Necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficients

We give necessary conditions for the positivity of Littlewood-Richardson coefficients and SXP coefficients. We deduce necessary conditions for the positivity of the plethystic coefficients. Explicitly, our main result states that if $S^\lambda(V)$ appears as a summand in the decomposition into irreducibles of $S^\mu(S^\nu(V))$, then $\nu$'s diagram is contained in $\lambda$'s diagram.Comment: 11 pages, 7 figure

Positivity of the symmetric group characters is as hard as the polynomial time hierarchy

We prove that deciding the vanishing of the character of the symmetric group is $C_=P$-complete. We use this hardness result to prove that the the square of the character is not contained in $\#P$, unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof we conclude that deciding positivity of the character is $PP$-complete under many-one reductions, and hence $PH$-hard under Turing-reductions.Comment: 15 pages, 1 figur

The Newton polytope of the Kronecker product

We study the Kronecker product of two Schur functions $s_\lambda\ast s_\mu$, defined as the image of the characteristic map of the product of two $S_n$ irreducible characters. We prove special cases of a conjecture of Monical--Tokcan--Yong that its monomial expansion has a saturated Newton polytope. Our proofs employ the Horn inequalities for positivity of Littlewood-Richardson coefficients and imply necessary conditions for the positivity of Kronecker coefficients.Comment: 25 page

Condiciones de positividad y sucesiones gancho+columna de coeficientes pletísticos

In this work we aim to study and better understand the coefficients of the plethystic operation on the symmetric functions. We will try to be as self-contained as possible, beginning with the basic definitions of symmetric functions and some proven formulas on them. We give a condition on the positivity of the coefficient [sμ](pn◦sλ) of the plethysm between a power sum function pn and a Schur function sλ, namely, λ⊆μ. We then study the coefficients of the Schur expansion of a plethystic family of functions sn1◦sn2◦...◦snk=∑aλsλwhenλis a partition of the form (α,2β,1γ) (hook+column). We completely characterize the cases2◦sn◦sm. Fixing aγ, and letting β vary, we associate a sequence of coefficients (a0,a1,...,aβ,...) to each functionf, and prove that f=s2◦s2◦···s2◦sn◦sm always yields a symmetric sequence. Finally, we make some remarks and conjectures regarding unimodality and asymptotic normality of these sequences.En este trabajo pretendemos estudiar y entender la operación pletística sobre las funciones simétricas. Intentaremos ser autocontenidos, empezando por las definiciones más básicas de qué es una función simétrica y enunciando resultados conocidos sobre ellas. Damos una condición de positividad del coeficiente [sμ](pn◦sλ) del pletismo entre una función suma de potencias pn y una función de Schu rsλ, concretamente λ⊆μ. Estudiaremos seguidamente la expansión sobre la base de Schur the una familia pletística de funciones sn1◦sn2◦...◦snk=∑aλsλ para particiones λ de la forma (α,2β,1γ) (gancho+columna). Caracterizamos completamente el casos2◦sn◦sm. Fijando un γ y dejando β variar, asociamos una sucesión finita (a0,a1,...,aβ,...) de coeficientes a cada función f, y probamos que f=s2◦s2◦···s2◦sn◦sm siempre está asociada a una sucesión simétrica. Finalmente, damos algunos comentarios y conjeturas sobre la unimodalidad y la normalidad asintótica de dichas sucesiones.Dans ce travail nous avons l’intention d’étudier et de mieux comprendre le pléthysme des fonctions symétriques. Nous commençons par donner les définitions les plus basiques des fonctions symétriques, et par énoncer les principaux résultats les concernant, pour en donner une présentation complète. Nous donnons une condition sur la positivité du coefficient [sμ](pn ◦sλ) du pléthysme d’une fonction symétrique somme de puissances pn avec une fonction de Schur sλ, plus concrètement, λ ⊆ μ. Puis, nous étudions le développement dans la base de Schur d’une famille de pléthysmes sn1 ◦ sn2 ◦ ... ◦ snk = ∑ aλsλ quand λ est une partition de la forme (α, 2β , 1γ ) (équerre+colonne). Nous donnons une formule explicite pour le cas s2 ◦ sn ◦ sm. En fixant γ et en faisant varier β, on peut associer à chaque fonction f une suite finie (a0, a1, ..., aβ , ...) de ses coefficients. Nous montrons que les suites associées à f =s2 ◦ s2 ◦ · · · s2 ◦ sn ◦ sm sont toujours symétriques. Finalement, nous faisons quelques remarques sur l’unimodalité et normalité asymptotique de ces suites.Universidad de Sevilla. Grado en Matemática

The partition algebra and the plethysm coefficients II: ramified plethysm

The plethysm coefficient $p(\nu, \mu, \lambda)$ is the multiplicity of the Schur function $s_\lambda$ in the plethysm product $s_\nu \circ s_\mu$. In this paper we use Schur--Weyl duality between wreath products of symmetric groups and the ramified partition algebra to interpret an arbitrary plethysm coefficient as the multiplicity of an appropriate composition factor in the restriction of a module for the ramified partition algebra to the partition algebra. This result implies new stability phenomenon for plethysm coefficients when the first parts of $\nu$, $\mu$ and $\lambda$ are all large. In particular, it gives the first positive formula in the case when $\nu$ and $\lambda$ are arbitrary and $\mu$ has one part. Corollaries include new explicit positive formulae and combinatorial interpretations for the plethysm coefficients $p((n-b,b), (m), (mn-r,r))$, and $p((n-b,1^b), (m), (mn-r,r))$ when $m$ and $n$ are large

What is in# P and what is not?

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 algebraic 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

The Computational Complexity of Plethysm Coefficients

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 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