9 research outputs found
Connections between conjectures of Alon-Tarsi, Hadamard-Howe, and integrals over the special unitary group
We show the Alon-Tarsi conjecture on Latin squares is equivalent to a very
special case of a conjecture made independently by Hadamard and Howe, and to
the non-vanishing of some interesting integrals over SU(n). Our investigations
were motivated by geometric complexity theory.Comment: 7 page
Stability of the Levi-Civita tensors and an AlonâTarsi type theorem
We show that the Levi-Civita tensors are semistable in the sense of Geometric Invariant Theory, which is equivalent to an analogue of the AlonâTarsi conjecture on Latin squares. The proof uses the connection of Taoâs slice rank with semistable tensors. We also show an application to an asymptotic saturation-type version of Rotaâs basis conjecture
There are asymptotically the same number of Latin squares of each parity
A Latin square is reduced if its first row and column are in natural order. For Latin squares of a particular order n there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order n â â
Implementing Geometric Complexity Theory: On the Separation of Orbit Closures via Symmetries
Understanding the difference between group orbits and their closures is a key
difficulty in geometric complexity theory (GCT): While the GCT program is set
up to separate certain orbit closures, many beautiful mathematical properties
are only known for the group orbits, in particular close relations with
symmetry groups and invariant spaces, while the orbit closures seem much more
difficult to understand. However, in order to prove lower bounds in algebraic
complexity theory, considering group orbits is not enough.
In this paper we tighten the relationship between the orbit of the power sum
polynomial and its closure, so that we can separate this orbit closure from the
orbit closure of the product of variables by just considering the symmetry
groups of both polynomials and their representation theoretic decomposition
coefficients. In a natural way our construction yields a multiplicity
obstruction that is neither an occurrence obstruction, nor a so-called
vanishing ideal occurrence obstruction. All multiplicity obstructions so far
have been of one of these two types.
Our paper is the first implementation of the ambitious approach that was
originally suggested in the first papers on geometric complexity theory by
Mulmuley and Sohoni (SIAM J Comput 2001, 2008): Before our paper, all existence
proofs of obstructions only took into account the symmetry group of one of the
two polynomials (or tensors) that were to be separated. In our paper the
multiplicity obstruction is obtained by comparing the representation theoretic
decomposition coefficients of both symmetry groups.
Our proof uses a semi-explicit description of the coordinate ring of the
orbit closure of the power sum polynomial in terms of Young tableaux, which
enables its comparison to the coordinate ring of the orbit.Comment: 47 page
On the complexity of evaluating highest weight vectors
Geometric complexity theory (GCT) is an approach towards separating algebraic complexity classes through algebraic geometry and representation theory. Originally Mulmuley and Sohoni proposed (SIAM J Comput 2001, 2008) to use occurrence obstructions to prove Valiant's determinant vs permanent conjecture, but recently B\"urgisser, Ikenmeyer, and Panova (Journal of the AMS 2019) proved this impossible. However, fundamental theorems of algebraic geometry and representation theory grant that every lower bound in GCT can be proved by the use of so-called highest weight vectors (HWVs). In the setting of interest in GCT (namely in the setting of polynomials) we prove the NP-hardness of the evaluation of HWVs in general, and we give efficient algorithms if the treewidth of the corresponding Young-diagram is small, where the point of evaluation is concisely encoded as a noncommutative algebraic branching program! In particular, this gives a large new class of separating functions that can be efficiently evaluated at points with low (border) Waring rank