We prove the lower bound R(Mm) ≥ 3 2 m 2 − 2 on the border rank of m × m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense the 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 Sym d 3 (C n ) * in terms of combinatorial objects, called obstruction designs. This description results from analyzing the process of polarization and Schur-Weyl duality.

Mulmuley and Sohoni [28, 29] 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) × GL(W2) × GL(W3) acting on the tensor product W = W1 ⊗ W2 ⊗ W3 of complex finite dimensional vector spaces. Let Gs = SL(W1) × SL(W2) × SL(W3). A key idea from [29] is that the irreducible Gsrepresentations 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, as suggested in [28, 29], 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 [28, 29]. 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.

Abstract. We show that the problem of deciding positivity of Kronecker coefficients is NP-hard. Previously, this problem was conjectured to be in P, just as for the Littlewood-Richardson coefficients. Our result establishes in a formal way that Kronecker coefficients are more difficult than Littlewood-Richardson coefficients, unless P = NP. We also show that there exists a #P -formula for a particular subclass of Kronecker coefficients whose positivity is NP-hard to decide. This is an evidence that, despite the hardness of the positivity problem, there may well exist a positive combinatorial formula for the Kronecker coefficients. Finding such a formula is a major open problem in representation theory and algebraic combinatorics. Finally, we consider the existence of the partition triples (λ, µ, π) such that the Kronecker coefficient k λ µ,π = 0 but the Kronecker coefficient k lλ lµ,lπ > 0 for some integer l > 1. Such "holes" are of great interest as they witness the failure of the saturation property for the Kronecker coefficients, which is still poorly understood. Using insight from computational complexity theory, we turn our hardness proof into a positive result: We show that not only do there exist many such triples, but they can also be found efficiently. Specifically, we show that, for any 0 < ≤ 1, there exists 0 < a < 1 such that, for all m, there exist Ω(2 m a ) partition triples (λ, µ, µ) in the Kronecker cone such that: (a) the Kronecker coefficient k λ µ,µ is zero, (b) the height of µ is m, (c) the height of λ is ≤ m , and (d) |λ| = |µ| ≤ m 3 . The proof of the last result illustrates the effectiveness of the explicit proof strategy of GCT.

For several objects of interest in geometric complexity theory, namely for the determinant, the permanent, the product of variables, the power sum, the unit tensor, and the matrix multiplication tensor, we introduce and study a fundamental SL-invariant function that relates the coordinate ring of the orbit with the coordinate ring of its closure. For the power sums we can write down this fundamental invariant explicitly in most cases. Our constructions generalize the two Aronhold invariants on ternary cubics. For the other objects we identify the invariant function conditional on intriguing combinatorial problems much like the well-known Alon-Tarsi conjecture on Latin squares. We provide computer calculations in small dimensions for these cases. As a main tool for our analysis, we determine the stabilizers, and we establish the polystability of all the mentioned forms and tensors (including the generic ones).