$n$-Valued Groups, Kronecker Sums, and Wendt Matrices

In 1971, V.,M. Buchstaber and S.,P. Novikov proposed a construction motivated by the theory of characteristic classes. This construction describes a multiplication in which the product of any pair of elements is a multiset of $n$ points. The axiomatic definition of $n$-valued groups and the first results of their algebraic theory were obtained in a subsequent series of works by V.,M. Buchstaber. At present, the theory of $n$-valued (formal, finite, discrete, topological, and algebro-geometric) groups and its applications in various areas of mathematics and mathematical physics is being developed by a number of authors.

In the talk, for each $n$, the notion of classes of symmetric $n$-algebraic $n$-valued groups will be introduced. For $n = 2, 3$, a description of the universal objects of these classes will be presented.

An important class of $n$-valued $n$-algebraic groups is given by the groups $\mathbb{G}_n$ over the field of complex numbers $\mathbb{C}$. It will be shown that the $n$-valued multiplication $x * y = [z_1,\ldots,z_n]$ in $\mathbb{G}_n$ is realized in terms of the eigenvalues of the Kronecker sum of the companion matrices of the Frobenius polynomials $t^n - x$ and $t^n - y$ in the variable $t$, where $z = t^n$. $(n \times n)$-matrices $W_n(z; x,y)$ will be introduced, whose determinant is an integral homogeneous symmetric polynomial $p_n(z; x,y)$ defining the law $x*y$. The matrix $W_n(1; (-1)^n, 1)$ is the classical Wendt matrix, which was introduced in 1894 in connection with Fermat’s Last Theorem.

The talk is based on the results of the preprints arXiv:2505.04296 and arXiv:2508.04454.