Invariant rings of the special orthogonal group have nonunimodal h-vectors

2024/06/20 by Aldo Conca, Anurag K. Singh, Conca, Aldo +3 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Matrix Theory and Algorithms #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2406.14439

openalex publication_date 2024/06/20 · openalex created_date 2024/06/22 · openalex updated_date 2026/07/28

Abstract

For K an infinite field of characteristic other than two, consider the action of the special orthogonal group SO_t(K) on a polynomial ring via copies of the regular representation. When K has characteristic zero, Boutot's theorem implies that the invariant ring has rational singularities; when K has positive characteristic, the invariant ring is F-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for SO_t(K) as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example it readily yields the a-invariant and information on the Hilbert series. Indeed, we use this to show that the h-vector of the invariant ring for SO_t(K) need not be unimodal.

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