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arXiv · 2401.05291

The miracle of integer eigenvalues

Richard Kenyon·Maxim Kontsevich·Oleg Ogievetsky·Cosmin Pohoata·Will Sawin·Senya Shlosman
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Abstract

For partially ordered sets $X$ we consider the square matrices $M^{X}$ with rows and columns indexed by linear extensions of the partial order on $X$. Each entry $\left( M^{X}\right)_{PQ}$ is a formal variable defined by a pedestal of the linear order $Q$ with respect to linear order $P$. We show that all the eigenvalues of any such matrix $M^{X}$ are $\mathbb{Z}$-linear combinations of those variables.

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Richard Kenyon, Maxim Kontsevich, Oleg Ogievetsky, Cosmin Pohoata, Will Sawin, Senya Shlosman. 2024-03-13. The miracle of integer eigenvalues. https://arxiv.org/abs/2401.05291

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