Complete GL(V)-decomposition of the kernel of $\Psi_{6,6}$: the degree-6 equations of the 6th Chow variety

For a complex vector space $V$ with $\dim V \ge 6$, let $\Psi_{6,6}: \mathrm{Sym}^6(\mathrm{Sym}^6 V) \to \mathrm{Sym}^6(\mathrm{Sym}^6 V)$ be the canonical symmetrization self-map obtained by embedding symmetric tensors into $V^{\otimes 36}$, reordering the tensor factors, and projecting back. Cheung, Ikenmeyer, and Mkrtchyan (2017) determined only part of its kernel: among partitions $\lambda$ of 36 with exactly six rows and plethysm multiplicity exactly 1 in $\mathrm{Sym}^6(\mathrm{Sym}^6 V)$, exactly six types occur in $\ker\Psi_{6,6}$. The problem is to determine $\mathrm{mult}_\lambda(\ker\Psi_{6,6})$ for every partition $\lambda$ of 36 with at most six rows — equivalently, the complete decomposition of the degree-6 homogeneous part of the vanishing ideal of the 6th Chow variety as a $GL(V)$-representation.

Background

A partition $\lambda=(\lambda_1\ge\lambda_2\ge\cdots)$ of an integer $n$ is a nonincreasing list of positive integers summing to $n$; it indexes the irreducible polynomial representations ${\lambda}$ of the general linear group $GL(V)$ (Schur modules, obtained by applying the Schur functor ${\lambda}$ to $V$). When a representation is a symmetric power $\mathrm{Sym}^a$ of another symmetric power $\mathrm{Sym}^b V$, its decomposition into irreducibles ${\lambda}$ is called a plethysm, and the multiplicities $\mathrm{mult}_{\lambda}(\mathrm{Sym}^a(\mathrm{Sym}^b V))$ are plethysm coefficients; no positive combinatorial formula (in the spirit of the Littlewood–Richardson rule) is known for them in general. Each isotypic component contains, up to scaling, a unique highest weight vector of type $\lambda$ per copy of ${\lambda}$: a vector fixed by the upper-triangular unipotent group that the diagonal torus rescales by the character $\alpha_1^{\lambda_1}\cdots\alpha_d^{\lambda_d}$. Writing $\mathrm{HWV}_{\lambda}(W)$ for the space of such vectors in $W$, one has $\dim\mathrm{HWV}_{\lambda}(W)=\mathrm{mult}_{\lambda}(W)$.

Foulkes (1950) conjectured that for $a\le b$ the multiplicities in $\mathrm{Sym}^a(\mathrm{Sym}^b V)$ are at most those in $\mathrm{Sym}^b(\mathrm{Sym}^a V)$, every $\lambda$. The conjecture is known for $a\le 2$ (explicit formulas of Thrall 1942), for $a\le 3$ (Dent and Siemons 2000), and for $a=4$ via injectivity of the canonical map $\Psi_{4,4}$ (Müller and Neunhöffer 2005) plus a propagation theorem of McKay (2008): if $\Psi_{a,b-1}$ is injective then so is $\Psi_{a,c}$ for all $c\ge b$. Here $\Psi_{a,b}:\mathrm{Sym}^a(\mathrm{Sym}^b V)\to\mathrm{Sym}^b(\mathrm{Sym}^a V)$ is the $GL(V)$-equivariant map given by embedding symmetric tensors in $V^{\otimes ab}$, reordering tensor factors, and projecting back. Hadamard conjectured $\Psi_{a,b}$ injective for all $a\le b$ (the Foulkes–Howe conjecture, endorsed by Howe in 1987); Müller and Neunhöffer disproved it by computer, finding $\ker\Psi_{5,5}\ne 0$, and Cheung–Ikenmeyer–Mkrtchyan (2017) determined $\ker\Psi_{5,5}$ completely, proved $\Psi_{5,6}$ injective, and thereby established Foulkes' conjecture for $a=5$. When $a=b$ the two sides of Foulkes' inequality coincide, so the interest of $\ker\Psi_{6,6}$ lies elsewhere: by a theorem of Hadamard, $\ker\Psi_{a,b}$ is exactly the degree-$a$ homogeneous part of the vanishing ideal of the $b$-th Chow variety $\mathcal{C}_b$, the $GL(V)$-orbit closure of the product of linear forms $X_1\cdots X_b$ in $\mathrm{Sym}^b V$ — the variety of completely decomposable forms. Chow varieties are studied in algebraic geometry and commutative algebra, and their equations supply candidate obstruction representations in geometric complexity theory, where $\mathcal{C}_b$ is a subvariety of the determinant orbit closure. Cheung–Ikenmeyer–Mkrtchyan showed there are no equations for $\mathcal{C}_6$ in degrees up to 5, and computed part of the degree-6 part of its ideal.

Problem Statement

Determine completely the kernel of the canonical symmetrization map $\Psi_{6,6}:\mathrm{Sym}^6(\mathrm{Sym}^6 V)\longrightarrow\mathrm{Sym}^6(\mathrm{Sym}^6 V),$ where $\Psi_{6,6}$ embeds symmetric tensors into $V^{\otimes 36}$, reorders the tensor factors by swapping the two levels of the $6\times 6$ block grouping, and projects back (with $\dim V\ge 6$). Concretely: for every partition $\lambda$ of 36 with at most six rows, give the multiplicity $\mathrm{mult}_{\lambda}(\ker\Psi_{6,6})\;=\;\dim\ker\big(\Psi_{6,6}\text{ restricted to }\mathrm{HWV}_{\lambda}(\mathrm{Sym}^6(\mathrm{Sym}^6 V))\big),$ i.e. the full decomposition of $\ker\Psi_{6,6}$ into irreducible $GL(V)$-representations ${\lambda}$. Equivalently, determine the degree-6 homogeneous part of the vanishing ideal of the 6th Chow variety $\mathcal{C}_6=\overline{GL(V)\cdot X_1X_2\cdots X_6}\subset\mathrm{Sym}^6 V$ as a $GL(V)$-module. The answer must at minimum be the explicit list of all partitions $\lambda\vdash 36$ with $\mathrm{mult}_{\lambda}(\ker\Psi_{6,6})>0$ together with their multiplicities; it must extend, and be consistent with, the known partial result that the six partitions $(20,7,6,1,1,1)$, $(9,8,8,7,3,1)$, $(16,7,5,3,3,2)$, $(15,9,4,3,3,2)$, $(11,6,6,6,4,3)$, $(11,9,7,3,3,3)$ occur with multiplicity 1 in the kernel and that no other partition with exactly six rows and plethysm multiplicity exactly 1 occurs.

References

1. Man-Wai Cheung, Christian Ikenmeyer, Sevak Mkrtchyan, "Symmetrizing tableaux and the 5th case of the Foulkes conjecture", Journal of Symbolic Computation 80 (2017) 833–843. DOI: 10.1016/j.jsc.2016.09.002, https://doi.org/10.1016/j.jsc.2016.09.002 (also arXiv:1509.03944, https://arxiv.org/abs/1509.03944)
2. H. O. Foulkes, "Concomitants of the Quintic and Sextic Up To Degree Four in the Coefficients of the Ground Form", Journal of the London Mathematical Society s1-25 (1950) 205–209. DOI: 10.1112/jlms/s1-25.3.205, https://doi.org/10.1112/jlms/s1-25.3.205
3. R. M. Thrall, "On Symmetrized Kronecker Powers and the Structure of the Free Lie Ring", American Journal of Mathematics 64 (1942) 371–388. DOI: 10.2307/2371691, https://doi.org/10.2307/2371691
4. Suzie C. Dent, Johannes Siemons, "On a Conjecture of Foulkes", Journal of Algebra 226 (2000) 236–249. DOI: 10.1006/jabr.1999.8169, https://doi.org/10.1006/jabr.1999.8169
5. Jürgen Müller, Max Neunhöffer, "Some Computations Regarding Foulkes' Conjecture", Experimental Mathematics 14 (2005) 277–283. DOI: 10.1080/10586458.2005.10128928, https://doi.org/10.1080/10586458.2005.10128928
6. Tom McKay, "On Plethysm conjectures of Stanley and Foulkes", Journal of Algebra 319 (2008) 2050–2071. DOI: 10.1016/j.jalgebra.2007.12.003, https://doi.org/10.1016/j.jalgebra.2007.12.003
7. Michel Brion, "Stable properties of plethysm: on two conjectures of Foulkes", Manuscripta Mathematica 80 (1993) 347–371. DOI: 10.1007/BF03026558, https://doi.org/10.1007/BF03026558
8. Christian Ikenmeyer, "On McKay's propagation theorem for the Foulkes conjecture", preprint (2015). arXiv:1509.04957, https://arxiv.org/abs/1509.04957
9. Chris Bowman, Rowena Paget, "The partition algebra and the plethysm coefficients I: Stability and Foulkes' conjecture", Journal of Algebra 655 (2024) 110–138. DOI: 10.1016/j.jalgebra.2023.08.042, https://doi.org/10.1016/j.jalgebra.2023.08.042 (also arXiv:1809.08128, https://arxiv.org/abs/1809.08128)
10. Thomas Kahle, Mateusz Michałek, "Obstructions to Combinatorial Formulas for Plethysm", The Electronic Journal of Combinatorics 25(1) (2018) #P1.41. DOI: 10.37236/6597, https://doi.org/10.37236/6597
11. Nick Fischer, Christian Ikenmeyer, "The Computational Complexity of Plethysm Coefficients", Computational Complexity 29 (2020), article 8. DOI: 10.1007/s00037-020-00198-4, https://doi.org/10.1007/s00037-020-00198-4
12. Christian Ikenmeyer, Igor Pak, "What is in #P and what is not?", 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 860–871. DOI: 10.1109/FOCS54457.2022.00087, https://doi.org/10.1109/FOCS54457.2022.00087 (also arXiv:2204.13149, https://arxiv.org/abs/2204.13149)
13. Claudiu Raicu, Steven V. Sam, Jerzy Weyman, "On Some Modules Supported in the Chow Variety", Vietnam Journal of Mathematics 50 (2022) 501–521. DOI: 10.1007/s10013-021-00527-2, https://doi.org/10.1007/s10013-021-00527-2 (also arXiv:2108.10910, https://arxiv.org/abs/2108.10910)
14. Paul Breiding, Reuven Hodges, Christian Ikenmeyer, Mateusz Michałek, "Equations for GL Invariant Families of Polynomials", Vietnam Journal of Mathematics 50 (2022) 545–556. DOI: 10.1007/s10013-022-00549-4, https://doi.org/10.1007/s10013-022-00549-4 (also arXiv:2110.06608, https://arxiv.org/abs/2110.06608)
15. Étienne Tétreault, "A new order on integer partitions", arXiv preprint (first posted 2020, revised 2022). arXiv:2010.15333, https://arxiv.org/abs/2010.15333