arXiv subjects

Apoorva Khare

Publications and source records attributed to Apoorva Khare.

At least 37 records · Page 2Linked to original sources

arXiv2023

Probability inequalities for strongly left-invariant metric semigroups/monoids, including all Lie groups

Recently, a general version of the Hoffmann-Jorgensen inequality was shown jointly with Rajaratnam [Ann. Probab. 2017], which (a) improved the result even for real-valued variables, but also (b) simultaneously unified and extended several versions in the Banach space literature, including that by Hitczenko and Montgomery-Smith [Ann. Probab. 2001], as well as special cases and variants of results by Johnson-Schechtman [Ann. Probab. 1989] and Klass-Nowicki [Ann. Probab. 2000], in addition to the original versions by Kahane and Hoffmann-Jorgensen. Moreover, our result with Rajaratnam was in a primitive framework: over all semigroups with a bi-invariant metric; this includes Banach spaces as well as compact and abelian Lie groups. In this note we show the result even more generally: over every semigroup $\mathscr{G}$ with a strongly left- (or right-)invariant metric. We also prove some applications of this inequality over such $\mathscr{G}$, extending Banach space-valued versions by Hitczenko and Montgomery-Smith [Ann. Probab. 2001] and by Hoffmann-Jorgensen [Studia Math. 1974]. Furthermore, we show several other stochastic inequalities - by Ottaviani-Skorohod, Mogul'skii, and Levy-Ottaviani - as well as Levy's equivalence, again over $\mathscr{G}$ as above. This setting of generality for $\mathscr{G}$ subsumes not only semigroups with bi-invariant metric (thus extending the previously shown results), but it also means that these results now hold over all Lie groups (equipped with a left-invariant Riemannian metric). We also explain why this primitive setting of strongly left/right-invariant metric semigroups $\mathscr{G}$ is equivalent to that of left/right-invariant metric monoids $\mathscr{G}_\circ$: each such $\mathscr{G}$ embeds in some $\mathscr{G}_\circ$.

math.PR↗

arXiv2023

The Khinchin-Kahane and Levy inequalities for abelian metric groups, and transfer from normed (abelian semi)groups to Banach spaces

The Khinchin-Kahane inequality is a fundamental result in the probability literature, with the most general version to date holding in Banach spaces. Motivated by modern settings and applications, we generalize this inequality to arbitrary metric groups which are abelian. If instead of abelian one assumes the group's metric to be a norm (i.e., $\mathbb{Z}_{>0}$-homogeneous), then we explain how the inequality improves to the same one as in Banach spaces. This occurs via a "transfer principle" that helps carry over questions involving normed metric groups and abelian normed semigroups into the Banach space framework. This principle also extends the notion of the expectation to random variables with values in arbitrary abelian normed metric semigroups $\mathscr{G}$. We provide additional applications, including studying weakly $\ell_p$ $\mathscr{G}$-valued sequences and related Rademacher series. On a related note, we also formulate a "general" Levy inequality, with two features: (i) It subsumes several known variants in the Banach space literature; and (ii) We show the inequality in the minimal framework required to state it: abelian metric groups.

math.PR↗

arXiv2023

From Cauchy's determinant formula to bosonic and fermionic immanant identities

Cauchy's determinant formula (1841) involving $\det ((1-u_i v_j)^{-1})$ is a fundamental result in symmetric function theory. It has been extended in several directions, including a determinantal extension by Frobenius [J. reine angew. Math. 1882] involving a sum of two geometric series in $u_i v_j$. This theme also resurfaced in a matrix analysis setting in a paper by Horn [Trans. Amer. Math. Soc. 1969] - where the computations are attributed to Loewner - and in recent works by Belton-Guillot-Khare-Putinar [Adv. Math. 2016] and Khare-Tao [Amer. J. Math. 2021]. These formulas were recently unified and extended in [Trans. Amer. Math. Soc. 2022] to arbitrary power series, with commuting/bosonic variables $u_i, v_j$. In this note we formulate analogous permanent identities, and in fact, explain how all of these results are a special case of a more general identity, for any character - in fact, any complex class function - of any finite group that acts on the bosonic variables $u_i$ and on the $v_j$ via signed permutations. (We explain why larger linear groups do not work, via a - perhaps novel - "symmetric function" characterization of signed permutation matrices that holds over any integral domain.) We then provide fermionic analogues of these formulas, as well as of the closely related Cauchy product identities.

math.RT↗

arXiv2023

Recovering affine-linearity of functions from their restrictions to affine lines

Motivated by recent results of Tao-Ziegler [Discrete Anal. 2016] and Greenfeld-Tao (2022 preprint) on concatenating affine-linear functions along subgroups of an abelian group, we show three results on recovering affine-linearity of functions $f : V \to W$ from their restrictions to affine lines, where $V,W$ are $\mathbb{F}$-vector spaces and $\dim V \geqslant 2$. First, if $\dim V < |\mathbb{F}|$ and $f : V \to \mathbb{F}$ is affine-linear when restricted to affine lines parallel to a basis and to certain "generic" lines through $0$, then $f$ is affine-linear on $V$. (This extends to all modules $M$ over unital commutative rings $R$ with large enough characteristic.) Second, we explain how a classical result attributed to von Staudt (1850s) extends beyond bijections: if $f : V \to W$ preserves affine lines $\ell$, and if $f(v) \not\in f(\ell)$ whenever $v \not\in \ell$, then this also suffices to recover affine-linearity on $V$, but up to a field automorphism. In particular, if $\mathbb{F}$ is a prime field $\mathbb{Z}/p\mathbb{Z}$ ($p>2$) or $\mathbb{Q}$, or a completion $\mathbb{Q}_p$ or $\mathbb{R}$, then $f$ is affine-linear on $V$. We then quantitatively refine our first result above, via a weak multiplicative variant of the additive $B_h$-sets initially explored by Singer [Trans. Amer. Math. Soc. 1938], Erdos-Turan [J. London Math. Soc. 1941], and Bose-Chowla [Comment. Math. Helv. 1962]. Weak multiplicative $B_h$-sets occur inside all rings with large enough characteristic, and in all infinite or large enough finite integral domains/fields. We show that if $R$ is among any of these classes of rings, and $M = R^n$ for some $n \geqslant 3$, then one requires affine-linearity on at least $\binom{n}{\lceil n/2 \rceil}$-many generic lines to deduce the global affine-linearity of $f$ on $R^n$. Moreover, this bound is sharp.

math.AC↗

arXiv2022

Preservers of totally positive kernels and Polya frequency functions

Fractional powers and polynomial maps preserving structured totally positive matrices, one-sided Polya frequency functions, or totally positive kernels are treated from a unifying perspective. Besides the stark rigidity of the polynomial transforms, we unveil an ubiquitous separation between discrete and continuous spectra of such inner fractional powers. Classical works of Schoenberg, Karlin, Hirschman, and Widder are completed by our classification. Concepts of probability theory, multivariate statistics, and group representation theory naturally enter into the picture.

math.FA↗

arXiv2022

The lattice of nil-Hecke algebras over real and complex reflection groups

Associated to every complex reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, and which are obtained by killing all braid words that are "sufficiently long", as well as some integer power of each generator. These include usual nil-Coxeter algebras, nil-Temperley-Lieb algebras, and their variants, and lead to symmetric semigroup module categories which necessarily cannot be monoidal. Motivated by classical work of Coxeter (1957) and the Broue-Malle-Rouquier freeness conjecture [Crelle 1998], and continuing beyond work of the second author [Trans. Amer. Math. Soc. 2018], we obtain a complete classification of the finite-dimensional nil-Hecke algebras for all complex reflection groups $W$. These comprise the usual nil-Coxeter algebras for $W$ of finite type, their "fully commutative" analogues for $W$ of FC-finite type, three exceptional algebras (of types $F_4,H_3,H_4$), and three exceptional series (of types $B_n$ and $A_n$, two of them novel). In particular, we find the first - and only two - finite-dimensional nil-Hecke algebras over discrete complex reflection groups; this breaks from the nil-Coxeter case (where no braid words are further killed, and) where Marin [J. Pure Appl. Alg. 2014] and Khare [Trans. Amer. Math. Soc. 2018] showed that such algebras do not exist. In addition to these algebras, and also algebraic connections (to PBW deformations and non-monoidal tensor categories), we further uncover combinatorial bases of algebras, both known (fully commutative elements) and novel ($\bar{12}$-avoiding signed permutations). Our classification draws from and brings together results of Popov [Comm. Math. Inst. Utrecht 1982], Stembridge [J. Alg. Combin. 1996, 1998], Malle [Transform. Groups} 1996], Postnikov via Gowravaram-Khovanova (2015), Hart [J. Group Th. 2017], and Khare [Trans. Amer. Math. Soc. 2018].

math.RA↗

arXiv2022

Hirschman-Widder densities

Hirschman and Widder introduced a class of Pólya frequency functions given by linear combinations of one-sided exponential functions. The members of this class are probability densities, and the class is closed under convolution but not under pointwise multiplication. We show that, generically, a polynomial function of such a density is a Pólya frequency function only if the polynomial is a homothety, and also identify a subclass for which each positive-integer power is a Pólya frequency function. We further demonstrate connections between the Maclaurin coefficients, the moments of these densities, and the recovery of the density from finitely many moments, via Schur polynomials.

math.CA↗

arXiv2022

The weights of simple modules in Category $\mathscr{O}$ for Kac-Moody algebras

We give the first positive formulas for the weights of every simple highest weight module $L(λ)$ over an arbitrary Kac-Moody algebra. Under a mild condition on the highest weight, we also express the weights of $L(λ)$ as an alternating sum similar to the Weyl-Kac character formula. To obtain these results, we show the following data attached to a highest weight module are equivalent: (i) its integrability, (ii) the convex hull of its weights, (iii) the Weyl group symmetry of its character, and (iv) when a localization theorem is available, its behavior on certain codimension one Schubert cells. We further determine precisely when the above datum determines the weights themselves. Moreover, we use condition (iv) to relate localizations of the convex hull of the weights with the introduction of poles of the corresponding $D$-module on certain divisors, which answers a question of Brion. Many of these results are new even in finite type. We prove similar assertions for highest weight modules over a symmetrizable quantum group.

math.RT↗

arXiv2022

Matrix compression along isogenic blocks

A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation that Hadamard entrywise functional calculus preserves isogenic blocks has already proved to be of paramount importance for thresholding large correlation matrices. The proposed isogenic stratification of the set of complex matrices bears similarities to the Schubert cell stratification of a homogeneous algebraic manifold. An array of potential applications to current investigations in computational matrix analysis is briefly mentioned, touching concepts such as symmetric statistical models, hierarchical matrices and coherent matrix organization induced by partition trees.

math.RA↗

arXiv2022

Blowup polynomials and delta-matroids of graphs

For every finite simple connected graph $G = (V,E)$, we introduce an invariant, its blowup-polynomial $p_G(\{ n_v : v \in V \})$. This is obtained by dividing the determinant of the distance matrix of its blowup graph $G[{\bf n}]$ (containing $n_v$ copies of $v$) by an exponential factor. We show that $p_G({\bf n})$ is indeed a polynomial function in the sizes $n_v$, which is moreover multi-affine and real-stable. This associates a hitherto unexplored delta-matroid to each graph $G$; and we provide a second unexplored one for each tree. As another consequence, we obtain a new characterization of complete multipartite graphs, via the homogenization at $-1$ of $p_G$ being completely/strongly log-concave, i.e., Lorentzian. (These results extend to weighted graphs.) Finally, we show $p_G$ is indeed a graph invariant, i.e., $p_G$ and its symmetries (in the variables ${\bf n}$) recover $G$ and its isometries, respectively.

math.CO↗

arXiv2021

Totally positive kernels, Polya frequency functions, and their transforms

The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A.M. Whitney's density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Pólya frequency functions, and Pólya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.

math.FA↗

arXiv2021

On the sign patterns of entrywise positivity preservers in fixed dimension

Given $I\subset\mathbb{C}$ and an integer $N>0$, a function $f:I\to\mathbb{C}$ is entrywise positivity preserving on positive semidefinite (p.s.d.) matrices $A=(a_{jk})\in I^{N\times N}$, if the entrywise application $f[A]=(f(a_{jk}))$ of $f$ to $A$ is p.s.d. for all such $A$. Such preservers in all dimensions have been classified by Schoenberg and Rudin as being absolutely monotonic [Duke Math. J. 1942, 1959]. In fixed dimension $N$, results akin to work of Horn and Loewner [Trans. AMS 1969] show the first $N$ nonzero Maclaurin coefficients of a positivity preserver $f$ are positive; and the last $N$ coefficients are also positive if $I$ is unbounded. However, little was known about the other coefficients: the only examples to date for unbounded domains $I$ were absolutely monotonic, so work in all dimensions; and for bounded $I$ examples of non-absolutely monotonic preservers were very few (and recent). In this paper, we completely characterize the sign patterns of the Maclaurin coefficients of positivity preservers in fixed dimension $N$, over bounded and unbounded domains $I$. In particular, the above Horn-type conditions cannot be improved upon. This also yields the first polynomials which preserve positivity on p.s.d. matrices in $I^{N\times N}$ but not in $I^{(N+1)\times (N+1)}$. We obtain analogous results for real exponents using the Harish-Chandra-Itzykson-Zuber formula. We then go from qualitative bounds, which suffice to understand all possible sign patterns, to exact quantitative bounds. As an application, we extend our previous qualitative and quantitative results to understand preservers of total non-negativity in fixed dimension - including their sign patterns. We deduce several further applications, including extending a Schur polynomial conjecture by Cuttler-Greene-Skandera to obtain a novel characterization of weak majorization for real tuples.

math.CA↗

arXiv2021

Smooth entrywise positivity preservers, a Horn-Loewner master theorem, and symmetric function identities

A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 1969] says that given an integer $n \geq 1$, if the entrywise application of a smooth function $f : (0,\infty) \to \mathbb{R}$ preserves the set of $n \times n$ positive semidefinite matrices with positive entries, then $f$ and its first $n-1$ derivatives are non-negative on $(0,\infty)$. In a recent joint work with Belton-Guillot-Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg-Rudin characterization of dimension-free positivity preservers [Duke Math. J. 1942, 1959]. In recent works with Belton-Guillot-Putinar [Adv. Math. 2016] and with Tao [Amer. J. Math., in press] we used local, real-analytic versions at the origin of the Horn-Loewner condition, and discovered unexpected connections between entrywise polynomials preserving positivity and Schur polynomials. In this paper, we unify these two stories via a Master Theorem (Theorem A) which (i) simultaneously unifies and extends all of the aforementioned variants; and (ii) proves the positivity of the first $n$ nonzero Taylor coefficients at individual points rather than on all of $(0,\infty)$. A key step in the proof is a new determinantal / symmetric function calculation (Theorem B), which shows that Schur polynomials arise naturally from considering arbitrary entrywise maps that are sufficiently differentiable. Of independent interest may be the following application to symmetric function theory: we extend the Schur function expansion of Cauchy's (1841) determinant (whose matrix entries are geometric series $1 / (1 - u_j v_k)$), as well as of a determinant of Frobenius [J. reine angew. Math. 1882] (whose matrix entries are a sum of two geometric series), to arbitrary power series, and over all commutative rings.

math.CA↗

arXiv2021

The $p$-norm of circulant matrices

In this note we study the induced $p$-norm of circulant matrices $A(n,\pm a, b)$, acting as operators on the Euclidean space $\mathbb{R}^n$. For circulant matrices whose entries are nonnegative real numbers, in particular for $A(n,a,b)$, we provide an explicit formula for the $p$-norm, $1 \leq p \leq \infty$. The calculation for $A(n,-a,b)$ is more complex. The 2-norm is precisely determined. As for the other values of $p$, two different categories of upper and lower bounds are obtained. These bounds are optimal at the end points (i.e. $p=1$ and $p = \infty$) as well as at $p=2$.

math.FA↗

arXiv2021

Moment-sequence transforms

We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, with possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions.

math.CA↗

arXiv2021

Covering modules by proper submodules

A classical problem in the literature seeks the minimal number of proper subgroups whose union is a given finite group. A different question, with applications to error-correcting codes and graph colorings, involves covering vector spaces over finite fields by (minimally many) proper subspaces. In this note we cover $R$-modules by proper submodules for commutative rings $R$, thereby subsuming and recovering both cases above. Specifically, we study the smallest cardinal number $\aleph$, possibly infinite, such that a given $R$-module is a union of $\aleph$-many proper submodules. (1) We completely characterize when $\aleph$ is a finite cardinal; this parallels for modules a 1954 result of Neumann. (2) We also compute the covering (cardinal) numbers of finitely generated modules over quasi-local rings and PIDs, recovering past results for vector spaces and abelian groups respectively. (3) As a variant, we compute the covering number of an arbitrary direct sum of cyclic monoids. Our proofs are self-contained.

math.AC↗

arXiv2021

Multivariate blowup-polynomials of graphs

In recent joint work (2021), we introduced a novel multivariate polynomial attached to every metric space - in particular, to every finite simple connected graph $G$ - and showed it has several attractive properties. First, it is multi-affine and real-stable (leading to a hitherto unstudied delta-matroid for each graph $G$). Second, the polynomial specializes to (a transform of) the characteristic polynomial $χ_{D_G}$ of the distance matrix $D_G$; as well as recovers the entire graph, where $χ_{D_G}$ cannot do so. Third, the polynomial encodes the determinants of a family of graphs formed from $G$, called the blowups of $G$. In this short note, we exhibit the applicability of these tools and techniques to other graph-matrices and their characteristic polynomials. As a particular case, we will see that the adjacency characteristic polynomial $χ_{A_G}$ is in fact the shadow of a richer multivariate blowup-polynomial, which is similarly multi-affine and real-stable. Moreover, this polynomial encodes not only the aforementioned three properties, but also yields additional information for specific families of graphs.

math.CO↗

arXiv2021

Multiply positive functions, critical exponent phenomena, and the Jain-Karlin-Schoenberg kernel

We prove the converse to a result of Karlin [Trans. AMS 1964], and also strengthen his result and two results of Schoenberg [Ann. of Math. 1955]. One of the latter results concerns zeros of Laplace transforms of multiply positive functions. The other results study which powers $α$ of two specific kernels are totally non-negative of order $p\geq 2$ (denoted TN$_p$); both authors showed this happens for $α\geq p-2$, and Schoenberg proved that it does not for $α p-2$, and is not TN$_p$ for every non-integer $α\in(0,p-2)$. In particular, these results reveal 'critical exponent' phenomena in total positivity. We also prove the converse to a 1968 result of Karlin, revealing yet another critical exponent phenomenon - for Laplace transforms of all Polya Frequency (PF) functions. We further classify the powers preserving all TN$_p$ Hankel kernels on intervals, and isolate individual kernels encoding these powers. We then transfer results on preservers by Polya-Szego (1925), Loewner/Horn (1969), and Khare-Tao (in press), from positive matrices to Hankel TN$_p$ kernels. Another application constructs individual matrices encoding the Loewner convex powers. This complements Jain's results (2020) for Loewner positivity, which we strengthen to total positivity. Remarkably, these (strengthened) results of Jain, those of Schoenberg and Karlin, the latter's converse, and the above Hankel kernels all arise from a single symmetric rank-two kernel and its powers: $\max(1+xy,0)$. We also provide a novel characterization of PF functions and sequences of order $p\geq 3$, following Schoenberg's 1951 result for $p=2$. We correct a small gap in his paper, in the classification of discontinuous PF functions.

math.FA↗