Universality of high-strength tensors

Abstract

A theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the same degree in a bounded number of variables. Using entirely different techniques, we extend this theorem to arbitrary polynomial functors. As a corollary of our work, we show that specialisation 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.

Groups GL(∞) over finite fields and multiplications of double cosets

“…Remarks. 1) Theorem 3.1 is a very special case of Harman, Snowden [10], on cubic spaces see also [13], [3], [4]. In our degree of generality, the statement admits a trivial probabilistic proof, we need probabilistic construction and present it in the next division.…”

Stable representation theory: beyond the classical groups

“…Let π n : A λ → A λ {K n } be the natural map. By [BDDE,Corollary 2.6.3], the restriction of π n to GL • x is surjective on k-points. We can thus find g n ∈ GL such that π n (g n x) = ỹ.…”