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Greta Panova

Publications and source records attributed to Greta Panova.

At least 37 records · Page 2Linked to original sources
arXiv2023

Computational Complexity in Algebraic Combinatorics

Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives and brought us some beautiful formulas and combinatorial interpretations. The flagship hook-length formula counts the number of Standard Young Tableaux, which also gives the dimension of the irreducible Specht modules of the Symmetric group. The elegant Littlewood-Richardson rule gives the multiplicities of irreducible GL-modules in the tensor products of GL-modules. Such formulas and rules have inspired large areas of study and development beyond Algebra and Combinatorics, becoming applicable to Integrable Probability and Statistical Mechanics, and Computational Complexity Theory. We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity Theory, that could formally explain the beauty we see and the difficulties we encounter in finding further formulas and ``combinatorial interpretations''. A 85-year-old such problem asks for a positive combinatorial formula for the Kronecker coefficients of the Symmetric group, another one pertains to the plethysm coefficients of the General Linear group. In the opposite direction, the study of Kronecker and plethysm coefficients leads to the disproof of the wishful approach of Geometric Complexity Theory (GCT) towards the resolution of the algebraic P vs NP Millennium problem, the VP vs VNP problem. In order to make GCT work and establish computational complexity lower bounds, we need to understand representation theoretic multiplicities in further detail, possibly asymptotically.

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arXiv2023

On the cross-product conjecture for the number of linear extensions

We prove a weak version of the cross--product conjecture: ${F}(k+1,\ell) {F}(k,\ell+1) \geq (\frac12+\varepsilon) {F}(k,\ell) {F}(k+1,\ell+1)$, where ${F}(k,\ell)$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are $k$ and $\ell$ apart, respectively, and where $\varepsilon>0$ depends on the poset. We also prove the converse inequality and disprove the {generalized cross--product conjecture}. The proofs use geometric inequalities for mixed volumes and combinatorics of words.

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arXiv2023

Effective poset inequalities

We explore inequalities on linear extensions of posets and make them effective in different ways. First, we study the Björner--Wachs inequality and generalize it to inequalities on order polynomials and their $q$-analogues via direct injections and FKG inequalities. Second, we give an injective proof of the Sidorenko inequality with computational complexity significance, namely that the difference is in $\#P$. Third, we generalize the Sidorenko inequality to posets with small chain intersections and give complexity theoretic applications.

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arXiv2023

Complexity and asymptotics of structure constants

Kostka, Littlewood-Richardson, Kronecker, and plethysm coefficients are fundamental quantities in algebraic combinatorics, yet many natural questions about them stay unanswered for more than 80 years. Kronecker and plethysm coefficients lack ``nice formulas'', a notion that can be formalized using computational complexity theory. Beyond formulas and combinatorial interpretations, we can attempt to understand their asymptotic behavior in various regimes, and inequalities they could satisfy. Understanding these quantities has applications beyond combinatorics. On the one hand, the asymptotics of structure constants is closely related to understanding the [limit] behavior of vertex and tiling models in statistical mechanics. More recently, these structure constants have been involved in establishing computational complexity lower bounds and separation of complexity classes like VP vs VNP, the algebraic analogs of P vs NP in arithmetic complexity theory. Here we discuss the outstanding problems related to asymptotics, positivity, and complexity of structure constants focusing mostly on the Kronecker coefficients of the symmetric group and, less so, on the plethysm coefficients. This expository paper is based on the talk presented at the Open Problems in Algebraic Combinatorics coneference in May 2022.

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arXiv2023

All Kronecker coefficients are reduced Kronecker coefficients

We settle the question of where exactly the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of a question by Stanley from 2000 and a question by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. Moreover, as a corollary, we deduce that deciding the positivity of reduced Kronecker coefficients is $NP$-hard, and computing them is $\#P$-hard under parsimonious many-one reductions.

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arXiv2023

Chromatic symmetric functions of Dyck paths and q-rook theory

The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian-Wachs $q$-analogue have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are also curiously related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet (2013). For the $q$-analogue, these results have been generalized by Abreu-Nigro (2020) and Guay-Paquet (private communication), using $q$-hit numbers. Among our main results is a new proof of Guay-Paquet's elegant identity expressing the $q$-CSFs in a CSF basis with $q$-hit coefficients. We further show its equivalence to the Abreu-Nigro identity expanding the $q$-CSF in the elementary symmetric functions. In the course of our work we establish that the $q$-hit numbers in these expansions differ from the originally assumed Garsia-Remmel $q$-hit numbers by certain powers of $q$. We prove new identities for these $q$-hit numbers, and establish connections between the three different variants.

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arXiv2022

Durfee squares, symmetric partitions and bounds on Kronecker coefficients

We resolve two open problems on Kronecker coefficients $g(λ,μ,ν)$ of the symmetric group. First, we prove that for partitions $λ,μ,ν$ with fixed Durfee square size, the Kronecker coefficients grow at most polynomially. Second, we show that the maximal Kronecker coefficients $g(λ,λ,λ)$ for self-conjugate partitions $λ$ grow superexponentially. We also give applications to explicit special cases.

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arXiv2022

The cross-product conjecture for width two posets

The cross--product conjecture (CPC) of Brightwell, Felsner and Trotter (1995) is a two-parameter quadratic inequality for the number of linear extensions of a poset $P= (X, \prec)$ with given value differences on three distinct elements in $X$. We give two different proofs of this inequality for posets of width two. The first proof is algebraic and generalizes CPC to a four-parameter family. The second proof is combinatorial and extends CPC to a $q$-analogue. Further applications include relationships between CPC and other poset inequalities, including a new $q$-analogue of the Kahn--Saks inequality.

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arXiv2022

Extensions of the Kahn--Saks inequality for posets of width two

The Kahn--Saks inequality is a classical result on the number of linear extensions of finite posets. We give a new proof of this inequality for posets of width two using explicit injections of lattice paths. As a consequence we obtain a $q$-analogue, a multivariate generalization and an equality condition in this case. We also discuss the equality conditions of the Kahn--Saks inequality for general posets and prove several implications between conditions conjectured to be equivalent.

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arXiv2021

Sorting probability for large Young diagrams

For a finite poset $P=(X,\prec)$, let $\mathcal{L}_P$ denote the set of linear extensions of $P$. The sorting probability $δ(P)$ is defined as \[δ(P) \, := \, \min_{x,y\in X} \, \bigl| \mathbf{P} \, [L(x)\leq L(y) ] \ - \ \mathbf{P} \, [L(y)\leq L(x) ] \bigr|\,, \] where $L \in \mathcal{L}_P$ is a uniform linear extension of $P$. We give asymptotic upper bounds on sorting probabilities for posets associated with large Young diagrams and large skew Young diagrams, with bounded number of rows.

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arXiv2021

Hook formulas for skew shapes IV. Increasing tableaux and factorial Grothendieck polynomials

We present a new family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes our formulas generalize the classical hook-length formula and Stanley's formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its $q$-analogues, which were studied in previous papers of the series.

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arXiv2021

Chromatic symmetric functions of Dyck paths and q-rook theory (extended abstract)

The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian-Wachs $q$-analogue have important connections to Hessenberg varieties, diagonal harmonics, and LLT polynomials. In the case of, so-called, abelian Dyck paths they are also curiously related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet (2013). For the $q$-analogue, these results have been generalized by Abreu-Nigro (2020) and Guay-Paquet (private communication), using $q$-hit numbers, which are a variant of the ones introduced by Garsia and Remmel. Among our main results is a new proof of Guay-Paquet's elegant identity expressing the $q$-CSFs in a CSF basis with $q$-hit coefficients. We further show its equivalence to the Abreu-Nigro identity expanding the $q$-CSF in the elementary symmetric functions. This is the FPSAC extended abstract version. The full version is at ArXiv: 2104.07599.

math.CO↗
arXiv2020

Hook formulas for skew shapes III. Multivariate and product formulas

We give new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of the certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur functions, extensively studied in the first two papers in the series "Hook formulas for skew shapes" [arxiv:1512.08348, arxiv:1610.04744]. We also apply our technology to obtain determinantal and product formulas for the partition function of certain weighted lozenge tilings, and give various probabilistic and asymptotic applications.

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arXiv2020

Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications

The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In 2015 we gave two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. In this paper we give an elementary proof of Naruse's formula based on the case of border strips. For special border strips, we obtain curious new formulas for the Euler and $q$-Euler numbers in terms of certain Dyck path summations.

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arXiv2020

Breaking down the reduced Kronecker coefficients

We resolve three interrelated problems on \emph{reduced Kronecker coefficients} $\overline{g}(α,β,γ)$. First, we disprove the \emph{saturation property} which states that $\overline{g}(Nα,Nβ,Nγ)>0$ implies $\overline{g}(α,β,γ)>0$ for all $N>1$. Second, we esimate the maximal $\overline{g}(α,β,γ)$, over all $|α|+|β|+|γ| = n$. Finally, we show that computing $\overline{g}(λ,μ,ν)$ is strongly $\# P$-hard, i.e. $\#P$-hard when the input $(λ,μ,ν)$ is in unary.

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arXiv2020

Upper bounds on Kronecker coefficients with few rows

We present three different upper bounds for Kronecker coefficients $g(λ,μ,ν)$ in terms of Kostka numbers, contingency tables and Littlewood--Richardson coefficients. We then give various examples, asymptotic applications, and compare them with existing lower bounds.

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