Plethysm Coefficients Research Articles

The partition algebra and the plethysm coefficients I: Stability and Foulkes' conjecture

We propose a new approach to study plethysm coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. This provides an explanation of the stability properties of plethysm and Kronecker coefficients in a simple and uniform fashion for the first time. We prove the strengthened Foulkes' conjecture for stable plethysm coefficients in an elementary fashion.

Some stable plethysms

In this note, we prove some new stability results for plethysm coefficients. As special cases, we verify a conjecture of Wildon, and show the stability of sequences recently predicted by Bessenrodt, Bowman and Paget [Trans. Amer. Math. Soc. 375 (2022), pp. 5151–5194] to be weakly increasing.

The classification of multiplicity-free plethysms of Schur functions

We classify and construct all multiplicity-free plethystic products of Schur functions. We also compute many new (infinite) families of plethysm coefficients, with particular emphasis on those near maximal in the dominance ordering and those of small Durfee size.

The Computational Complexity of Plethysm Coefficients

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 NP-hard and that computing plethysm coefficients is #P-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.

Cyclic sieving and plethysm coefficients

A combinatorial expression for the coefficient of the Schur function s λ s_{\lambda } in the expansion of the plethysm p n / d d ∘ s μ p_{n/d}^d \circ s_{\mu } is given for all d d dividing n n for the cases in which n = 2 n=2 or λ \lambda is rectangular. In these cases, the coefficient ⟨ p n / d d ∘ s μ , s λ ⟩ \langle p_{n/d}^d \circ s_{\mu }, s_{\lambda } \rangle is shown to count, up to sign, the number of fixed points of an ⟨ s μ n , s λ ⟩ \langle s_{\mu }^n, s_{\lambda } \rangle -element set under the d d th power of an order- n n cyclic action. If n = 2 n=2 , the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if λ \lambda is rectangular, the action is a certain power of Schützenberger and Shimozono’s jeu-de-taquin promotion. This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case n = 2 n=2 is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function.

A geometric approach to the stabilisation of certain sequences of Kronecker coefficients

We give another proof, using tools from Geometric Invariant Theory, of a result due to Sam and Snowden in (J Algebraic Comb 43(1):1–10, 2016), concerning the stability of Kronecker coefficients. This result states that some sequences of Kronecker coefficients eventually stabilise, and our method gives a nice geometric bound from which the stabilisation occurs. We perform the explicit computation of such a bound on two examples, one being the classical case of Murnaghan’s stability. Moreover, we see that our techniques apply to other coefficients arising in representation theory: namely to some plethysm coefficients, as well as multiplicities for tensor products of representations of the hyperoctahedral group.

Rectangular Kronecker coefficients and plethysms in geometric complexity theory

We prove that in the geometric complexity theory program the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial.Moreover, we prove the positivity of rectangular Kronecker coefficients for a large class of partitions where the side lengths of the rectangle are at least quadratic in the length of the partition. We also compare rectangular Kronecker coefficients with their corresponding plethysm coefficients, which leads to a new lower bound for rectangular Kronecker coefficients. Moreover, we prove that the saturation of the rectangular Kronecker semigroup is trivial, we show that the rectangular Kronecker positivity stretching factor is 2 for a long first row, and we completely classify the positivity of rectangular limit Kronecker coefficients that were introduced by Manivel in 2011.

Stability properties of the plethysm: A combinatorial approach

An important family of structural constants in the theory of symmetric functions and in the representation theory of symmetric groups and general linear groups are the plethysm coefficients. In 1950, Foulkes observed that they have some stability properties: certain sequences of plethysm coefficients are eventually constant. Such stability properties were proven by Brion with geometric techniques, and by Thibon and Carré by means of vertex operators.In this paper we present a new approach to prove such stability properties. Our proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients related to the complete homogeneous basis of symmetric functions. We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope.

Rectangular Symmetries for Coefficients of Symmetric Functions

We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share symmetries related to the operations of taking complements with respect to rectangles and adding rectangles.

Stability properties of Plethysm: new approach with combinatorial proofs (Extended abstract)

Plethysm coefficients are important structural constants in the theory of symmetric functions and in the representations theory of symmetric groups and general linear groups. In 1950, Foulkes observed stability properties: some sequences of plethysm coefficients are eventually constants. Such stability properties were proven by Brion with geometric techniques and by Thibon and Carré by means of vertex operators. In this paper we present a newapproach to prove such stability properties. This new proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients (related to the complete homogeneous basis of symmetric functions). We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope. Les coefficients du pléthysme sont des constantes de structure importantes de la théorie des fonctions symétriques, ainsi que de la théorie de la représentation des groupes symétriques et des groupes généraux linéaires. En 1950, Foulkes a observé pour ces coefficients de phénomènes de stabilité: certaines suites de coefficients du pléthysme sont stationnaires. De telles propriétés ont été démontrées par Brion, au moyen de techniques géométriques, et par Thibon et Carré, au moyen d’opérateurs vertex. Dans ce travail, nous présentons une nouvelle approche, purement combinatoire, pour démontrer des propriétés de stabilité de ce type. Nous décomposons les coefficients du pléthysme comme somme alternées de coefficients de pléthysme d’un autre type (liés à la base des fonctions symétriques sommes complètes), qui comptent les points entiers dans des polytopes. Nous démontrons la stabilité des suites de ces coefficients en exhibant des bijections entres les ensembles de points entiers des polytopes correspondants.

Invariance properties for coefficients of symmetric functions

We show that several of the main structural constants for symmetric functions (Littlewood-Richardsoncoefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share invarianceproperties related to the operations of taking complements with respect to rectangles and adding rectangles. Nous montrons que plusieurs des principales constantes de structure de la théorie des fonctions symétriques(les coefficients de Littlewood–Richardson, les coefficients de Kronecker, les coefficients du pléthysme, et les polynômesde Kostka–Foulkes) ont en commun des symétries décrites par des opérations de complémentation dans des rectangleset d’ajout de rectangles pour les partitions qui les étiquettent.

Combinatorial proof for a stability property of plethysm coefficients

Plethysm coefficients are important structural constants in the representation theory of the symmetric groups and general linear groups. Remarkably, some sequences of plethysm coefficients stabilize (they are ultimately constants). In this paper we give a new proof of such a stability property, proved by Brion with geometric representation theory techniques. Our new proof is purely combinatorial: we decompose plethysm coefficients as a alternating sum of terms counting integer points in polytopes, and exhibit bijections between these sets of integer points.

Eigenvalue Distributions of Reduced Density Matrices

Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution's support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.

An algorithm for computing plethysm coefficients

The plethysm of two Schur functions can be expressed as a sum of Schur functions with nonnegative integer coefficients. Current algorithms for computing plethysms are designed to compute the whole expansion. However, in some applications only a few coefficients are of interest. In this work, we develop an algorithm for calculating individual plethysm coefficients. We also give a simple result concerning the zero coefficients which is obtained from the combinatorial properties of the Kostka and the inverse Kostka numbers.

Plethysm and vertex operators

We derive some stability properties and recurrence relations for plethysm coefficients.

Group theoretical classification of states of a molecular system at definite states of its constituent parts

AbstractA method is developed for determining the allowed states of a many‐particle system at definite states of the subsystems (impurity ion complexes in crystals, nucleon clusters in nuclei, polyatomic molecules). The advantage of the method proposed here over those previously developed is that the present one is independent of the number of particles in the subsystems. Depending on the value of spin Sa the subsystems are considered either as bosons or fermions. The representation character expression whose expansion gives all the terms 2s+1Γ(α) with a definite value of total spin of the system is written as the product of permutation and orbital factors. The orbital factor is defined in terms of characters of the local point group representations. Some alternative ways of calculating the permutation factor are discussed: a) with the use of tables of reduction of the unitary transformation group U2Sa+1 in a three‐dimensional rotation group; b) through the plethysm coefficients; and c) in terms of 3nj‐symbols of three‐dimensional rotation group. Some examples of application of the method to various systems are presented.