n+1}\) has Coxeter matrix \[ \begin{cases} m_{ii}=1 \\ m_{i,i+1} =2 \\ m_{ij} = 3, \vert i-j \vert >1. \end{cases} \]

Of course, there are strict restrictions on which matrices can be Coxeter matrices. The classification of Coxeter matrices can be found in [Cox35], and most probabilists could use this classification as a black box.

3 Hecke Algebras

The Hecke algebra is a q–deformation of (the group algebra) of a Coxeter group, analogously to the quantum group being a q–deformation of (the universal enveloping algebra) of a Lie algebra. First, note that if \( m_{ij}=3\) then \[ s_is_js_is_js_is_j=1 \] can be rewritten as \[ s_is_js_i = s_js_is_j \] since \(s_i^2=1\) for all i. By rewriting this way, the equation can be justifiably called a braid relation.

More precisely, given a Coxeter group \(W\) with generating set \(S\), the Hecke algebra \(H_q(W)\) (or \(H(W,q)\)) is a \(q\)-deformation of the group algebra \(\mathbb {C}[W]\). It is generated by elements \(\{T_i : 1 \leq i \leq n\}\) satisfying:

• The braid relations: \((T_i T_j T_i \cdots ) = (T_j T_i T_j \cdots )\) with \(m_{ij}\) factors on each side, corresponding to the Coxeter relations, and
• The quadratic relation: \((T_i - q)(T_i + 1) = 0\) for each i.

When \(q = 1\), the quadratic relation becomes \(T_i^2 = 1\), and the Hecke algebra specializes to the group algebra \(\mathbb {C}[W]\).

As an example of a Hecke algebra, we have that \(H_q(S_{n+1})\) is generated by \(T_1,\ldots ,T_n\) with relations \[ (T_i - q)(T_i + 1) = 0, \quad T_i T_{i+1} T_i = T_{i+1} T_i T_{i+1}. \]

For the type B Hecke algebra, which deforms the hyperoctohedral group, there is in fact a two–parameter deformation. This algebra is generated by \(T_0, T_1, \ldots , T_{n-1}\) subject to the relations \begin{align*} T_i T_{i+1} T_i &= T_{i+1} T_i T_{i+1}, \quad 1 \leq i < n-1 \\ T_i T_j &= T_j T_i, \quad |i - j| \geq 2, \\ T_0 T_1 T_0 T_1 &= T_1 T_0 T_1 T_0, \\ (T_i - q^{-1})(T_i + q) &= 0, \quad 1 \leq i < n-1 \\ (T_0 - Q^{-1})(T_0 + Q) &= 0. \end{align*}

The relation \(T_0T_1T_0T_1 = T_1T_0T_1T_0\) is related to the reflection equation ([Che84, Skl88]). Type B algebras in the context of reflecting boundary conditions have occurred in [?, Kua22]. Alexey Bufetov also has several recent papers on Hecke algebras in probability, such as [Buf20, BC24, BC24b, BB24, BN22]

4 \(R\)–matrices and representations

For any vector space \(V\), there is a natural representation of the symmetric group \(S_n\) on the n–fold tensor product \(V^{\otimes n}\) given by \[ \sigma \cdot (v_1 \otimes \cdots \otimes v_n) = v_{\sigma (1)} \otimes \cdots \otimes \cdots v_{\sigma (n)}. \] In the case of the Hecke algebra \(H_q(S_n)\), we need a R–matrix \( R: V\otimes V \rightarrow V\otimes V\) satisfying

• The braid relations \(R_{12}R_{23}R_{12} = R_{23}R_{12}R_{23} \)
• The quadratic relation: \((R - q)(R + 1) = 0\) for each i.

In fact, such a \(R\) exists (see [Jim86, ?, FRT88]). Thus there exists a representation of the Hecke algebra \(H_q(S_n)\) on \(V^{\otimes n}\) given by having the algebra element \(T_i\) act as \(R_{i,i+1}\).

Given a solution to a braid relation, generalizing that to a solution to the Yang–Baxter equation is called (Yang)–Baxterization. Generally speaking, this is not an easy task and goes back to e.g. [?, ?, ZGB91, Li93].

5 Schur–Weyl Duality

Since the main “body” of this survey is on quantum groups, we should relate Hecke algebras to quantum groups. The classical Schur–Weyl duality is a duality between the action of the symmetric group \(S_n\) and the action of the general linear group \(GL(V)\) on the \(n\)–fold tensor product \(V^{\otimes n}\). The theorem then states

The quantum version of Schur–Weyl duality was discovered in [Jim86]:

References