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Damir Yeliussizov

Publications and source records attributed to Damir Yeliussizov.

At least 19 recordsLinked to original sources
arXiv2026

On equidistribution theorem for plane partitions

We prove equidistribution of two pairs of statistics on boxed plane partitions: (volume, trace) and (corner-hook volume, number of corners). The proof relies on different 3d visualizations of the corresponding non-intersecting path systems. In particular, we obtain a new visual proof for a volume generating function of plane partitions. We also introduce a new statistic called the cohook area on ordinary partitions, and prove that it is equidistributed with the area of partitions.

math.CO↗
arXiv2026

Inhomogeneous $q$-Whittaker polynomials II: ring theorem and positive specializations

We study inhomogeneous $q$-Whittaker polynomials which extend both $q$-Whittaker and stable Grothendieck polynomials. We prove that inhomogeneous $q$-Whittaker polynomials (in countably many variables) form a basis of certain commutative ring extending the ring of symmetric functions to a subring of its completion. We then describe positive specializations of that ring and relate them with a subset of Macdonald-positive specializations of the ring of symmetric functions. We also show some related probability distributions obtained from positive specializations of inhomogeneous $q$-Whittaker polynomials.

math.CO↗
arXiv2026

Highest weight vectors of tensors

We study highest weight vectors for symmetric and alternating spaces of tensors, whose dimensions are given by generalized Kronecker coefficients. We describe the algebraic relations for classical constructions of corresponding spanning sets of highest weight vectors. The proof is based on important duality that we discover for these highest weight spaces and vectors. As applications of duality, we also give conceptual interpretations to power expansions of Cayley's first hyperdeterminant and its dual exterior form.

math.CO↗
arXiv2025

Flagged Hamel--Goulden formulas

We obtain Hamel--Goulden-type ribbon decomposition determinantal formulas for flagged supersymmetric Schur functions. As an application, we derive corresponding new determinantal formulas dual refined canonical stable Grothendieck polynomials. These results generalize and produce a number of new determinantal formulas for these symmetric functions including Jacobi--Trudi and skew Giambelli-type determinants.

math.CO↗
arXiv2025

Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability

We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter $β=1$ arising in K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of the corresponding Grothendieck polynomial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process), we describe the limiting permuton and fluctuations around it as the order $n$ of the permutation grows to infinity. The fluctuations are of order $n^{\frac13}$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for $β=1$ Grothendieck polynomials, and provide bounds for general $β$.

math.PR↗
arXiv2023

Some unimodal sequences of Kronecker coefficients

We conjecture unimodality for some sequences of generalized Kronecker coefficients and prove it for partitions with at most two columns. The proof is based on a hard Lefschetz property for corresponding highest weight spaces. We also study more general Lefschetz properties, show implications to a higher-dimensional analogue of the Alon--Tarsi conjecture on Latin squares and give related positivity results.

math.CO↗
arXiv2023

Fundamental invariants of tensors, Latin hypercubes, and rectangular Kronecker coefficients

We study polynomial SL-invariants of tensors, mainly focusing on fundamental invariants which are of smallest degrees. In particular, we prove that certain 3-dimensional analogue of the Alon--Tarsi conjecture on Latin cubes considered previously by Bürgisser and Ikenmeyer, implies positivity of (generalized) Kronecker coefficients at rectangular partitions and as a result provides values for degree sequences of fundamental invariants.

math.CO↗
arXiv2022

Determinantal formulas for dual Grothendieck polynomials

We prove Jacobi-Trudi-type determinantal formulas for skew dual Grothendieck polynomials which are $K$-theoretic deformations of Schur polynomials. We also prove a bialternant-type formula analogous to the classical definition of Schur polynomials.

math.CO↗
arXiv2021

Saturation of Rota's basis conjecture

We prove an asymptotic saturation-type version of Rota's basis conjecture. It relies on the connection of Tao's slice rank with unstable tensors from geometric invariant theory.

math.CO↗
arXiv2021

Tensor slice rank and Cayley's first hyperdeterminant

Cayley's first hyperdeterminant is a straightforward generalization of determinants for tensors. We prove that nonzero hyperdeterminants imply lower bounds on some types of tensor ranks. This result applies to the slice rank introduced by Tao and more generally to partition ranks introduced by Naslund. As an application, we show upper bounds on some generalizations of colored sum-free sets based on constraints related to order polytopes.

math.CO↗
arXiv2021

Random plane partitions and corner distributions

We explore some probabilistic applications arising in connections with $K$-theoretic symmetric functions. For instance, we determine certain corner distributions of random lozenge tilings and plane partitions. We also introduce some distributions that are naturally related to the corner growth model. Our main tools are dual symmetric Grothendieck polynomials and normalized Schur functions.

math.CO↗
arXiv2020

Enumeration of plane partitions by descents

We study certain bijection between plane partitions and $\mathbb{N}$-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating functions are similar to classical MacMahon's formulas; one of these statistics is equidistributed with the usual volume. We also show natural connections with the longest increasing subsequences of words.

math.CO↗
arXiv2020

MacMahon's statistics on higher-dimensional partitions

We study some combinatorial properties of higher-dimensional partitions which generalize plane partitions. We present a natural bijection between $d$-dimensional partitions and $d$-dimensional arrays of nonnegative integers. This bijection has a number of important applications. We introduce a statistic on $d$-dimensional partitions, called the corner-hook volume, whose generating function has the formula of MacMahon's conjecture. We obtain multivariable formulas whose specializations give analogues of various formulas known for plane partitions. We also introduce higher-dimensional analogues of dual Grothendieck polynomials which are quasisymmetric functions and whose specializations enumerate higher-dimensional partitions of a given shape. Finally, we show probabilistic connections with a directed last passage percolation model in $\mathbb{Z}^d$.

math.CO↗
arXiv2020

Dual Grothendieck polynomials via last-passage percolation

The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous $K$-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.

math.CO↗
arXiv2020

Positive specializations of symmetric Grothendieck polynomials

It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes the Edrei-Thoma theorem. In this paper we study positive specializations of symmetric Grothendieck polynomials, $K$-theoretic deformations of Schur polynomials.

math.CO↗
arXiv2018

Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs

Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.

math.CO↗
arXiv2018

On the largest Kronecker and Littlewood--Richardson coefficients

We give new bounds and asymptotic estimates for Kronecker and Littlewood--Richardson coefficients. Notably, we resolve Stanley's questions on the shape of partitions attaining the largest Kronecker and Littlewood--Richardson coefficients. We apply the results to asymptotics of the number of standard Young tableaux of skew shapes.

math.CO↗