Christian Ikenmeyer

Geometric complexity theory and tensor rank

Mulmuley and Sohoni [GCT1, SICOMP 2001; GCT2, SICOMP 2008] proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tensors, in particular for matrix multiplication. We thus study specific orbit closure problems for the group G =GL(W1) x GL(W2) x GL(W3) acting on the tensor product W=W1 ⊗ W2 ⊗ W3 of complex finite dimensional vector spaces. Let Gs =SL(W1) x SL(W2) x SL(W3). A key idea from [GCT2] is that the irreducible Gs-representations occurring in the coordinate ring of the G-orbit closure of a stable tensor w ∈ W are exactly those having a nonzero invariant with respect to the stabilizer group of w. However, we prove that by considering Gs-representations, only trivial lower bounds on border rank can be shown. It is thus necessary to study G-representations, which leads to geometric extension problems that are beyond the scope of the subgroup restriction problems emphasized in [GCT1, GCT2] and its follow up papers. We prove a very modest lower bound on the border rank of matrix multiplication tensors using G-representations. This shows at least that the barrier for Gs-representations can be overcome. To advance, we suggest the coarser approach to replace the semigroup of representations of a tensor by its moment polytope. We prove first results towards determining the moment polytopes of matrix multiplication and unit tensors.

Explicit lower bounds via geometric complexity theory

We prove the lower bound R Mm) ≥ 3/2 m2-2 on the border rank of m x m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense of Mulmuley and Sohonis geometric complexity theory (GCT) program. While this bound is weaker than the one recently obtained by Landsberg and Ottaviani, these are the first significant lower bounds obtained within the GCT program. Behind the proof is an explicit description of the highest weight vectors in Symd⊗3 (Cn)* in terms of combinatorial objects, called obstruction designs. This description results from analyzing the process of polarization and Schur-Weyl duality.

Nonvanishing of Kronecker coefficients for rectangular shapes

We prove that for any partition ( 1;:::; d2) of size ‘d there exists k 1 such that the tensor square of the irreducible representation of the symmetric group Sk‘d with respect to the rectangular partition (k‘;:::;k‘) contains the irreducible representation corresponding to the stretched partition (k 1;:::;k d 2). We also prove a related approximate version of this statement in which the stretching factor k is eectively bounded in terms of d. We further discuss the consequences for geometric complexity theory which provided the motivation for this work.

Even partitions in plethysms

Abstract We prove that for all natural numbers k , n , d with k ⩽ d and every partition λ of size kn with at most k parts there exists an irreducible GL d ( C ) -representation of highest weight 2λ in the plethysm Sym k ( Sym 2 n C d ) . This gives an affirmative answer to a conjecture by Weintraub [Steven H. Weintraub, Some observations on plethysms, J. Algebra 129 (1) (1990) 103–114]. Our investigation is motivated by questions of geometric complexity theory and uses ideas from quantum information theory.

Equations for Lower Bounds on Border Rank

We present new methods for determining polynomials in the ideal of the variety of bilinear maps of border rank at most r. We apply these methods to several cases including the case r=6 in the space of bilinear maps . This space of bilinear maps includes the matrix multiplication operator M 2 for 2×2 matrices. We show that these newly obtained polynomials do not vanish on the matrix multiplication operator M 2, which gives a new proof that the border rank of the multiplication of 2×2 matrices is seven. Other examples are considered along with an explanation of how to implement the methods.

Deciding Positivity of Littlewood--Richardson Coefficients

Starting with Knutson and Taos hive model [J. Amer. Math. Soc., 12 (1999), pp. 1055--1090] we characterize the Littlewood--Richardson coefficient

Permanent versus Determinant: Not via Saturations

{c_{\lambda,\mu}^{\nu}}

Rectangular Kronecker Coefficients and Plethysms in Geometric Complexity Theory

of given partitions

Binary determinantal complexity

\lambda,\mu,\nu\in\mathbb{N}^n

No Occurrence Obstructions in Geometric Complexity Theory

as the number of capacity achieving hive flows on the honeycomb graph. Based on this, we design a polynomial time algorithm for deciding