Projective Joint Spectra and Characters of representations of $\tilde{A}_n$

For a tuple of square complex-valued $N\times N$ matrices $A_1,\dots,A_n$ the determinant of their linear combination $x_1A_1+\cdots +x_nA_n$, which is called \textit{a pencil}, is a homogeneous polynomial of degree $N$ in $\C[x_1,...x_n]$. Zero-set of this polynomial is an algebraic set in the projective space $\C\Po^{n-1}$. This set is called the determinantal hypersurface or determinantal manifold of the tuple $(A_1,...,A_n)$. It was shown in Cuckovic, Stessin, Tchernev (2021) that if $G$ is a non-special Coxeter group of type $A,B$, or $D$, $\rho_1$ and $\rho_2$ are two linear representations of $G$, and the determinantal hypersurfaces of images of the Coxeter generators of $G$ under $\rho_1$ and $\rho_2$ coincide as divisors in the projective space, the characters of $\rho_1$ and $\rho_2$ are equal, and, therefore, $\rho_1$ and $\rho_2$ are equivalent. In Peebles, Stessin, Tchernev (in preparation) this result was extended in the characters part to affine Coxeter groups of types $B,C$, and $D$. It was shown there that each such group contains a finite subset such that, if the determinantal hypersurfaces of the images of this set under two finite-dimensional representations coincide as divisors in the projective space, the characters of these representations are equal. Notably, the affine Coxeter groups of $A$ type are not covered by this result, as their combinatorics is quite different.mIn this paper we explicitly construct a finite set in $\tilde{A}_n$ having the same property. We also show that every group which is a semidirect product of a fine group and a finitely generated abelian group contains a finite subset with the similar property: for every finite-dimensonal representation of the group, the determinantal hypersurface of images of the set determines the representation character.