arXiv subjects

Daniel Soskin

Publications and source records attributed to Daniel Soskin.

12 recordsLinked to original sources

arXiv2026

Bounded ratios for Lorentzian polynomials

We study multiplicative inequalities among the coefficients of Lorentzian polynomials through the notion of bounded ratios. Our main structural result completely characterizes the cone of bounded ratios for Lorentzian polynomials of degree $n$ in $k$ variables. We show that the dual of the cone of bounded ratios is generated by equivalence classes of M-convex functions modulo affine functions. For ternary Lorentzian forms of arbitrary degree $n\ge3$, we show that the cone of bounded ratios is generated by triangular ratios and determine the optimal bounding constant of every bounded ratio. Furthermore, we characterize the pairs $(n,k)$ for which the cone of bounded ratios can be computed by tropicalizing products of $n$ linear forms in $k$ variables with nonnegative coefficients.

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arXiv2026

On the largest Littlewood--Richardson coefficient

We study partitions which attain the largest Littlewood-Richardson coefficient. More precisely, we prove that the largest $c^\lambda_{\mu\nu}$ is attained at partitions such that $\mu \subseteq \nu$ and $\nu/\mu$ is a disjoint union of squares. We conjecture that for $n \ge 16$, all largest $c^\lambda_{\mu\nu}$ must satisfy this property. We confirm this conjecture numerically, for $16 \le n \le 45$.

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arXiv2026

Quantum determinants in polynomial time

We give an algebraic branching program of polynomial size which computes Cayley determinant of right quantum matrices. This is a rare example of an efficient computation of a noncommutative determinant, and the first such example for quantum groups. We extend the results to the $q$-Cayley determinant of $q$-right quantum matrices, as well as to their multiparameter generalization. The proofs are entirely combinatorial, as we relate Cayley, Moore and Valiant determinants using bijections/involutions on words. We then employ the celebrated determinant construction of Mahajan and Vinay (SODA'97), to obtain the results.

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arXiv2026

Correlation inequalities for Schur positivity

We generalize the Ahlswede--Daykin inequality (1978) to a Schur positive \emph{ADS inequality}, which also contains the Lam--Postnikov--Pylyavskyy inequality (2007) as a special case. We then present a number of further generalizations and applications. Notably, we resolve Mihalcea's conjecture on log-supermodularity of stable Grothendieck polynomials.

math.CO↗

arXiv2025

Hadamard Products of dual Jacobi-Trudi matrices

We study positivity properties of Hadamard products of Jacobi-Trudi matrices. Mal\'{o} proved that the Hadamard (entrywise) product of two totally positive upper-triangular Toeplitz matrices whose Toeplitz sequences are the coefficient sequences of real-rooted polynomials with nonpositive zeros is again totally positive. Sokal conjectured that this result can be strengthened to total monomial positivity for the Hadamard product of Jacobi-Trudi matrices. In this paper we show that Temperley-Lieb immanants are Schur positive for Hadamard products of Jacobi-Trudi matrices given by ribbon-like skew shapes. In particular, we affirm Sokal's conjecture for minors given by ribbon-like skew shapes. Moreover, we provide a manifestly positive Schur expansion for Temperley-Lieb immanants evaluated on the Hadamard product of Jacobi-Trudi matrices indexed by ribbons. In addition, for the ribbon case, we construct a corresponding representation, offering a representation-theoretic proof of the Schur positivity.

math.CO↗

arXiv2025

Bounded ratios for Lorentzian matrices

We study multiplicative inequalities among entries of Lorentzian matrices, referred to as bounded ratios. These inequalities can be viewed as generalizations of the classical Alexandrov--Fenchel inequalities for mixed volumes. Our main structural result identifies the cone of all bounded ratios on Lorentzian matrices with the dual of the cut cone, a finitely generated integral polyhedral cone extensively studied in metric geometry and graph theory. We examine in detail the pentagonal ratio, which first appears for Lorentzian matrices of size at least five. For Lorentzian matrices of size three, we determine the optimal bounding constants across the entire cone of bounded ratios, obtaining an explicit entropy-like formula. We conjecture that any normalized bounded ratio is, in fact, bounded by 2.

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arXiv2024

Multiplicative Inequalities In Cluster Algebras Of Finite Type

Generalizing the notion of a multiplicative inequality among minors of a totally positive matrix, we describe, over full rank cluster algebras of finite type, the cone of Laurent monomials in cluster variables that are bounded as a real-valued function on the positive locus of the cluster variety. We prove that the extreme rays of this cone are the u-variables of the cluster algebra. Using this description, we prove that all bounded ratios are bounded by 1 and give a sufficient condition for all such ratios to be subtraction free. This allows us to show in Gr(2, n), Gr(3, 6), Gr(3, 7), Gr(3, 8) that every bounded Laurent monomial in Pl\"ucker coordinates factors into a positive integer combination of so-called primitive ratios. In Gr(4, 8) this factorization does not exists, but we provide the full list of extreme rays of the cone of bounded Laurent monomials in Pl\"ucker coordinates.

math.CO↗

arXiv2024

Generalized Diagonals in Positive Semi-Definite Matrices

We describe all inequalities among generalized diagonals in positive semi-definite matrices. These turn out to be governed by a simple partial order on the symmetric group. This provides an analogue of results of Drake, Gerrish, and Skandera on inequalities among generalized diagonals in totally nonnegative matrices.

math.CO↗

arXiv2024

Permanental inequalities for totally positive matrices

We characterize ratios of permanents of (generalized) submatrices which are bounded on the set of all totally positive matrices. This provides a permanental analog of results of Fallat, Gekhtman, and Johnson [{\em Adv.\ Appl.\ Math.} {\bf 30} no.\ 3, (2003) pp.\ 442--470] concerning ratios of matrix minors. We also extend work of Drake, Gerrish, and the first author [{\em Electron.\ J.\ Combin.,} {\bf 11} no.\ 1, (2004) Note 6] by characterizing the differences of monomials in $\mathbb{Z}[x_{1,1},x_{1,2},...,x_{n,n}]$ which evaluate positively on the set of all totally positive $n \times n$ matrices.

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arXiv2023

Pl\"ucker inequalities for weakly separated coordinates in totally nonnegative Grassmannian

We show that the partial sums of the long Pl\"ucker relations for pairs of weakly separated Pl\"ucker coordinates oscillate around $0$ on the totally nonnegative part of the Grassmannian. Our result generalizes the classical oscillating inequalities by Gantmacher--Krein (1941) and recent results on totally nonnegative matrix inequalities by Fallat--Vishwakarma (2023). In fact we obtain a characterization of weak separability, by showing that no other pair of Pl\"ucker coordinates satisfies this property. Weakly separated sets were initially introduced by Leclerc and Zelevinsky and are closely connected with the cluster algebra of the Grassmannian. Moreover, our work connects several fundamental objects such as weak separability, Temperley--Lieb immanants, and Pl\"ucker relations, and provides a very general and natural class of additive determinantal inequalities on the totally nonnegative part of the Grassmannian.

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arXiv2022

Barrett-Johnson inequalities for totally nonnegative matrices

Given a matrix $A$, let $A_{I,J}$ denote the submatrix of $A$ determined by rows $I$ and columns $J$. Fischer's Inequalities state that for each $n \times n$ Hermitian positive semidefinite matrix $A$, and each subset $I$ of $\{1,\dotsc,n\}$ and its complement $I^c$, we have $\det(A) \leq \det(A_{I,I})\det(A_{I^c,I^c})$. Barrett and Johnson (Linear Multilinear Algebra 34, 1993) extended these to state inequalities for sums of products of principal minors whose orders are given by nonincreasing integer sequences $(λ_1,\dotsc,λ_r)$, $(μ_1,\dotsc,μ_s)$ summing to $n$. Specifically, if $λ_1+\cdots+λ_i\leq μ_1+\cdots+μ_i$ for all $i$, then $$ λ_1!\cdotsλ_r! \sum_{(I_1,\dotsc,I_r)} \det(A_{I_1,I_1}) \cdots \det(A_{I_r,I_r}) ~\geq~ μ_1!\cdotsμ_s! \sum_{(J_1,\dotsc,J_s)} \det(A_{J_1,J_1}) \cdots \det(A_{J_s,J_s}), $$ where sums are over sequences of disjoint subsets of $\{1,\dotsc,n\}$ satisfying $|I_k| = λ_k$, $|J_k| = μ_k$. We show that these inequalities hold for totally nonnegative matrices as well.

math.CO↗

arXiv2022

On bounded ratios of minors of totally positive matrices

We provide several examples of bounded Laurent monomials of minors of a totally positive matrix, which can not be factored into a product of so called primitive ratios, thus showing that the conjecture about factorization of bounded ratios stated in [3] by Fallat, Gekhtman, and Johnson does not hold. However, all found examples satisfy subtraction-free conjecture stated also in [3]. In addition, we show that the set of all bounded ratios form a polyhedral cone of dimension $\binom{2n}{n}-2n$.

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