Spectra and eigenvectors of the Segre transformation

TL;DR

This paper analyzes the spectral properties of bilinear transformations related to the Segre product of sequences, providing explicit formulas for eigenvalues and eigenvectors, with implications for algebraic structures like Hilbert series.

Contribution

It offers an explicit description of the transformations' spectra and eigenvectors associated with Segre products, revealing their diagonalizability and integral eigenvalues.

Findings

Transformation matrices are diagonalizable with integral eigenvalues.

Explicit formulas for eigenvectors are derived.

Conjecture on the real-rootedness of iterated Segre products' generating functions.

Abstract

Given two sequences $\fa=(a_n)_{n\geq 0}$ and $\fb=(b_n)_{n\geq 0}$ of complex numbers such that their generating series are of the form $\sum_{n\geq 0}a_n t^n=\frac{\fh(\fa)(t)}{(1-t)^{d_{\fa}}}$ and $\sum_{n\geq 0}b_n t^n=\frac{\fh(\fb)(t)}{(1-t)^{d_{\fb}}}$, where $\fh(\fa)(t)$ and $\fh(\fb)(t)$ are polynomials, we consider their Segre product $\fa\ast\fb=(a_nb_n)_{n\geq 0}$. We are interested in the bilinear transformations that compute the coefficient sequence of $\fh(\fa\ast\fb)(t)$ from those of $\fh(\fa)(t)$ and $\fh(\fb)(t)$, where $\sum_{n\geq 0}a_nb_n t^n=\frac{\fh(\fa\ast\fb)(t)}{(1-t)^{d_{\fa}+d_{\fb}-1}}$. The motivation to study this problem comes from commutative algebra as the Hilbert series of the Segre product of two standard graded algebras equals the Segre product of the two individual Hilbert series. We provide an explicit description of these transformations and compute their spectra. In particular, we show that the transformation matrices are diagonalizable with integral eigenvalues. We also provide explicit formulae for the eigenvectors of the transformation matrices. Finally, we present a conjecture concerning the real-rootedness of $\fh(\fa^{\ast r})(t)$ if $r$ is large enough, where $\fa^{\ast r}=\fa\ast\cdots\ast \fa$ is the $r$\textsuperscript{th} Segre product of the sequence $\fa$ and the coefficients of $\fh(\fa)(t)$ are assumed to be non-negative.