Let $A = \bigoplus_{i=0}^{+\infty} A_i$ be a graded bialgebra in a braided monoidal category $\mathcal V$. Then, according to Kapranov–Schechtman's article "Parabolic induction and perverse sheaves on $\mathfrak h/W$", section 8.A, the datum of $A$ is equivalent to the data of certain $\mathcal V$-valued mixed Bruhat sheaves, that is for each $n$ a mixed Bruhat sheaf $E_n(A)$ on $\mathfrak h_n/S_n$ where $\mathfrak h_n$ is a Cartan subalgebra of $\mathfrak{gl}_n$.
The stratification of $\mathfrak h/W$ is the stratification obtained from the double Coxeter complex (see 1.A and proposition 1.8). By 8.A, we know that this complex is indexed by contingency matrices of order $n$ for $\mathfrak g = \mathfrak{gl}_n$.
By the Theorem 2.6, such Bruhat sheaves on $\mathfrak h_n/W$ are given by the following datum: for each contingency matrix $M$ of order $n$, an object $E_n(A)(M) \in \mathcal V$, and certain maps associated to matrices $M,N$ such that $M \leq' N$ or $M \leq'' N$ (by definition 2.1). If $M = (m_{ij})$, according to 8.A we should moreover have $E_n(A)(M) = \bigotimes_{i,j} B_{m_{ij}}$.
I'm trying to work it out for $n=3$. In the mixed Bruhat sheaf picture, the complex stratum $\{ \alpha_1 = 0\} \subset \mathfrak h/W$ is refined into $4$ strata, indexed by the following contingency matrices : $M_1 = \begin{pmatrix} 2 & 0 \\ 0 & 1\end{pmatrix}$ and $M_2 = \begin{pmatrix} 1 & 1 \\ 1 & 0\end{pmatrix}$, $M_3 = \begin{pmatrix} 2 \\ 1\end{pmatrix}$ and $M_4 = \begin{pmatrix} 2 & 1\end{pmatrix}$. It seems to me that by definition, the inclusion of the 1-dimensional stratum into the 2-dimensional stratum are anodyne (definition 1.9). Hence the maps $$E_n(A)(M_1) \cong E_n(A)(M_3) \cong E_n(A)(M_2)$$ should be isomorphisms (from definition 2.1). However, even as vector spaces, there is no reason why $A_2 \otimes A_1$ should be isomorphic to $A_1 \otimes A_1 \otimes A_1$. What is wrong?
Remark: the set of all contingency matrices corresponding to the double Coxeter complex for $S_3$ is written in Kapranov and Schechtman - Contingency tables with variable margins (with an appendix by Pavel Etingof) page 5.
