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Universality of high-strength tensors. (English) Zbl 1490.14097

In the last few years several conjectures and new results were proved for form of fixed degree when the number of variables go to infinity. In the paper over an algegraically closed field \(K\) with characteristic zero. Take a degree \(d\) form \(f\in K[x_1,\dots ,x_n]\), \(f\ne 0\). The strength \(\mathrm{str}(f)\) of \(f\) is the minimal positive integer \(k\) such that \(f=g_1h_1+\cdots +g_kh_k\) for some positive degree forms. D. Kazhdan and T. Ziegler [Geom. Funct. Anal. 30, No. 4, 1063–1096 (2020; Zbl 1460.14138)] proved that form of fixed degree \(d\) of sufficiently high strength specializes to any given degree \(d\) form in a bounded number of variables by substituting linear forms for its variables. In this important paper the theorem is extended (in a completely different way) to arbitrary polynomial functors. As a corollary of their theory they get specialization induces a quasi-order on elements in polynomial functors, and that among the elements with a dense orbit there are unique smallest and largest equivalence classes in this quasi-order. The characteristic \(0\) assumption comes from the use of Schur functors.

MSC:

14R20 Group actions on affine varieties
15A21 Canonical forms, reductions, classification
15A69 Multilinear algebra, tensor calculus
20G05 Representation theory for linear algebraic groups

Citations:

Zbl 1460.14138

References:

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