A Note on Normalizations of Orbit Closures
We give a negative answer to a question by J.M. Landsberg on the nature of normalizations of orbit closures. A counterexample originates from the study of complex, ternary, cubic forms.
math.AG↗
arXiv subjects
Jesko Hüttenhain
Publications and source records attributed to Jesko Hüttenhain.
The Stabilizer of Elementary Symmetric Polynomials
We study the r-th elementary symmetric polynomial in $n$ variables with 2<r<n. There are two kinds of linear transformations on the parameter space that leave this polynomial invariant: Namely, any permutation of the variables and simultaneous scaling by any r-th root of unity. We prove that there are no other linear transformations with this property.
math.AC↗
The boundary of the orbit of the 3 by 3 determinant polynomial
We consider the 3 by 3 determinant polynomial and we describe the limit points of the set of all polynomials obtained from the determinant polynomial by linear change of variables. This answers a question of J. M. Landsberg.
math.AG↗
Permanent versus determinant: not via saturations
Let Det_n denote the closure of the GL_{n^2}(C)-orbit of the determinant polynomial det_n with respect to linear substitution. The highest weights (partitions) of irreducible GL_{n^2}(C)-representations occurring in the coordinate ring of Det_n form a finitely generated monoid S(Det_n). We prove that the saturation of S(Det_n) contains all partitions lambda with length at most n and size divisible by n. This implies that representation theoretic obstructions for the permanent versus determinant problem must be holes of the monoid S(Det_n).
cs.CC↗
Binary Determinantal Complexity
We prove that for writing the 3 by 3 permanent polynomial as a determinant of a matrix consisting only of zeros, ones, and variables as entries, a 7 by 7 matrix is required. Our proof is computer based and uses the enumeration of bipartite graphs. Furthermore, we analyze sequences of polynomials that are determinants of polynomially sized matrices consisting only of zeros, ones, and variables. We show that these are exactly the sequences in the complexity class of constant free polynomially sized (weakly) skew circuits.
cs.CC↗