The Computational Complexity of Plethysm Coefficients

Abstract

In two papers, Bürgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity. In this paper, we show that deciding positivity of plethysm coefficients is -hard and that computing plethysm coefficients is #-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way, we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017). This makes our work the first to apply techniques from discrete tomography to the study of plethysm coefficients. Quite surprisingly, that interpretation also leads to new equalities between certain plethysm coefficients and Kronecker coefficients.

Equations for GL Invariant Families of Polynomials

“…However, they are known up to d ≤ 5 [23] and for fixed d and c there are algorithms to find such decompositions. For general d and c = 3 even the task of deciding positivity of a λ is NP-hard, see [19].…”

An Evaluation of Supervised Dimensionality Reduction For Large Scale Data

“…When a penalty term is fine-tuned, it creates a new optimization issue for each distinct value of a parameter. In other words, LOLR's computational complexity [8] is a 0 (one zero). ( whether the projection dimensions are computed using a random technique or not.…”

Partial symmetries of iterated plethysms

“…e understanding of the plethystic coe cients is a notoriously hard problem [6,8,13]. However, the presence of symmetries gives us a be er grasp of these coe cients.…”