Speaker
Description
The first results on transformations preserving matrix invariants is due to Frobenius. This result describes the structure of linear maps $T$ preserving the determinant function, i.e., $\det X = \det T(X)$ for all $X$. Later on there were several extension of this result which are due to Diedonnie, Schur, Dynkin and others.
In 1913 Cullis and then in 1966 independently Radi\'c introduced an analog of the determinant function for the rectangular matrices. Namely, given an injection $\sigma \in S^I_K,$ we denote by ${\rm sgn}_{nk}(\sigma)$ the product
$$s \cdot (-1)^{\sum_{l = 1}^{k} (\sigma(l) - l)} ,$$
where $s={\rm sgn}(\pi)$ is the sign of the permutation
$$
\pi =
\begin{pmatrix}
i_1 & \ldots & i_k\\
\sigma(1) & \ldots & \sigma(k)
\end{pmatrix},
$$
here $\sigma(K)={i_1, \ldots, i_k}$, and $i_1<i_2<\ldots <i_k$.
Let $n \ge k$, $X \in M_{nk} (\mathbb F)$. Then Cullis determinant $\det_{n\, k}(X)$ of $X$ is defined to be the function:
$$
{\det}_{n\, k} (X) = \sum_{\sigma \in S_{k}^{I}} {\rm sgn}_{nk} (\sigma) X_{(\sigma(1),1)}X_{(\sigma(2),2)}\ldots X_{(\sigma(k),k)}.
$$
Cullis determinant has many interesting properties of the usual determinant as well as many new properties. It is interesyting as from theoretical point of view, as due to the number of applications. In this talk we discuss an analog of Frobenius theorem for Cullis determinant, namely we characterize linear preservers of this function. It appears that the preservers depend on the parity of $k+n$. Our proof is based on analogs of Flander's results and matroid theory.
The talk is based on the series of joint works with Andrey Yurkov.