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Nolan Wallach

Publications and source records attributed to Nolan Wallach.

15 recordsLinked to original sources
arXiv2025

Compact, connected, complex manifolds that admit a compact transitive group of holomorphic automorphisms

The purpose of this paper is to develop a Lie algebraic approach to obtain new proofs of important results of H.-C. Wang, Tits and Wolf-Wang-Ziller on compact complex homogeneous manifolds emphasizing only those that admit a transitive compact group of biholomorphic transformations. The method only uses some standard results in Lie theory. The new approach provides a method of associating a canonical abelian Lie algebra with a given integrable complex structure on a compact Lie algebra which extends the earlier work of Samelson and Pittie.

math.DG↗
arXiv2021

Stability theorems for multiplicities in graded $S_n$-modules

In this paper, we prove several stability theorems for multiplicities of naturally defined representations of symmetric groups. The first such theorem states that if we consider the diagonal action of the symmetric group $S_{m+r}$ on $k$ sets of $m+r$ variables, then the dimension of the invariants of degree $m$ is the same as the dimension of the invariants of degree $m$ for $S_{m}$ acting on $k$ sets of $m$ variables. Building on this stability, the last section looks at the Hilbert series of coinvariants of the polynomial ring in $k$ sets of $m$ variables. We address a conjecture that the Hilbert series, in degrees no more than $m$, can be computed by a truncated power series expression. Using some auxiliary results and manipulations of power series, we show that if this holds for $k$ and $m$, then the truncation gives the correct Hilbert series up to degree $m$ for $k$ sets of $n \geq m$ variables. This shows the validity of the conjecture up to certain degrees. We also provide a new equivalent conjecture regarding Gr\"{o}bner bases. The second type of stability result is for Weyl modules. We prove that the dimension of the $S_{m+r}$ invariants for a Weyl module ${}_{m+r}F^{\lambda}$ (the Schur-Weyl dual of the $S_{|\lambda|}$ module $V^{\lambda}$) with $\left\vert \lambda \right\vert \leq m$ is of the same dimension as the space of $S_{m}$ invariants for ${}_{m}F^{\lambda}$. Multigraded versions of the first type of result are given, as are multigraded generalizations to non-trivial modules of symmetric groups.

math.RT↗
arXiv2018

Unentangled Measurements and Frame Functions

Gleason's theorem asserts the equivalence of von Neumann's density operator formalism of quantum mechanics and frame functions, which are functions on the pure states that sum to 1 on any orthonormal basis of Hilbert space of dimension at least 3. The unentangled frame functions are initially only defined on unentangled (that is, product) states in a multi-partite system. The third author's Unentangled Gleason's Theorem shows that unentangled frame functions determine unique density operators if and only if each subsystem is at least 3-dimensional. In this paper, we determine the structure of unentangled frame functions in general. We first classify them for multi-qubit systems, and then extend the results to factors of varying dimensions including countably infinite dimensions (separable Hilbert spaces). A remarkable combinatorial structure emerges, suggesting possible fundamental interpretations.

quant-ph↗
arXiv2016

Local Distinguishability of Generic Unentangled Orthonormal Bases

An orthonormal basis consisting of unentangled (pure tensor) elements in a tensor product of Hilbert spaces is an Unentangled Orthogonal Basis (UOB). In general, for $n$ qubits, we prove that in its natural structure as a real variety, the space of UOB is a bouquet of products of Riemann spheres parametrized by a class of edge colorings of hypercubes. Its irreducible components of maximum dimension are products of $2^n-1$ two-spheres. Using a theorem of Walgate and Hardy, we observe that the UOB whose elements are distinguishable by local operations and classical communication (called locally distinguishable or LOCC distinguishable UOB) are exactly those in the maximum dimensional components. Bennett et al, in their in-depth study of quantum nonlocality without entanglement, include a specific 3 qubit example UOB which is not LOCC distinguishable; we construct certain generalized counterparts of this UOB in $n$ qubits.

quant-ph↗
arXiv2011

Action of the conformal group on steady state solutions to Maxwell's equations and background radiation

The representation of the conformal group (PSU(2,2)) on the space of solutions to Maxwell's equations on the conformal compactification of Minkowski space is shown to break up into four irreducible unitarizable smooth Fréchet representations of moderate growth. An explicit inner product is defined on each representation. The frequency spectrum of each of these representations is analyzed. These representations have notable properties; in particular they have positive or negative energy, they are of type $A_{\frak q}(λ)$ and are quaternionic. Physical implications of the results are explained.

math-ph↗
arXiv2010

Bessel Models for General Admissible Induced Representations: The Compact Stabilizer Case

A holomorphic continuation of Jacquet type integrals for parabolic subgroups with abelian nilradical is studied. Complete results are given for generic characters with compact stabilizer and arbitrary representations induced from admissible representations. A description of all of the pertinent examples is given. These results give a complete description of the Bessel models corresponding to compact stabilizer.

math.RT↗
arXiv2010

On the algebraic set of singular elements in a complex simple Lie algebra

Let $G$ be a complex simple Lie group and let $\g = \hbox{\rm Lie}\,G$. Let $S(\g)$ be the $G$-module of polynomial functions on $\g$ and let $\hbox{\rm Sing}\,\g$ be the closed algebraic cone of singular elements in $\g$. Let ${\cal L}\s S(\g)$ be the (graded) ideal defining $\hbox{\rm Sing}\,\g$ and let $2r$ be the dimension of a $G$-orbit of a regular element in $\g$. Then ${\cal L}^k = 0$ for any $k<r$. On the other hand, there exists a remarkable $G$-module $M\s {\cal L}^r$ which already defines $\hbox{\rm Sing}\,\g$. The main results of this paper are a determination of the structure of $M$.

math.RT↗
arXiv2008

Invariants, Kronecker Products, and Combinatorics of Some Remarkable Diophantine Systems (Extended Version)

This work lies across three areas (in the title) of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link consists of a single formal power series with a multifaced interpretation. The deeper exploration of this link yielded results as well as methods for solving some numerical problems in each of these separate areas.

math.CO↗
arXiv2007

On a classification of the gradient shrinking solitons

The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. In dimension three we also generalize the result by removing the $κ$-non-collapsing assumption. In high dimension this new method allows us to prove a classification result on gradient shrinking solitons with vanishing Weyl curvature tensor, which includes the rotationally symmetric ones.

math.DG↗
arXiv2007

On 4-dimensional gradient shrinking solitons

In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional gradient shrinking solitons to dimension four.

math.DG↗
arXiv2005

Gelfand-Zeitlin theory from the perspective of classical mechanics II

In this paper, Part II, of a two part paper we apply the results of [KW], Part I, to establish, with an explicit dual coordinate system, a commutative analogue of the Gelfand-Kirillov theorem for M(n), the algebra of $n\times n$ complex matrices. The function field F(n) of M(n) has a natural Poisson structure and an exact analogue would be to show that F(n) is isomorphic to the function field of a $n(n-1)$-dimensional phase space over a Poisson central rational function field in $n$ variables. Instead we show that this the case for a Galois extension, $F(n, {\frak e})$, of F(n). The techniques use a maximal Poisson commutative algebra of functions arising from Gelfand-Zeitlin theory, the algebraic action of a $n(n-1)/2$--dimensional torus on $F(n, {\frak e})$, and the structure of a Zariski open subset of M(n) as a $n(n-1)/2$--dimensional torus bundle over a $n(n+1)/2$--dimensional base space of Hessenberg matrices.

math.SG↗
arXiv2005

Nice Parabolic Subalgebras of Reductive Lie Algebras

This paper gives a classification of parabolic subalgebras of simple Lie algebras over $\CC$ that are complexifications of parabolic subalgebras of real forms for which Lynch's vanishing theorem for generalized Whittaker modules is non-vacuous. The paper also describes normal forms for the admissible characters in the sense of Lynch (at least in the quasi-split cases) and analyzes the important special case when the parabolic is defined by an even embedded TDS (three dimensional simple Lie algebra).

math.RT↗
arXiv2004

Gelfand-Zeitlin theory from the perspective of classical mechanics. I

A commutative Poisson subalgebra of the Poisson algebra of polynomials on the Lie algebra of n x n matrices over ${\Bbb C}$ is introduced which is the Poisson analogue of the Gelfand-Zeitlin subalgebra of the universal enveloping algebra. As a commutative algebra it is a polynomial ring in $n(n+1)/2$ generators, $n$ of which can be taken to be basic generators of the polynomial invariants. Any choice of the next $n(n-1)/2$ generators yields a Lie algebra of vector fields that generates a global holomorphic action of the additive group ${\Bbb C}^{n(n -1)/2}$. This paper proves several remarkable properties of this group action and relates it to the theory of orthogonal polynomials.

math.SG↗