Speaker
Description
We start with the basic notions of a hyperplane arrangement on $\mathbb{R}^n$ and then explain the braid arrangement on $\mathbb{R}^n$, which consists of $\frac{n(n-1)}{2}$ hyperplanes $H_{ij}=\{(x_1,x_2,\dots, x_n)\in \mathbb{R}^n\mid x_i=x_j\}$, $1\le i < j\le n$. Each region $P_\sigma$ of this arrangement is in one-to-one correspondence with a permutation $\sigma$ as follows
\begin{align}
P_{\sigma} =\{(x_1,x_2,\dots, x_n)\in \mathbb{R}^n\mid x_{\sigma_1}<x_{\sigma_2}<\dots < x_{\sigma_n} \}.
\end{align}
If we introduce the orientation of the braid arrangement, we obtain the oriented braid arrangement on $\mathbb{R}^n$ consisting of $n(n-1)$ open half-spaces
\begin{equation}
H_{ij}^+=\{(x_1,x_2,\dots, x_n)\in \mathbb{R}^ n\mid x_i>x_j\},
\end{equation}
\begin{equation}
H_{ij}^-=\{(x_1,x_2,\dots, x_n)\in \mathbb{R}^ n\mid x_i<x_j\}
\end{equation}
for all ${1\le i < j\le n}.$
Then to every open half-space ${H_{ij}^+}$ we associate a weight $q_{ij}$ and similarly to every open half-space ${H_{ij}^-}$ we associate a weight $q_{ji}$ in the polynomial ring in complex variables $q_{ij}$, ${1\le i\ne j\le n}$.
The quantum bilinear form $\mathit{B}^*_n$ is defined on the regions of that arrangement by
$$\mathit{B}^*_n(P_{\sigma},P_{\tau})=\prod_{(a,b)\in I(\tau^{-1}\sigma)} q_{\sigma(a)\sigma(b)}$$
with $I(\tau^{-1}\sigma)={(a,b)\mid a<b, \ \tau^{-1}\sigma(a)>\tau^{-1}\sigma(b) }$.
The matrix $${B}^*_n=\left({B}^*_n(P_{\sigma},P_{\tau})_{\sigma,\tau\in S_n}\right)$$ is known as the Varchenko matrix of the oriented braid arrangement.
To calculate the inverse of the matrix $${B}^*_n,$$ we need to use some special matrices and their factorizations in the form of simpler matrices. To simplify the calculation of the matrices, we first introduce a twisted group algebra $\mathcal{A}(S_{n}) = R_{n}\rtimes {\mathbb{C}}[S_n]$ of the symmetric group $S_{n}$ with coefficients in the polynomial ring $R_{n}$ of all polynomials in $n^2$ variables $X_{a\,b}$ over the set of complex numbers and then use a natural representation of some elements of the algebra ${\mathcal{A}(S_{n})}$ on the generic weight subspaces of the multiparametric quon algebra ${\mathcal{B}}$. In this way, we directly obtain the corresponding matrices of the quantum bilinear form.
