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Small Littlewood-Richardson coefficients [article]

Christian Ikenmeyer
2012 arXiv   pre-print
This graph was first introduced by Bürgisser and Ikenmeyer in arXiv:1204.2484, where its connectedness was proved.  ... 
arXiv:1209.1521v1 fatcat:dgk676z62vcdvdb5psbvzt4tgq

Geometric Complexity Theory and Tensor Rank [article]

Peter Buergisser, Christian Ikenmeyer
2010 arXiv   pre-print
Mulmuley and Sohoni (GCT1 in SICOMP 2001, GCT2 in 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(W_1)× GL(W_2)× GL(W_3) acting on the tensor product
more » ... =W_1⊗ W_2⊗ W_3 of complex finite dimensional vector spaces. Let G_s = SL(W_1)× SL(W_2)× SL(W_3). A key idea from GCT2 is that the irreducible G_s-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 G_s-representations, as suggested in GCT1-2, 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 GCT1-2. 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 G_s-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.
arXiv:1011.1350v1 fatcat:ifxavm6dsndlxifhq6ybceoupa

On McKay's propagation theorem for the Foulkes conjecture [article]

Christian Ikenmeyer
2015 arXiv   pre-print
We translate the main theorem in Tom McKay's paper "On plethysm conjectures of Stanley and Foulkes" (J. Alg. 319, 2008, pp. 2050-2071) to the language of weight spaces and projections onto invariant spaces of tensors, which makes its proof short and elegant.
arXiv:1509.04957v1 fatcat:hcwqewydxrh5poadj4dgo3qquy

The Computational Complexity of Plethysm Coefficients [article]

Nick Fischer, Christian Ikenmeyer
2020 arXiv   pre-print
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).  ...  In two papers, B\"urgisser 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  ... 
arXiv:2002.00788v1 fatcat:5r2dprrxdbfs7gf5emhr5dzzui

Hyperpfaffians and Geometric Complexity Theory [article]

Christian Ikenmeyer, Michael Walter
2020 arXiv   pre-print
The hyperpfaffian polynomial was introduced by Barvinok in 1995 as a natural generalization of the well-known Pfaffian polynomial to higher order tensors. We prove that the hyperpfaffian is the unique smallest degree SL-invariant on the space of higher order tensors. We then study the hyperpfaffian's computational complexity and prove that it is VNP-complete. This disproves a conjecture of Mulmuley in geometric complexity theory about the computational complexity of invariant rings.
arXiv:1912.09389v2 fatcat:5gxeb27warc2bpuq6pbh6hahr4

Karchmer-Wigderson Games for Hazard-free Computation [article]

Christian Ikenmeyer, Balagopal Komarath, Nitin Saurabh
2021 arXiv   pre-print
We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone
more » ... xity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth 2 that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a significant step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity.
arXiv:2107.05128v1 fatcat:a777vs2krnev7gpgyy4jtr45yy

Fundamental invariants of orbit closures [article]

Peter Bürgisser, Christian Ikenmeyer
2015 arXiv   pre-print
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
more » ... ernary 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).
arXiv:1511.02927v2 fatcat:25lw5bcrvbbkhevciakr5b5wgu

What is in #P and what is not? [article]

Christian Ikenmeyer, Igor Pak
2022 arXiv   pre-print
For several classical nonnegative integer functions, we investigate if they are members of the counting complexity class #P or not. We prove #P membership in surprising cases, and in other cases we prove non-membership, relying on standard complexity assumptions or on oracle separations. We initiate the study of the polynomial closure properties of #P on affine varieties, i.e., if all problem instances satisfy algebraic constraints. This is directly linked to classical combinatorial proofs of
more » ... gebraic identities and inequalities. We investigate #TFNP and obtain oracle separations that prove the strict inclusion of #P in all standard syntactic subclasses of #TFNP-1.
arXiv:2204.13149v1 fatcat:mbx6hsldbnf5nku2d7o5gws3we

Equations for GL invariant families of polynomials [article]

Paul Breiding, Christian Ikenmeyer, Mateusz Michałek, Reuven Hodges
2021 arXiv   pre-print
We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer d. It outputs the ideal of that family intersected with the space of homogeneous polynomials of degree d. Our motivation comes from open problems, which ask to find equations for varieties of cubic and quartic symmetroids. The algorithm relies on a database of specific Young tableaux and highest weight polynomials.
more » ... provide the database and the implementation of the database construction algorithm. Moreover, we provide a julia implementation to run the algorithm using the database, so that more varieties of homogeneous polynomials can easily be treated in the future.
arXiv:2110.06608v1 fatcat:dpddnpsjzfcffbd3aq7rkbq5ru

The Computational Complexity of Plethysm Coefficients

Nick Fischer, Christian Ikenmeyer
2020 Computational Complexity  
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).  ...  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  ...  Switching from tensors to polynomials, the role of Kronecker coefficients in Bürgisser & Ikenmeyer (2011) ; is now taken by plethysm coefficients (see e.g., Bläser & Ikenmeyer 2018, Sec. 12.4(i) ).  ... 
doi:10.1007/s00037-020-00198-4 fatcat:kl75seyn2vfatj7eqyfhio6w4y

On the complexity of the permanent in various computational models [article]

Christian Ikenmeyer, J.M. Landsberg
2016 arXiv   pre-print
We answer a question in [Landsberg, Ressayre, 2015], showing the regular determinantal complexity of the determinant det_m is O(m^3). We answer questions in, and generalize results of [Aravind, Joglekar, 2015], showing there is no rank one determinantal expression for perm_m or det_m when m >= 3. Finally we state and prove several "folklore" results relating different models of computation.
arXiv:1610.00159v1 fatcat:mtnhcjo5qvhxbdm5xcoqf6l7ku

The Saxl Conjecture and the Dominance Order [article]

Christian Ikenmeyer
2015 arXiv   pre-print
In 2012 Jan Saxl conjectured that all irreducible representations of the symmetric group occur in the decomposition of the tensor square of the irreducible representation corresponding to the staircase partition. We make progress on this conjecture by proving the occurrence of all those irreducibles which correspond to partitions that are comparable to the staircase partition in the dominance order. Moreover, we use our result to show the occurrence of all irreducibles corresponding to hook
more » ... itions. This generalizes results by Pak, Panova, and Vallejo from 2013.
arXiv:1410.6549v2 fatcat:cysgx2fslfg7jdcr2x4uagfg5i

Deciding Positivity of Littlewood-Richardson Coefficients [article]

Peter Bürgisser, Christian Ikenmeyer
2013 arXiv   pre-print
Starting with Knutson and Tao's hive model (in J. Amer. Math. Soc., 1999) we characterize the Littlewood-Richardson coefficient c_λ,μ^ν of given partitions λ,μ,ν∈ N^n as the number of capacity achieving hive flows on the honeycomb graph. Based on this, we design a polynomial time algorithm for deciding c_λ,μ^ν >0. This algorithm is easy to state and takes O(n^3 ν_1) arithmetic operations and comparisons. We further show that the capacity achieving hive flows can be seen as the vertices of a
more » ... ected graph, which leads to new structural insights into Littlewood-Richardson coefficients.
arXiv:1204.2484v2 fatcat:erpm3pswvfckhjotm5kgd7ukuy

Geometric complexity theory and matrix powering [article]

Fulvio Gesmundo, Christian Ikenmeyer, Greta Panova
2017 arXiv   pre-print
, FOCS 2016 and B\"urgisser, Ikenmeyer Panova, FOCS 2016).  ...  This padding was recently used heavily to show no-go results for the method of shifted partial derivatives (Efremenko, Landsberg, Schenck, Weyman, 2016) and for geometric complexity theory (Ikenmeyer Panova  ... 
arXiv:1611.00827v2 fatcat:fx56mhc4hfcclnden452h5bjsa

On the relative power of reduction notions in arithmetic circuit complexity [article]

Christian Ikenmeyer, Stefan Mengel
2016 arXiv   pre-print
We show that the two main reduction notions in arithmetic circuit complexity, p-projections and c-reductions, differ in power. We do so by showing unconditionally that there are polynomials that are VNP-complete under c-reductions but not under p-projections. We also show that the question of which polynomials are VNP-complete under which type of reductions depends on the underlying field.
arXiv:1609.05942v1 fatcat:f7l3fdoqyvbpdawnp4wappo5xu
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