Geometric interpretation of the Desnanot–Jacobi Identity

Question

Given a square $n\times n$ matrix $M$, let $M_i^j$ denote the $(n-1)\times(n-1)$ matrix obtained from M by omitting the $i$-th row and $j$-th column of $M$.

The Desnanot–Jacobi Identity states $$\det(M)\det(M^{1,n}_{1,n})=\det(M^1_1)\det(M^n_n)-\det(M^n_1)\det(M^1_n).$$

If you view $M\in A\otimes B$ where $A$ and $B$ are $n$-dimensional vector spaces, then $\det(M)\in \Lambda^nA\otimes\Lambda^nB\subset S^n(A\otimes B)$ and the determinant of a $k\times k$ minor is an element of $\Lambda^kA\otimes\Lambda^kB\subset S^k(A\otimes B)$.

With this interpretation of matrices, determinants, and determinants of minors of matrices, I would like to have a geometric interpretation of the Desnanot–Jacobi Identity.

Answer 1

One way to understand this identity is as a Plücker relation for coordinates of the graph of $M$ in the exterior power $\Lambda^n K^{2n}=\Lambda^n(K^n\oplus K^n)$ ($K$=coefficient field), namely those of the $n$-vector $(e_1\oplus M\,e_1)\wedge\dots\wedge(e_n\oplus M\,e_n)$ in the standard basis $e_{i_1}\wedge \dots\wedge e_{i_n}$, $1\leq i_1 <\dots < i_n\leq 2n$.

The family of Plücker coordinates $z_I$ ($I\subset [2n]:=[1,2n]$ of cardinality $n$) identifies with that of all minors of $M$ --- see below , and the Plücker relations read

$$\sum_{i\in I\setminus J} \pm z_{I\setminus i} z_{J\cup i}= 0$$ where $I$, $J$ are subsets of $[2n]:=\{1,2,\dots,2n\}$ of cardinalities $n+1$, $n-1$ respectively.

Denoting $x':=x+n$ for $x\in [n]$, the identification of minors with $z$ coordinates takes the following form :

for $A$, $B\subset [n]$ of same cardinality, $$z_{([n]\setminus A)\cup B'}=\det(M_{A,B}).$$

Taking $I=\{n,1',\dots,n'\}$, $J=\{1,2',\dots,(n-1)'\}$ in the above Plücker relation (so that $I\setminus J=\{n,1',n'\}$), one gets Desnanot-Jacobi-Dodgson/Lewis Caroll identity.

I am not sure if this counts as a geometric explanation though.

Answer 2

It is equivalent (in type A) to the relationship between generalized minors $\Delta^{\omega_i}(g)$ for $g\in G$ [Fomin–Zelevinski 1999 - Double Bruhat cells and total positivity]. Here $\omega_i$ is the weight of a highest weight representation of $G$. This relationship is proven in Koroteev–Zeitlin - q-Opers, QQ-systems, and Bethe Ansatz II: Generalized Minors.

One can rewrite the Lewis Carroll identity in terms of so-called Baxter Q-functions which appear a lot in the literature on integrable systems.