Searcharxiv⌕ Search

arXiv subjects

Nolan R. Wallach

Publications and source records attributed to Nolan R. Wallach.

At least 19 recordsLinked to original sources
arXiv2024

Hilbert and Fréchet bundle versions of the Harish-Chandra and Whittaker Plancherel Theorems

This paper, in particular, gives a complete proof of the direct integral version of the Whittaker Plancherel Theorem. The main emphasis is on certain Hilbert and Fréchet vector bundles over a space that has a submersion onto the tempered dual. This allows for an approach to the Plancherel Theorems (both for L^2 and the Whittaker case) that is representation theoretic, bypasses the need for Harish-Chandra's Eisenstein Integrals and yields a proof the direct integral decompositions without invoking the abstract theory.

math.RT↗
arXiv2023

On the Whittaker Plancherel Theorem for Real Reductive Groups

The purpose of this article is to give the first complete proof of the Whittaker Plancherel Theorem. The proof uses Harish-Chandra's Plancherel Theorem for a real reductive group and its exposition can be used as an introduction to Harish-Chandra's ideas. The proof follows the basic ideas in the author's original attempt in his second volume on real reductive groups. An error in the calculation of the Whittaker Transform of a Harish-Chandra wave packet is fixed using a result of Raphaël Beuzart-Plessis.

math.RT↗
arXiv2023

The spherical Whittaker Inversion Theorem and the quantum non-periodic Toda Lattice

In this paper the spherical case of the Whittaker Inversion Theorem is given a relatively self-contained proof. This special case can be used as a help in deciphering the handling of the continuous spectrum in the proof of the full theorem. It also leads directly to the solution of the quantum non-periodic Toda Lattice. This is also explained in detail in this paper.

math.RT↗
arXiv2022

On a question related to a basic convergence theorem of Harish-Chandra

In his first 1958 paper on zonal spherical functions Harish-Chandra proved an extremely delicate convergence theorem which was basic to his subsequent definition of his Schwartz space and his theory of cusp forms. This paper gives elementary proofs that a related integral converges for for groups of real rank one, several groups of real rank 2 (including $SO(n,2), Sp_4(R)$ and $Sp_4(C)$), $GL(n,R)$ and $GL(n,C)$. In fact, a stronger result has been proved in raphael . Applications of the question are also studied.

math.RT↗
arXiv2021

Harmonic differential forms for pseudo-reflection groups II. Bi-degree bounds

This paper studies three results that describe the structure of the super-coinvariant algebra of pseudo-reflection groups over a field of characteristic $0$. Our most general result determines the top component in total degree, which we prove for all Shephard--Todd groups $G(m, p, n)$ with $m \neq p$ or $m=1$. Our strongest result gives tight bi-degree bounds and is proven for all $G(m, 1, n)$, which includes the Weyl groups of types $A$ and $B$/$C$. For symmetric groups (i.e. type $A$), this provides new evidence for a recent conjecture of Zabrocki related to the Delta Conjecture of Haglund--Remmel--Wilson. Finally, we examine analogues of a classic theorem of Steinberg and the Operator Theorem of Haiman. Our arguments build on the type-independent classification of semi-invariant harmonic differential forms carried out in the first part of this series. In this paper we use concrete constructions including Gröbner and Artin bases for the classical coinvariant algebras of the pseudo-reflection groups $G(m, p, n)$, which we describe in detail. We also prove that exterior differentiation is exact on the super-coinvariant algebra of a general pseudo-reflection group. Finally, we discuss related conjectures and enumerative consequences.

math.CO↗
arXiv2020

The dependence on parameters of the inverse functor to the $K$-finite functor

An interpretation of the Casselman-Wallach (C-W) Theorem is that the $K$-finite functor is an isomorphism of categories from the category of finitely generated, admissible smooth Fréchet modules of moderate growth to the category of Harish-Chandra modules for a real reductive group, $G$ (here $K$ is a maximal compact subgroup of G).In this paper we study the dependence of this functor on parameters. Our main result implies that holomorphic dependence implies holomorphic dependence. The work uses results from the excellent thesis of van der Noort. Also a remarkable family of Universal Harish-Chandra modules developed in this paper plays a key role.

math.RT↗
arXiv2020

Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

We give a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Our "top-down" approach uses the methods of Cartan's exterior calculus and is in some sense dual to related work of Solomon, Orlik--Solomon, and Shepler describing (semi-)invariant differential forms. We apply our results to a recent conjecture of Zabrocki which provides a representation theoretic-model for the Delta conjecture of Haglund--Remmel--Wilson in terms of a certain non-commutative coinvariant algebra for the symmetric group. In particular, we verify the alternating component of a specialization of Zabrocki's conjecture.

math.CO↗
arXiv2019

Some implications of a conjecture of Zabrocki to the action of $S_{n}$ on polynomial differential forms

The symmetric group acts on polynomial differential forms on $\mathbb{R}^{n}$ through its action by permuting the coordinates. In this paper the $S_{n}% $-invariants are shown to be freely generated by the elementary symmetric polynomials and their exterior derivatives. A basis of the alternants in the quotient of the ideal generated by the homogeneous invariants of positive degree is given. In addition, the highest bigraded degrees are given for the quotient. All of these results are consistent with predictions derived by Garsia and Romero from a recent conjecture of Zabrocki.

math.CO↗
arXiv2019

Dependence on parameters of CW globalizations of families of Harish-Chandra modules and the meromorphic continuation of $C^{\infty}$ Eisenstein series

The first main result is that the Casselman-Wallach Globalization of a real analytic family of Harish-Chandra modules is continuous in the parameter. Our proof of this result uses results from the thesis of Vincent van der Noort in several critical ways. In his thesis the holomorphic version of the result was proved in the case when the parameter space is a one dimensional complex manifold up to a branched covering. The second main result is a proof of the meromorphic continuation of $C^{\infty}$ Eisenstein series.using Langlands' results in the $K$ finite case as an application of the methods in the proof of the first part.

math.RT↗
arXiv2018

Principal orbit type theorems for reductive algebraic group actions and the Kempf--Ness Theorem

The main result asserts: Let $G$ be a reductive, affine algebraic group and let $(ρ,V)$ be a regular representation of $G$. Let $X$ be an irreducible $\mathbb{C}^{ \times } G$ invariant Zariski closed subset such that $G$ has a closed orbit that has maximal dimension among all orbits (this is equivalent to: generic orbits are closed). Then there exists an open subset, $W$,of $X$ in the metric topology which is dense with complement of measure $0$ such that if $x ,y \in W$ then $\left (\mathbb{C}^{ \times } G\right )_{x}$ is conjugate to $\left (\mathbb{C}^{ \times } G\right )_{y}$. Furthermore, if $G x$ is a closed orbit of maximal dimension and if $x$ is a smooth point of $X$ then there exists $y \in W$ such that $\left (\mathbb{C}^{ \times } G\right )_{x}$ contains a conjugate of $\left (\mathbb{C}^{ \times } G\right )_{y}$. The proof involves using the Kempf-Ness theorem to reduce the result to the principal orbit type theorem for compact Lie groups.

math.AG↗
arXiv2018

Transformations among Pure Multipartite Entangled States via Local Operations Are Almost Never Possible

Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure multipartite multilevel states. Combined with the analogous results for qubit states shown by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization of almost all local transformations among multipartite pure states. We show that nontrivial LOCC transformations among generic, fully entangled, pure states are almost never possible. Thus, almost all multipartite states are isolated. They can neither be deterministically obtained from local-unitary-inequivalent (LU-inequivalent) states via local operations, nor can they be deterministically transformed to pure, fully entangled LU-inequivalent states. In order to derive this result, we prove a more general statement, namely, that, generically, a state possesses no nontrivial local symmetry. We discuss further consequences of this result for the characterization of optimal, probabilistic single copy and probabilistic multi-copy LOCC transformations and the characterization of LU-equivalence classes of multipartite pure states.

quant-ph↗
arXiv2017

Almost all multipartite qubit quantum states have trivial stabilizer

The stabilizer group of an n-qubit state ψis the set of all matrices of the form g=g_1\otimes\cdots\otimes g_n, with g_1,...,g_n being any 2x2 invertible complex matrices, that satisfy gψ=ψ. We show that for 5 or more qubits, except for a set of states of zero measure, the stabilizer group of multipartite entangled states is trivial; that is, containing only the identity element. We use this result to show that for 5 or more qubits, the action of deterministic local operations and classical communication (LOCC) can almost always be simulated simply by local unitary (LU) operations. This proves that almost all n-qubit states with n>4 are isolated, that is they can neither be reached nor converted into any other (n-partite entangled), LU-inequivalent state via deterministic LOCC. We also find a simple and elegant expression for the maximal probability to convert one multi-qubit entangled state to another for this generic set of states.

quant-ph↗
arXiv2016

On Neeman's gradient flows

In his brilliant but sketchy paper on the strucure of quotient varieties of affine actions of reductive algebraic groups over C, Amnon Neeman introduced a gradiant flow with remarkable properties. The purpose of this paper is to study several applications of this flow. In particular we prove that the cone on a Zariski closed subset of n-1 dimensional real projective space is a deformation retract of n dimensional Euclidean space. We also give an exposition of an extension to real reductive algebraic group actions of Schwarz's excellent explanation of Neeman's sketch of a proof of his deformation theorem. This exposition precisely explains the use of Lojasiewicz gradient inequality. The result described above for cones makes use of these ideas.

math.AG↗
arXiv2014

Fréchet completions of moderate growth old and (somewhat) new results

This article has two objectives. The first is to give a guide to the proof of the (so-called) Casselman-Wallach theorem as it appears in Real Reductive Groups II. The emphasis will be on one aspect of the original proof that leads to the new result in this paper which is the second objective. We show how a theorem of van der Noort combined with a clarification of the original argument in my book lead to a theorem with parameters (an alternative is one announced by Berstein and Krötz). This result gives a new proof of the meromorphic continulation of the smooth Eisenstein series.

math.RT↗
arXiv2013

Classification of multipartite entanglement in all dimensions

We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such an SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits we also provide much simpler expressions.

quant-ph↗
arXiv2012

On Symmetric SL-Invariant Polynomials in Four Qubits

We find the generating set of SL-invariant polynomials in four qubits that are also invariant under permutations of the qubits. The set consists of four polynomials of degrees 2,6,8, and 12, for which we find an elegant expression in the space of critical states. In addition, we show that the Hyperdeterminant in four qubits is the only SL-invariant polynomial (up to powers of itself) that is non-vanishing precisely on the set of generic states.

math-ph↗
arXiv2012

Shor's algorithm without partial fractions

The purpose of this note was to give a proof that Shor's algorithm for period search is polynomial using only the standard $2^{n}$ quantum Fourier thansform and some simple trigonometry. There is an error that was pointed out to the author by Pavel Wocjan.

quant-ph↗