Suvrit Sra · TU Munich, Department of Mathematics, School of CIT
Papers released version by version, each file timestamped with OpenTimestamps on the day it appears. Every version stays available. Where a paper is also on arXiv, its page links there.
One-paragraph summaries (PDF)Source repositoryarXiv listing
Combinatorics Matrix analysis Complex geometry Mathematical physics Real and complex analysis Probability Notes
View:
A fusion inequality for Macdonald plethysms
We prove that the plethysm $H_m[\widetilde H_\mu]$ of a Macdonald polynomial by the Hall–Littlewood polynomial $H_m$ lies above the Macdonald polynomial $\widetilde H_{m\mu}$ obtained by fusing $m$ copies of $\widetilde H_\mu$. More specifically, the quotient $(H_m[\widetilde H_\mu]-\widetilde H_{m\mu})/(1-q^m)$ is Schur positive with coefficients in $\mathbb N[q,t]$, and it is the character of an explicit module that we call the fusion defect. For one-row shapes the inequality places both $q$-Foulkes plethysms $H_b[H_a]$ and $H_a[H_b]$ above the same polynomial $H_{ab}$, and comparing their two defects we obtain a distorted version of Bergeron's $q$-analog of Foulkes' conjecture, in which the subtracted plethysm is shifted by $q^{a(a-1)\lfloor b/2\rfloor}$.
@misc{sra2026qfoulkes1-v1,
author = {Suvrit Sra},
title = {{The $q$-Foulkes problem I: A fusion inequality for Macdonald plethysms}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/qfoulkes1/v1}},
note = {Version v1, released 26 October 2026},
}Majorization for normalized Jack and Macdonald polynomials
We study the inequality $P_\lambda(x;q,t)/P_\lambda(1^n;q,t)\geq P_\mu(x;q,t)/P_\mu(1^n;q,t)$ for all $x\in\mathbb{R}_{>0}^n$ whenever the partition $\lambda$ majorizes $\mu$; Sra suggested it in 2015, and Chen, Khare and Sahi stated it as a conjecture in 2025. We prove the Jack case for every $n$ from Sun's positive Gelfand–Tsetlin integral and the Dixon–Anderson recursion. For Macdonald polynomials we prove the case $n=2$, the covers $(d)\succeq(d-1,1)$ for every $n$ when $t\leq q$, and, by exact certificates, the case $n=3$ with $|\lambda|\leq 8$. The HCIZ mechanism behind the Jack proof has no direct finite-$q$ analogue: at two explicit pairs $(q,t)$, no positive representation of the normalized ratio has a kernel multiplicative in $\lambda$, and normalizing at the principal specialization $t^\delta$ instead fails the inequality.
@misc{sra2026jackratio-v1,
author = {Suvrit Sra},
title = {{On majorization inequalities for normalized Jack and Macdonald polynomials}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/jackratio/v1}},
note = {Version v1, released 8 October 2026},
}Monotone Turán-ratios of Schur polynomials
We study the Turán-ratio $S_{\lambda,i}(x)^2/(S_{\lambda,i-1}(x)S_{\lambda,i+1}(x))$, where $S_{\lambda,i}(x):=s_\lambda(1^{[n-i]},x^{[i]})$, and $s_\lambda$ is the Schur polynomial in $n$ variables (indexed by partition $\lambda$) with $i$ arguments equal to $x$ and $n-i$ of them equal to $1$. Specifically, we prove that this Turán-ratio is strictly increasing for $x>1$ and $1\le i<n$ (unless all $n$ padded parts of $\lambda$ are equal, then it is identically one). For one-row shapes, our result answers a question of Ait-Haddou about complete homogeneous symmetric polynomials. For this case, we also offer a second proof through the zeros of Meixner polynomials that gives an explicit positive formula for the logarithmic derivative. We conclude with a Turán-ratio conjecture for Macdonald polynomials.
@misc{sra2026sym-turan-v1,
author = {Suvrit Sra},
title = {{Monotonicity of Turán-ratios of symmetric polynomials}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/sym-turan/v1}},
note = {Version v1, released 7 October 2026},
}McLeod's 1959 conjecture
We prove McLeod's (1959) conjecture that $(h_{\ell+r}/h_\ell)^{1/r}$ is subadditive for all $\ell\ge0$ and $r\ge1$, where $h_k$ is the complete homogeneous symmetric polynomial of degree $k$ (J. B. McLeod, "On four inequalities in symmetric functions", Proc. Edinburgh Math. Soc., 1959). We first prove that the ratio $h_k/h_{k-1}$ is convex, and our proof takes an unexpected detour through queuing theory. The geometric mean of $r$ consecutive such ratios equals McLeod's conjectured function, but geometric means of convex functions need not be convex. Our second and decisive step is a tensorization trick that couples neighboring ratios tightly enough to make their geometric mean convex.
@misc{sra2026mcleod-hk-v1,
author = {Suvrit Sra},
title = {{A Proof of McLeod's 1959 Conjecture on Complete Homogeneous Symmetric Ratios}},
year = {2026},
month = oct,
howpublished = {\url{https://arxiv.org/abs/2610.07302v1}},
eprint = {2610.07302v1},
archiveprefix = {arXiv},
note = {Also released as mcleod-hk v1 (5 October 2026) at \url{https://github.com/suvrit/preprints/tree/main/preprints/mcleod-hk/v1}},
}We prove a permanental analogue of Koteljanskii's inequality. If $A$ is an inverse $M$-matrix that becomes symmetric after a positive diagonal similarity, then $\mathrm{per}(A_{S\cup T})\,\mathrm{per}(A_{S\cap T})\ge\mathrm{per}(A_S)\,\mathrm{per}(A_T)$ for all $S,T\subseteq[n]$, where $A_S$ is the principal submatrix indexed by $S$. The proof expresses permanents as moments of a complex Gaussian vector and uses Ginibre's correlation inequality. Finally, an explicit counterexample shows that symmetry cannot be dropped.
@misc{sra2026perm-m-matrix-v1,
author = {Suvrit Sra},
title = {{A Koteljanskii inequality for permanents}},
year = {2026},
month = sep,
howpublished = {\url{https://arxiv.org/abs/2609.39979v1}},
eprint = {2609.39979v1},
archiveprefix = {arXiv},
note = {Also released as perm-m-matrix v1 (2 October 2026) at \url{https://github.com/suvrit/preprints/tree/main/preprints/perm-m-matrix/v1}},
}Let $A$ and $B$ be real symmetric matrices with nonnegative entries. We prove that $\rho(AB)^k\le\rho(A^kB^k)$ for the spectral radius $\rho(\cdot)$ and every integer $k\ge1$; for $B=uu^T$ this is the Blakley–Roy inequality. We prove it via the more general inequality $\|(AB)^m\|^{k}\le\|(A^kB^k)^m\|$ in the spectral norm, valid for all integers $k,m\ge1$. The even exponents case follows from standard tools; for odd exponents $k=2r+1$ the claim follows via the more general product inequality $(u^TC_1\cdots C_tv)^k\le(u^Tu)^r(v^Tv)^r\,u^TC_1^k\cdots C_t^kv$ for nonnegative symmetric $C_i$ and nonnegative $u,v$. Its proof is an entropy argument on paths that traverse each edge $k$ times, back and forth.
@misc{sra2026pourya-rho-v1,
author = {Suvrit Sra},
title = {{A Spectral Blakley--Roy Inequality for Nonnegative Matrices}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/pourya-rho/v1}},
note = {Version v1, released 15 October 2026},
}Tsallis metrics on density matrices
We study symmetrized divergences induced by centroids of $\alpha$-Tsallis relative entropies between density matrices $\sigma$ and $\rho$; up to scaling, the induced divergence equals $|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|$. Earlier work (S. Sra, "Metrics induced by Jensen–Shannon and related divergences on positive definite matrices," Linear Algebra Appl., 2021) studied such divergences and metrics induced by them. That paper allowed centroids to range over the positive definite cone, while we require that the centroid itself be a density matrix. This (more natural) choice makes the metricity in this paper considerably stronger than that of the aforementioned paper. Specifically, we prove that $|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|^{1/2}$ is a metric on density matrices for $1/2\le\alpha\le2$, $\alpha\ne1$. For $\alpha<1$, the proof uses a new four-point theorem for the S-divergence that might be of independent interest. For $\alpha>1$, we first prove the endpoint $\alpha=2$, and then transfer it to the other orders by a tensoring trick.
@misc{sra2026tsallis-v1,
author = {Suvrit Sra},
title = {{Tsallis Metrics on Density Matrices via the S-Divergence}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/tsallis/v1}},
note = {Version v1, released 13 October 2026},
}The $p$-Bures–Wasserstein distance and the Lin–van den Driessche conjecture
We study the $p$-Bures–Wasserstein distance $D_p(A,B)=(\mathrm{tr}\,A^p+\mathrm{tr}\,B^p-2\|A^{1/2}B^{1/2}\|_p^p)^{1/2}$ on positive semidefinite matrices; it equals the Bures–Wasserstein distance for $p=1$, and the Hilbert–Schmidt distance for $p=2$. We prove that $D_p$ is a metric for every $1\le p\le2$, by proving a stronger 2014 conjecture of Lin and van den Driessche on $3\times3$ block matrices. Our main tool is a partition inequality for Schatten norms, proved by complex interpolation. We also characterize Fréchet means for $D_p$ by a fixed-point equation and prove geometric convergence of the associated iteration.
@misc{sra2026pbures-v1,
author = {Suvrit Sra},
title = {{The $p$-Bures--Wasserstein distance via $3\times3$ block matrices (with application to $p$-Bures--Wasserstein barycenters)}},
year = {2026},
month = oct,
howpublished = {\url{https://arxiv.org/abs/2610.15889v1}},
eprint = {2610.15889v1},
archiveprefix = {arXiv},
note = {Also released as pbures v1 (11 October 2026) at \url{https://github.com/suvrit/preprints/tree/main/preprints/pbures/v1}},
}Lin's singular-value conjecture
Let $A$ and $B$ be positive semidefinite, and let $0\le t\le 1$. Lin (2017) conjectured that $\lambda_j((1-t)A+tB)\ge\sigma_j^{1/2}(A^{2(1-t)}B^{2t})$ for $j=1,\ldots,n$. The case $t=1/2$ is Drury's theorem, which answered a long-standing question of Bhatia and Kittaneh. Lin remarked that Drury's method does not seem to reach other values of $t$. We prove his conjecture for every $t\in[0,1]$. The proof builds on Drury's argument, but replaces his block-matrix step (unique to the midpoint $t=1/2$) by an eigenvalue count based on an inertia identity and a one-variable inequality.
@misc{sra2026linconj-v1,
author = {Suvrit Sra},
title = {{On Lin's Arithmetic--Geometric Means Conjecture for Singular Values}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/linconj/v1}},
note = {Version v1, released 8 October 2026},
}Bellman's 1959 claim on Hua–Bellman matrices
We study positive definiteness of Hua-Bellman matrices (i.e., matrices of the form $[\det(I-A_i^*A_j)^{-α}]_{ij}$, where $A_i$ and $A_j$ are strict contractions and $α\in\mathbb{R}$). We start by revisiting a 1959 work of Bellman (Representation theorems and inequalities for Hermitian matrices; Duke Mathematical J., 26(3), 1959) that studies positive definiteness of Hua-Bellman matrices and claims a strengthening of Hua's representation-theoretic results (Inequalities involving determinants; Acta Mathematica Sinica, 5 (1955)). We uncover a critical error in Bellman's proof that has surprisingly escaped notice to date, and we show by an explicit example that his claim itself is false. We then provide conditions under which $\det(I-A^*B)^{-α}$ is a positive definite function. For complex contractions our condition is the known sharp one, with the integer exponents below Hua's range reached by an elementary argument. For real symmetric contractions we recover the part of Bellman's claims that survives. Building on our result, we introduce a new Poincaré-like distance metric on noncommuting strict contractions, and yet another `log-det' based distance in the appendix.
@misc{sra2026hua-v1,
author = {Suvrit Sra},
title = {{Positive definite functions of noncommuting contractions, Hua-Bellman matrices, and a new distance metric}},
year = {2026},
month = sep,
howpublished = {\url{https://arxiv.org/abs/2112.00056v2}},
eprint = {2112.00056v2},
archiveprefix = {arXiv},
note = {Also released as hua v1 (2 October 2026) at \url{https://github.com/suvrit/preprints/tree/main/preprints/hua/v1}},
}Griffiths's 1969 conjecture on the positivity of Schur forms reduces to the positivity of the quantum permanent $\Phi_n$ on positive bistochastic maps. This conjecture was refuted by Du with a map for which $\Phi_9<0$. Therefore, we ask: how negative can $\Phi_n$ be? We prove a lower bound $\Phi_n\ge-3n/(3n^2+20n+103)$ (for $n\ge 8$), and an asymptotically stronger bound $\Phi_n\ge-1/n+(3e^2-7-o(1))\log n/n^2$. The key to our bounds is the orbital polynomial of the map, which turns $\Phi_n$ into a linear function of a distribution on partitions and positivity into linear constraints. We also construct explicit maps for $n=8, 9$ that improve on Du's example. Finally, if $d_*\in\{4,\ldots,8\}$ is the first dimension for which $\Phi_n<0$, the minimum $\mu_n$ of $\Phi_n$ over positive bistochastic maps satisfies $-\mu_n=\Omega(n^{-d_*})$; we conjecture the matching order $-\mu_n=O(n^{-d_*})$.
@misc{sra2026qperm-lower-v1,
author = {Suvrit Sra},
title = {{Lower bounds for the quantum permanent}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/qperm-lower/v1}},
note = {Version v1, released 24 October 2026},
}Schur forms under decomposable curvature
Griffiths's 1969 conjecture predicts positivity of Schur forms for Griffiths-positive vector bundles. Finski reduced its weak form to a sign question for the double mixed discriminant $\Phi_n(H)$ of a positive linear map. Du's rank-nine counterexample shows that $\Phi_n(H)\ge0$ fails in general. We prove $\Phi_n(H)\ge0$ for every decomposable map $H$, a sum of a completely positive and a completely copositive map, using a quaternionic Gaussian determinant representation. Combined with Finski's incidence formula, this inequality gives weak nonnegativity of every Schur form under decomposable curvature, without splitting assumptions. Beyond the decomposable cone, nonnegativity of $\Phi_k(H_{V,U})$ on every square compression is equivalent to $\Phi_n(H+S)\ge0$ for every entanglement-breaking map $S$. This class contains explicit indecomposable maps, but is nonconvex in every size $n\ge9$. We prove the nonconvexity by expressing Du's map as an average of four members, and the obstruction persists when all compression inequalities are strict.
@misc{sra2026qperm-v1,
author = {Suvrit Sra},
title = {{Weak positivity of Schur forms under decomposable curvature}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/qperm/v1}},
note = {Version v1, released 8 October 2026},
}Petz's conjecture on the Kubo–Mori scalar curvature
Petz (1994) conjectured that the scalar curvature of the Kubo–Mori metric on invertible density matrices increases as the state becomes more mixed. Dittmann (2000) reduced the conjecture to the concavity of an explicit symmetric function $h$ on $(0,\infty)^3$. We prove that $h$ is strictly concave, and hence Petz's conjecture. The proof writes $h=-1/(8M)$ with $M$ positive and homogeneous of degree one, and its central idea is to show that $t\mapsto 1/M(X+tY)$ is a Stieltjes function for all $X,Y>0$; Cauchy–Schwarz and homogeneity then give concavity of $h$ on $(0,\infty)^3$.
@misc{sra2026kubo-metric-v1,
author = {Suvrit Sra},
title = {{A Proof of Petz's Conjecture on the Scalar Curvature of the Kubo--Mori Metric}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/kubo-metric/v1}},
note = {Version v1, released 9 October 2026},
}The Gibilisco–Isola conjecture near the Kubo–Mori metric
Gibilisco and Isola conjectured that the scalar curvature of the Wigner–Yanase–Dyson metric with parameter $p$ on the invertible density matrices increases under mixing for all $p$ in some interval $(1,1+\varepsilon)$; the case $p\downarrow1$ is Petz's conjecture for the Kubo–Mori metric (resolved in a companion paper). We prove that for $p\in(1-\varepsilon,1+\varepsilon)$, with $\varepsilon>0$ independent of the dimension, the scalar curvature is a concave function of the eigenvalues and hence increases under mixing. By Dittmann's formula it suffices to prove that one symmetric function $H_p$ of three positive variables is concave, which we do by showing that $-1/H_p$ is a Stieltjes transform along every line in the positive cone, uniformly in $p$ near $1$. The value of $\varepsilon$ is not explicit.
@misc{sra2026wyd-v1,
author = {Suvrit Sra},
title = {{Wigner--Yanase--Dyson Metrics Have Monotone Curvature}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/wyd/v1}},
note = {Version v1, released 9 October 2026},
}Log-concavity and Pólya frequency of order ten for $1/\Xi$
We prove strict log-concavity of the positive functions $\Theta$ and $\Lambda$ with one-sided and bilateral Laplace transforms $1/\xi(\tfrac12+\sqrt s)$ and $1/\Xi$, respectively, where $\Xi(z)=\xi(\tfrac12+iz)$. For either function, log-concavity is the Pólya frequency property of order two (PF$_2$). Without assuming RH, but using known zero verification through height $3\cdot10^{12}$, we further prove that $\Lambda$ is a PF function of order ten (PF$_{10}$). We also prove local PF positivity for $\Theta$: for each fixed order, the ordered minors of $\Theta(x-y)$ are positive when all differences $x_i-y_j$ are sufficiently small and positive. Additionally, we prove the generalized Laguerre inequalities: if RH holds up to height $T>1/2$, then $L_n[\Xi](x)\ge0$ for every $n\ge0$ and $|x|\le T-1/2$; the case $n=1$ is due to Csordas, Ruttan and Varga (1991).
@misc{sra2026riem-xi-v1,
author = {Suvrit Sra},
title = {{On some positivity properties of the Riemann $\Xi$ function}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/riem-xi/v1}},
note = {Version v1, released 20 October 2026},
}Reverse regularity of the square-root theta kernel and Csordas's conjecture
We prove that the Hankel kernel $\Phi(\sqrt{x+y})$, where $\Phi$ is the Jacobi theta kernel whose cosine transform is the Riemann $\Xi$-function, is strictly reverse-regular through order four. The order is sharp: an interval enclosure of the fifth Hankel–Wronskian at $t=10^{-2}$ shows failure at order five. The third-order condition is Csordas's conjecture that $s'^2-ss''$ is strictly log-concave for $s(t)=\Phi(\sqrt t)$. The proof reduces the kernel minors to Wronskian signs and establishes the signs by exact polynomial estimates for $t\ge1$ and interval arithmetic on $[0,1]$. The fourth-order sign also yields three positive iterates of $s\mapsto s'^2-ss''$, the triple Turán inequalities for the normalized Taylor coefficients of $\Xi$, and quartic integral inequalities for $\Xi$.
@misc{sra2026csordas-v1,
author = {Suvrit Sra},
title = {{Sharp order-four sign regularity of a Jacobi theta kernel}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/csordas/v1}},
note = {Version v1, released 10 October 2026},
}A determinant inequality for matrix Gaussian series
We prove a functional extension of Gaussian operator-norm small-ball inequalities. Let $X=\sum_{i=1}^d g_iB_i$, where $g\sim N(0,I_d)$, the matrices $B_i\in\mathbb{R}^{d\times d}$ are symmetric, and $\sum_iB_i^2\preceq I_d$. For every even log-concave $f\colon\mathbb{R}\to[0,\infty)$, we obtain $\mathbb{E}\det f(X)\ge(\mathbb{E} f(\kappa G))^d$ at dilation $\kappa=13/5$, where $G$ is standard Gaussian. The proof reduces each function to one small-ball estimate, obtained from matrix-weighted Poincaré inequalities, and controls the rest by averages over the spectrum. We give an equivalent entropy formulation. An explicit counterexample shows that no universal dilation below $\sqrt{\pi/2}$ can work; we conjecture that this value is optimal.
@misc{sra2026gsd-v1,
author = {Suvrit Sra},
title = {{Beyond Small Balls: A Determinant Inequality for Functions of Matrix Gaussian Series}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/gsd/v1}},
note = {Version v1, released 19 October 2026},
}A fusion inequality for Macdonald plethysms
We prove that the plethysm $H_m[\widetilde H_\mu]$ of a Macdonald polynomial by the Hall–Littlewood polynomial $H_m$ lies above the Macdonald polynomial $\widetilde H_{m\mu}$ obtained by fusing $m$ copies of $\widetilde H_\mu$. More specifically, the quotient $(H_m[\widetilde H_\mu]-\widetilde H_{m\mu})/(1-q^m)$ is Schur positive with coefficients in $\mathbb N[q,t]$, and it is the character of an explicit module that we call the fusion defect. For one-row shapes the inequality places both $q$-Foulkes plethysms $H_b[H_a]$ and $H_a[H_b]$ above the same polynomial $H_{ab}$, and comparing their two defects we obtain a distorted version of Bergeron's $q$-analog of Foulkes' conjecture, in which the subtracted plethysm is shifted by $q^{a(a-1)\lfloor b/2\rfloor}$.
@misc{sra2026qfoulkes1-v1,
author = {Suvrit Sra},
title = {{The $q$-Foulkes problem I: A fusion inequality for Macdonald plethysms}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/qfoulkes1/v1}},
note = {Version v1, released 26 October 2026},
}Griffiths's 1969 conjecture on the positivity of Schur forms reduces to the positivity of the quantum permanent $\Phi_n$ on positive bistochastic maps. This conjecture was refuted by Du with a map for which $\Phi_9<0$. Therefore, we ask: how negative can $\Phi_n$ be? We prove a lower bound $\Phi_n\ge-3n/(3n^2+20n+103)$ (for $n\ge 8$), and an asymptotically stronger bound $\Phi_n\ge-1/n+(3e^2-7-o(1))\log n/n^2$. The key to our bounds is the orbital polynomial of the map, which turns $\Phi_n$ into a linear function of a distribution on partitions and positivity into linear constraints. We also construct explicit maps for $n=8, 9$ that improve on Du's example. Finally, if $d_*\in\{4,\ldots,8\}$ is the first dimension for which $\Phi_n<0$, the minimum $\mu_n$ of $\Phi_n$ over positive bistochastic maps satisfies $-\mu_n=\Omega(n^{-d_*})$; we conjecture the matching order $-\mu_n=O(n^{-d_*})$.
@misc{sra2026qperm-lower-v1,
author = {Suvrit Sra},
title = {{Lower bounds for the quantum permanent}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/qperm-lower/v1}},
note = {Version v1, released 24 October 2026},
}Log-concavity and Pólya frequency of order ten for $1/\Xi$
We prove strict log-concavity of the positive functions $\Theta$ and $\Lambda$ with one-sided and bilateral Laplace transforms $1/\xi(\tfrac12+\sqrt s)$ and $1/\Xi$, respectively, where $\Xi(z)=\xi(\tfrac12+iz)$. For either function, log-concavity is the Pólya frequency property of order two (PF$_2$). Without assuming RH, but using known zero verification through height $3\cdot10^{12}$, we further prove that $\Lambda$ is a PF function of order ten (PF$_{10}$). We also prove local PF positivity for $\Theta$: for each fixed order, the ordered minors of $\Theta(x-y)$ are positive when all differences $x_i-y_j$ are sufficiently small and positive. Additionally, we prove the generalized Laguerre inequalities: if RH holds up to height $T>1/2$, then $L_n[\Xi](x)\ge0$ for every $n\ge0$ and $|x|\le T-1/2$; the case $n=1$ is due to Csordas, Ruttan and Varga (1991).
@misc{sra2026riem-xi-v1,
author = {Suvrit Sra},
title = {{On some positivity properties of the Riemann $\Xi$ function}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/riem-xi/v1}},
note = {Version v1, released 20 October 2026},
}A determinant inequality for matrix Gaussian series
We prove a functional extension of Gaussian operator-norm small-ball inequalities. Let $X=\sum_{i=1}^d g_iB_i$, where $g\sim N(0,I_d)$, the matrices $B_i\in\mathbb{R}^{d\times d}$ are symmetric, and $\sum_iB_i^2\preceq I_d$. For every even log-concave $f\colon\mathbb{R}\to[0,\infty)$, we obtain $\mathbb{E}\det f(X)\ge(\mathbb{E} f(\kappa G))^d$ at dilation $\kappa=13/5$, where $G$ is standard Gaussian. The proof reduces each function to one small-ball estimate, obtained from matrix-weighted Poincaré inequalities, and controls the rest by averages over the spectrum. We give an equivalent entropy formulation. An explicit counterexample shows that no universal dilation below $\sqrt{\pi/2}$ can work; we conjecture that this value is optimal.
@misc{sra2026gsd-v1,
author = {Suvrit Sra},
title = {{Beyond Small Balls: A Determinant Inequality for Functions of Matrix Gaussian Series}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/gsd/v1}},
note = {Version v1, released 19 October 2026},
}Let $A$ and $B$ be real symmetric matrices with nonnegative entries. We prove that $\rho(AB)^k\le\rho(A^kB^k)$ for the spectral radius $\rho(\cdot)$ and every integer $k\ge1$; for $B=uu^T$ this is the Blakley–Roy inequality. We prove it via the more general inequality $\|(AB)^m\|^{k}\le\|(A^kB^k)^m\|$ in the spectral norm, valid for all integers $k,m\ge1$. The even exponents case follows from standard tools; for odd exponents $k=2r+1$ the claim follows via the more general product inequality $(u^TC_1\cdots C_tv)^k\le(u^Tu)^r(v^Tv)^r\,u^TC_1^k\cdots C_t^kv$ for nonnegative symmetric $C_i$ and nonnegative $u,v$. Its proof is an entropy argument on paths that traverse each edge $k$ times, back and forth.
@misc{sra2026pourya-rho-v1,
author = {Suvrit Sra},
title = {{A Spectral Blakley--Roy Inequality for Nonnegative Matrices}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/pourya-rho/v1}},
note = {Version v1, released 15 October 2026},
}Tsallis metrics on density matrices
We study symmetrized divergences induced by centroids of $\alpha$-Tsallis relative entropies between density matrices $\sigma$ and $\rho$; up to scaling, the induced divergence equals $|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|$. Earlier work (S. Sra, "Metrics induced by Jensen–Shannon and related divergences on positive definite matrices," Linear Algebra Appl., 2021) studied such divergences and metrics induced by them. That paper allowed centroids to range over the positive definite cone, while we require that the centroid itself be a density matrix. This (more natural) choice makes the metricity in this paper considerably stronger than that of the aforementioned paper. Specifically, we prove that $|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|^{1/2}$ is a metric on density matrices for $1/2\le\alpha\le2$, $\alpha\ne1$. For $\alpha<1$, the proof uses a new four-point theorem for the S-divergence that might be of independent interest. For $\alpha>1$, we first prove the endpoint $\alpha=2$, and then transfer it to the other orders by a tensoring trick.
@misc{sra2026tsallis-v1,
author = {Suvrit Sra},
title = {{Tsallis Metrics on Density Matrices via the S-Divergence}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/tsallis/v1}},
note = {Version v1, released 13 October 2026},
}The $p$-Bures–Wasserstein distance and the Lin–van den Driessche conjecture
We study the $p$-Bures–Wasserstein distance $D_p(A,B)=(\mathrm{tr}\,A^p+\mathrm{tr}\,B^p-2\|A^{1/2}B^{1/2}\|_p^p)^{1/2}$ on positive semidefinite matrices; it equals the Bures–Wasserstein distance for $p=1$, and the Hilbert–Schmidt distance for $p=2$. We prove that $D_p$ is a metric for every $1\le p\le2$, by proving a stronger 2014 conjecture of Lin and van den Driessche on $3\times3$ block matrices. Our main tool is a partition inequality for Schatten norms, proved by complex interpolation. We also characterize Fréchet means for $D_p$ by a fixed-point equation and prove geometric convergence of the associated iteration.
@misc{sra2026pbures-v1,
author = {Suvrit Sra},
title = {{The $p$-Bures--Wasserstein distance via $3\times3$ block matrices (with application to $p$-Bures--Wasserstein barycenters)}},
year = {2026},
month = oct,
howpublished = {\url{https://arxiv.org/abs/2610.15889v1}},
eprint = {2610.15889v1},
archiveprefix = {arXiv},
note = {Also released as pbures v1 (11 October 2026) at \url{https://github.com/suvrit/preprints/tree/main/preprints/pbures/v1}},
}Reverse regularity of the square-root theta kernel and Csordas's conjecture
We prove that the Hankel kernel $\Phi(\sqrt{x+y})$, where $\Phi$ is the Jacobi theta kernel whose cosine transform is the Riemann $\Xi$-function, is strictly reverse-regular through order four. The order is sharp: an interval enclosure of the fifth Hankel–Wronskian at $t=10^{-2}$ shows failure at order five. The third-order condition is Csordas's conjecture that $s'^2-ss''$ is strictly log-concave for $s(t)=\Phi(\sqrt t)$. The proof reduces the kernel minors to Wronskian signs and establishes the signs by exact polynomial estimates for $t\ge1$ and interval arithmetic on $[0,1]$. The fourth-order sign also yields three positive iterates of $s\mapsto s'^2-ss''$, the triple Turán inequalities for the normalized Taylor coefficients of $\Xi$, and quartic integral inequalities for $\Xi$.
@misc{sra2026csordas-v1,
author = {Suvrit Sra},
title = {{Sharp order-four sign regularity of a Jacobi theta kernel}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/csordas/v1}},
note = {Version v1, released 10 October 2026},
}Petz's conjecture on the Kubo–Mori scalar curvature
Petz (1994) conjectured that the scalar curvature of the Kubo–Mori metric on invertible density matrices increases as the state becomes more mixed. Dittmann (2000) reduced the conjecture to the concavity of an explicit symmetric function $h$ on $(0,\infty)^3$. We prove that $h$ is strictly concave, and hence Petz's conjecture. The proof writes $h=-1/(8M)$ with $M$ positive and homogeneous of degree one, and its central idea is to show that $t\mapsto 1/M(X+tY)$ is a Stieltjes function for all $X,Y>0$; Cauchy–Schwarz and homogeneity then give concavity of $h$ on $(0,\infty)^3$.
@misc{sra2026kubo-metric-v1,
author = {Suvrit Sra},
title = {{A Proof of Petz's Conjecture on the Scalar Curvature of the Kubo--Mori Metric}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/kubo-metric/v1}},
note = {Version v1, released 9 October 2026},
}The Gibilisco–Isola conjecture near the Kubo–Mori metric
Gibilisco and Isola conjectured that the scalar curvature of the Wigner–Yanase–Dyson metric with parameter $p$ on the invertible density matrices increases under mixing for all $p$ in some interval $(1,1+\varepsilon)$; the case $p\downarrow1$ is Petz's conjecture for the Kubo–Mori metric (resolved in a companion paper). We prove that for $p\in(1-\varepsilon,1+\varepsilon)$, with $\varepsilon>0$ independent of the dimension, the scalar curvature is a concave function of the eigenvalues and hence increases under mixing. By Dittmann's formula it suffices to prove that one symmetric function $H_p$ of three positive variables is concave, which we do by showing that $-1/H_p$ is a Stieltjes transform along every line in the positive cone, uniformly in $p$ near $1$. The value of $\varepsilon$ is not explicit.
@misc{sra2026wyd-v1,
author = {Suvrit Sra},
title = {{Wigner--Yanase--Dyson Metrics Have Monotone Curvature}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/wyd/v1}},
note = {Version v1, released 9 October 2026},
}Majorization for normalized Jack and Macdonald polynomials
We study the inequality $P_\lambda(x;q,t)/P_\lambda(1^n;q,t)\geq P_\mu(x;q,t)/P_\mu(1^n;q,t)$ for all $x\in\mathbb{R}_{>0}^n$ whenever the partition $\lambda$ majorizes $\mu$; Sra suggested it in 2015, and Chen, Khare and Sahi stated it as a conjecture in 2025. We prove the Jack case for every $n$ from Sun's positive Gelfand–Tsetlin integral and the Dixon–Anderson recursion. For Macdonald polynomials we prove the case $n=2$, the covers $(d)\succeq(d-1,1)$ for every $n$ when $t\leq q$, and, by exact certificates, the case $n=3$ with $|\lambda|\leq 8$. The HCIZ mechanism behind the Jack proof has no direct finite-$q$ analogue: at two explicit pairs $(q,t)$, no positive representation of the normalized ratio has a kernel multiplicative in $\lambda$, and normalizing at the principal specialization $t^\delta$ instead fails the inequality.
@misc{sra2026jackratio-v1,
author = {Suvrit Sra},
title = {{On majorization inequalities for normalized Jack and Macdonald polynomials}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/jackratio/v1}},
note = {Version v1, released 8 October 2026},
}Lin's singular-value conjecture
Let $A$ and $B$ be positive semidefinite, and let $0\le t\le 1$. Lin (2017) conjectured that $\lambda_j((1-t)A+tB)\ge\sigma_j^{1/2}(A^{2(1-t)}B^{2t})$ for $j=1,\ldots,n$. The case $t=1/2$ is Drury's theorem, which answered a long-standing question of Bhatia and Kittaneh. Lin remarked that Drury's method does not seem to reach other values of $t$. We prove his conjecture for every $t\in[0,1]$. The proof builds on Drury's argument, but replaces his block-matrix step (unique to the midpoint $t=1/2$) by an eigenvalue count based on an inertia identity and a one-variable inequality.
@misc{sra2026linconj-v1,
author = {Suvrit Sra},
title = {{On Lin's Arithmetic--Geometric Means Conjecture for Singular Values}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/linconj/v1}},
note = {Version v1, released 8 October 2026},
}Schur forms under decomposable curvature
Griffiths's 1969 conjecture predicts positivity of Schur forms for Griffiths-positive vector bundles. Finski reduced its weak form to a sign question for the double mixed discriminant $\Phi_n(H)$ of a positive linear map. Du's rank-nine counterexample shows that $\Phi_n(H)\ge0$ fails in general. We prove $\Phi_n(H)\ge0$ for every decomposable map $H$, a sum of a completely positive and a completely copositive map, using a quaternionic Gaussian determinant representation. Combined with Finski's incidence formula, this inequality gives weak nonnegativity of every Schur form under decomposable curvature, without splitting assumptions. Beyond the decomposable cone, nonnegativity of $\Phi_k(H_{V,U})$ on every square compression is equivalent to $\Phi_n(H+S)\ge0$ for every entanglement-breaking map $S$. This class contains explicit indecomposable maps, but is nonconvex in every size $n\ge9$. We prove the nonconvexity by expressing Du's map as an average of four members, and the obstruction persists when all compression inequalities are strict.
@misc{sra2026qperm-v1,
author = {Suvrit Sra},
title = {{Weak positivity of Schur forms under decomposable curvature}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/qperm/v1}},
note = {Version v1, released 8 October 2026},
}Monotone Turán-ratios of Schur polynomials
We study the Turán-ratio $S_{\lambda,i}(x)^2/(S_{\lambda,i-1}(x)S_{\lambda,i+1}(x))$, where $S_{\lambda,i}(x):=s_\lambda(1^{[n-i]},x^{[i]})$, and $s_\lambda$ is the Schur polynomial in $n$ variables (indexed by partition $\lambda$) with $i$ arguments equal to $x$ and $n-i$ of them equal to $1$. Specifically, we prove that this Turán-ratio is strictly increasing for $x>1$ and $1\le i<n$ (unless all $n$ padded parts of $\lambda$ are equal, then it is identically one). For one-row shapes, our result answers a question of Ait-Haddou about complete homogeneous symmetric polynomials. For this case, we also offer a second proof through the zeros of Meixner polynomials that gives an explicit positive formula for the logarithmic derivative. We conclude with a Turán-ratio conjecture for Macdonald polynomials.
@misc{sra2026sym-turan-v1,
author = {Suvrit Sra},
title = {{Monotonicity of Turán-ratios of symmetric polynomials}},
year = {2026},
month = oct,
howpublished = {Preprint, \url{https://github.com/suvrit/preprints/tree/main/preprints/sym-turan/v1}},
note = {Version v1, released 7 October 2026},
}McLeod's 1959 conjecture
We prove McLeod's (1959) conjecture that $(h_{\ell+r}/h_\ell)^{1/r}$ is subadditive for all $\ell\ge0$ and $r\ge1$, where $h_k$ is the complete homogeneous symmetric polynomial of degree $k$ (J. B. McLeod, "On four inequalities in symmetric functions", Proc. Edinburgh Math. Soc., 1959). We first prove that the ratio $h_k/h_{k-1}$ is convex, and our proof takes an unexpected detour through queuing theory. The geometric mean of $r$ consecutive such ratios equals McLeod's conjectured function, but geometric means of convex functions need not be convex. Our second and decisive step is a tensorization trick that couples neighboring ratios tightly enough to make their geometric mean convex.
@misc{sra2026mcleod-hk-v1,
author = {Suvrit Sra},
title = {{A Proof of McLeod's 1959 Conjecture on Complete Homogeneous Symmetric Ratios}},
year = {2026},
month = oct,
howpublished = {\url{https://arxiv.org/abs/2610.07302v1}},
eprint = {2610.07302v1},
archiveprefix = {arXiv},
note = {Also released as mcleod-hk v1 (5 October 2026) at \url{https://github.com/suvrit/preprints/tree/main/preprints/mcleod-hk/v1}},
}Bellman's 1959 claim on Hua–Bellman matrices
We study positive definiteness of Hua-Bellman matrices (i.e., matrices of the form $[\det(I-A_i^*A_j)^{-α}]_{ij}$, where $A_i$ and $A_j$ are strict contractions and $α\in\mathbb{R}$). We start by revisiting a 1959 work of Bellman (Representation theorems and inequalities for Hermitian matrices; Duke Mathematical J., 26(3), 1959) that studies positive definiteness of Hua-Bellman matrices and claims a strengthening of Hua's representation-theoretic results (Inequalities involving determinants; Acta Mathematica Sinica, 5 (1955)). We uncover a critical error in Bellman's proof that has surprisingly escaped notice to date, and we show by an explicit example that his claim itself is false. We then provide conditions under which $\det(I-A^*B)^{-α}$ is a positive definite function. For complex contractions our condition is the known sharp one, with the integer exponents below Hua's range reached by an elementary argument. For real symmetric contractions we recover the part of Bellman's claims that survives. Building on our result, we introduce a new Poincaré-like distance metric on noncommuting strict contractions, and yet another `log-det' based distance in the appendix.
@misc{sra2026hua-v1,
author = {Suvrit Sra},
title = {{Positive definite functions of noncommuting contractions, Hua-Bellman matrices, and a new distance metric}},
year = {2026},
month = sep,
howpublished = {\url{https://arxiv.org/abs/2112.00056v2}},
eprint = {2112.00056v2},
archiveprefix = {arXiv},
note = {Also released as hua v1 (2 October 2026) at \url{https://github.com/suvrit/preprints/tree/main/preprints/hua/v1}},
}We prove a permanental analogue of Koteljanskii's inequality. If $A$ is an inverse $M$-matrix that becomes symmetric after a positive diagonal similarity, then $\mathrm{per}(A_{S\cup T})\,\mathrm{per}(A_{S\cap T})\ge\mathrm{per}(A_S)\,\mathrm{per}(A_T)$ for all $S,T\subseteq[n]$, where $A_S$ is the principal submatrix indexed by $S$. The proof expresses permanents as moments of a complex Gaussian vector and uses Ginibre's correlation inequality. Finally, an explicit counterexample shows that symmetry cannot be dropped.
@misc{sra2026perm-m-matrix-v1,
author = {Suvrit Sra},
title = {{A Koteljanskii inequality for permanents}},
year = {2026},
month = sep,
howpublished = {\url{https://arxiv.org/abs/2609.39979v1}},
eprint = {2609.39979v1},
archiveprefix = {arXiv},
note = {Also released as perm-m-matrix v1 (2 October 2026) at \url{https://github.com/suvrit/preprints/tree/main/preprints/perm-m-matrix/v1}},
}