Abstract |
We investigate various topics in the area of complexity and deformation theory of bilinear maps and answer some questions posed by V. Strassen in his papers [The asymptotic spectrum of tensors, J. reine angew. Math., 384:102–152] and [Degeneration and complexity of bilinear maps. J. reine angew. Math., 413:127–180]. The upper and lower support functionals yield points in the asymptotic spectrum associated with a set of bilinear maps, thus giving necessary conditions for degenerations. We determine the asymptotic growth of the value of the lower support functional on a generic bilinear map of n-dimensional spaces. We then use this to show that, in contrast to the upper support functional, the lower support functional is not additive. Strassen's theory of asymptotic spectra is based on the discovery that the asymptotic preorder allows an extension from the semiring of equivalence classes of bilinear maps to its Grothendieck ring. We prove that the degeneration order does not allow such an extension. In the last section, we study a question in the general setting of a linear algebraic group $G$ operating on a vector space $V$ over a field $k$. For objects in $V$ we define $k$-degeneration $<_k$, $k$-isomorphy $\sim_k$, and isomorphy ~ (defined with respect to the algebraic closure of $k$). We investigate the so-called strong partial order property (SPE): $d <_k f$, $d \sim f$ ⟶ $d \sim_k f$. We prove that (SPE) is always true if $k$ is algebraically closed, real closed, or a $p$-adic field. Moreover, we show that (SPE) is true if $G$ is a torus. We verify (SPE) for the operation of $\mathrmGl(n,k)$ on quadratic forms over finite fields $k$ and over the rationals. Finally, in the setting of bilinear maps, we are able to show that the implication of (SPE) is true for certain direct sums of matrix multiplications. |