I'm interesting in knowing whether a certain variety defined by maximal minors is irreducible. The specific construction is as follows: let $n \geq 2$ and let $R = \mathbb{C}[a_1,b_1,c_1,d_1,e_1,f_1,g_1,h_1,a_2, \ldots, h_n]$ be a polyonomial ring in $8n$ variables. For each $1 \leq i \leq n$ form the matrix $$M_i = \begin{pmatrix} a_i &0 & 0& b_i & c_i &0 \\0 & d_i &0&0 & e_i& f_i \\ e_i & c_i & a_i+d_i & g_i & 0 & h_i \end{pmatrix}.$$
Now $M$ is the $3n \times 6$ matrix formed by stacking $M_1, \ldots, M_n$ vertically. Let $K$ be the affine variety in $\mathbb{C}^{8n}$ defined by the set of maximal minors of $M$. Is $K$ irreducible?
I understand that, if the repetitions of $c_i$ and $e_i$ and the $a_i+d_i$ on the bottom row were replaced with new independent variables I'd be home and dry, thanks to a result of Guisti and Merle: the largest semi-perimeter of a block of zeros in $M$ is $3 \leq 6-4$. There's also some results of Eisenbud on quotients of determinantal varieties of "1- generic" matrices, but again my matrix seems to have too many zeros for those to apply either. Are there any more recent results which might apply to my matrix?
Edit: In response to Sasha's question below, I agree this is a very strange question to ask if it doesn't have some geometric meaning, so here's an idea of the context: let $B$ be the group of $3 \times 3$ invertible upper triangular matrices over $\mathbb{C}$, and let $V$ be the set of $3 \times 3$ upper triangular matrices over $\mathbb{C}$. B acts on $V$ by conjugation. A pair $A,A'$ of matrices in $V$ in the same $B$ orbit must have identical diagonal entries, so let $$A = \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33} \end{pmatrix}, A' = \begin{pmatrix} a_{11} & a'_{12} & a'_{13}\\ 0 & a_{22} & a'_{23} \\ 0 & 0 & a_{33} \end{pmatrix}.$$ Now let $$b = \begin{pmatrix} p & x & y\\ 0 & q & z \\ 0 & 0 & r \end{pmatrix} \in B.$$ One can show that $b^{-1}Ab = A'$ if and only if $Mv=0$ where
$$M = \begin{pmatrix} a_{11}-a_{22} &0 & 0& a_{12} & a'_{12} &0 \\0 & a_{22}-a_{33} &0&0 & a_{23}& a'_{23} \\ a_{23} & a'_{12} & a_{11}-a_{33} & -a_{13} & 0 & a'_{13} \end{pmatrix}$$ and $v = (-x,y,z,p,-q,r)$.
Now consider the action of $B$ on $V^n$ by simultaneous conjugation, so we regard the entries of $A$ and $A'$ as vectors. The graph $\Gamma$ of the action is the subvariety of $V^n \times V^n$ of pairs of matrices in $V^n$ lying in the same $B$-orbit. I want to compute the Zariski closure of $\Gamma$ (this gives some information about the number of invariants needed to separate orbits). A good candidate is the variety $K$ consisting of pairs of $n$-tuples of matrices such that the stack of associated matrices $M$ above has rank $<6$. Clearly $\Gamma = K \setminus L$ where $L$ is some closed subvariety of $K$, and I can show that $\overline{\Gamma}$ and $K$ have the same dimension. Therefore, if I could show that $K$ was irreducible, I would know that $\overline{\Gamma} = K$.
