SL(m,C)-equivariant and translation covariant continuous tensor valuations

Abstract

The space of continuous, SL(m, C)-equivariant, m ≥ 2, and translation covariant valuations taking values in the space of real symmetric tensors on C m ∼ = R 2m of rank r ≥ 0 is completely described. The classification involves the moment tensor valuation for r ≥ 1 and is analogous to the known classification of the corresponding tensor valuations that are SL(2m, R)equivariant, although the method of proof cannot be adapted. * AMS 2010 subject classification: Primary 52B45; Secondary 52A40

Geometric valuation theory

“…For classifications of matrix valuation on P n without regularity assumptions, see [106,107], and for tensor valuations on lattice polytopes, see [93]. Continuous tensor valuations on complex vector spaces are classified in [4].…”

SL($n$) contravariant vector valuations

“…Recently, Zeng and the second author [31] obtained a complete classification of SL(n) covariant vector valuations on P n without any additional assumptions. There are also some interesting characterizations of matrix and tensor valuations (see [1,11,20,25,26]). Surprisingly, classifications of SL(n) contravariant vector valuations are still missing on any spaces and with any conditions.…”