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2014 | Buch

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Symmetry: Representation Theory and Its Applications

In Honor of Nolan R. Wallach

herausgegeben von: Roger Howe, Markus Hunziker, Jeb F. Willenbring

Verlag: Springer New York

Buchreihe : Progress in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Nolan Wallach's mathematical research is remarkable in both its breadth and depth. His contributions to many fields include representation theory, harmonic analysis, algebraic geometry, combinatorics, number theory, differential equations, Riemannian geometry, ring theory, and quantum information theory. The touchstone and unifying thread running through all his work is the idea of symmetry. This volume is a collection of invited articles that pay tribute to Wallach's ideas, and show symmetry at work in a large variety of areas. The articles, predominantly expository, are written by distinguished mathematicians and contain sufficient preliminary material to reach the widest possible audiences. Graduate students, mathematicians, and physicists interested in representation theory and its applications will find many gems in this volume that have not appeared in print elsewhere. Contributors: D. Barbasch, K. Baur, O. Bucicovschi, B. Casselman, D. Ciubotaru, M. Colarusso, P. Delorme, T. Enright, W.T. Gan, A Garsia, G. Gour, B. Gross, J. Haglund, G. Han, P. Harris, J. Hong, R. Howe, M. Hunziker, B. Kostant, H. Kraft, D. Meyer, R. Miatello, L. Ni, G. Schwarz, L. Small, D. Vogan, N. Wallach, J. Wolf, G. Xin, O. Yacobi.

Inhaltsverzeichnis

Frontmatter

Unitary Hecke algebra modules with nonzero Dirac cohomology

Abstract

In this paper, we review the construction of the Dirac operator for graded affine Hecke algebras and calculate the Dirac cohomology of irreducible unitary modules for the graded Hecke algebra of gl(n).

Dan Barbasch, Dan Ciubotaru

On the nilradical of a parabolic subgroup

Abstract

We present various approaches to understanding the structure of the nilradical of parabolic subgroups in type A. In particular, we consider the complement of the open dense orbit and describe its irreducible components.

Karin Baur

Arithmetic invariant theory

Abstract

Let k be a field, let G be a reductive algebraic group over k, and let V be a linear representation of G. Geometric invariant theory involves the study of the k-algebra of G-invariant polynomials on V, and the relation between these invariants and the G-orbits on V, usually under the hypothesis that the base field k is algebraically closed. In favorable cases, one can determine the geometric quotient $V /\!/G = \mathrm{Spec}(\mathrm{Sym}^{{\ast}}(V ^{\vee })^{G})$ and can identify certain fibers of the morphism $V \rightarrow V/\!/G$ with certain G-orbits on V. In this paper we study the analogous problem when k is not algebraically closed. The additional complexity that arises in the orbit picture in this scenario is what we refer to as arithmetic invariant theory. We illustrate some of the issues that arise by considering the regular semisimple orbits—i.e., the closed orbits whose stabilizers have minimal dimension—in three arithmetically rich representations of the split odd special orthogonal group $G = \mathrm{SO}_{2n+1}$.

Manjul Bhargava, Benedict H. Gross

Structure constants of Kac–Moody Lie algebras

Abstract

This paper outlines an algorithm for computing structure constants of Kac–Moody Lie algebras. In contrast to the methods currently used for finite-dimensional Lie algebras, which rely on the additive structure of the roots, it reduces to computations in the extended Weyl group first defined by Jacques Tits in about 1966. The new algorithm has some theoretical interest, and its basis is a mathematical result generalizing a theorem of Tits about the finite-dimensional case. The explicit algorithm seems to be new, however, even in the finite-dimensional case. I include towards the end some remarks about repetitive patterns of structure constants, which I expect to play an important role in understanding the associated groups. That neither the idea of Tits nor the phenomenon of repetition has already been exploited I take as an indication of how little we know about Kac–Moody structures.

Bill Casselman

The Gelfand–Zeitlin integrable system and K-orbits on the flag variety

Abstract

In this paper, we provide an overview of the Gelfand–Zeitlin integrable system on the Lie algebra of n × n complex matrices $\mathfrak{g}\mathfrak{l}(n, \mathbb{C})$ introduced by Kostant and Wallach in 2006. We discuss results concerning the geometry of the set of strongly regular elements, which consists of the points where the Gelfand–Zeitlin flow is Lagrangian. We use the theory of $K_{n} = GL(n - 1,\mbox{ $\mathbb{C}$}) \times GL(1,\mbox{ $\mathbb{C}$})$-orbits on the flag variety $\mathcal{B}_{n}$ of $GL(n,\mbox{ $\mathbb{C}$})$ to describe the strongly regular elements in the nilfiber of the moment map of the system. We give an overview of the general theory of orbits of a symmetric subgroup of a reductive algebraic group acting on its flag variety, and illustrate how the general theory can be applied to understand the specific example of K _n and GL(n, ℂ).

Mark Colarusso, Sam Evens

Diagrams of Hermitian type, highest weight modules, and syzygies of determinantal varieties

Abstract

In this mostly expository paper, a natural generalization of Young diagrams for Hermitian symmetric spaces is used to give a concrete and uniform approach to a wide variety of interconnected topics including posets of noncompact roots, canonical reduced expressions, rational smoothness of Schubert varieties, parabolic Kazhdan–Lusztig polynomials, equivalences of categories of highest weight modules, BGG resolutions of unitary highest weight modules, and finally, syzygies and Hilbert series of determinantal varieties.

Thomas J. Enright, Markus Hunziker, W. Andrew Pruett

A conjecture of Sakellaridis–Venkatesh on the unitary spectrum of spherical varieties

Abstract

We describe the spectral decomposition of certain spherical varieties of low rank, verifying a recent conjecture of Sakellaridis and Venkatesh in these cases.

Wee Teck Gan, Raul Gomez

Proof of the 2-part compositional shuffle conjecture

Abstract

In a recent paper [9] J. Haglund, J. Morse and M. Zabrocki advanced a refinement of the Shuffle Conjecture of Haglund et. al. [8]. They introduce the notion of “touch composition” of a Dyck path, whose parts yield the positions where the path touches the diagonal. They conjectured that the polynomial $\big\langle \nabla \mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}}\,1\,\ h_{\mu _{1}}h_{\mu _{2}}\cdots h_{\mu _{l}}\big\rangle$, where $\mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}}\,1$ is essentially a rescaled Hall–Littlewood polynomial and $\nabla $ is the Macdonald eigen-operator introduced in [1], enumerates by $t^{\mathrm{area}}q^{\mathrm{dinv}}$ the parking functions whose Dyck paths hit the diagonal by (p ₁, p ₂, …, p _k) and whose diagonal word is a shuffle of l increasing words of lengths μ ₁, μ ₂, …, μ _l. In this paper we prove the case l = 2 of this conjecture.

Adriano M. Garsia, Gouce Xin, Mike Zabrocki

On symmetric SL-invariant polynomials in four qubits

Abstract

We find the generating set of $\mathop{\mathrm{SL}}\nolimits$-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. These invariants are the degrees if the basic invariants of the invariants for F ₄, and in fact, the group plays an important role in this note. In addition, we show that the hyperdeterminant in four qubits is the only $\mathop{\mathrm{SL}}\nolimits$-invariant polynomial (up to powers of itself) that is non-vanishing precisely on the set of generic states.

Gilad Gour, Nolan R. Wallach

Finite maximal tori

Abstract

We define a “finite maximal torus” of a compact Lie group G to be a maximal finite abelian subgroup A of G. We introduce structure for finite maximal tori parallel to the classical structure for maximal tori, like roots and the Weyl group; and we recall a large number of (previously known) examples.

Gang Han, David A. Vogan Jr.

Sums of squares of Littlewood–Richardson coefficients and GL n -harmonic polynomials

Abstract

We consider the example from invariant theory concerning the conjugation action of the general linear group on several copies of the n × n matrices, and examine a symmetric function which stably describes the Hilbert series for the invariant ring with respect to the multigradation by degree. The terms of this Hilbert series may be described as a sum of squares of Littlewood–Richardson coefficients. A “principal specialization” of the gradation is then related to the Hilbert series of the K-invariant subspace in the GL_n-harmonic polynomials (in the sense of Kostant), where K denotes a block diagonal embedding of a product of general linear groups. We also consider other specializations of this Hilbert series.

Pamela E. Harris, Jeb F. Willenbring

Polynomial functors and categorifications of Fock space

Abstract

Fix an infinite field k of characteristic p, and let $\mathfrak{g}$ be the Kac–Moody algebra $\mathfrak{s}\mathfrak{l}_{\infty }$ if p = 0 and $\widehat{\mathfrak{s}\mathfrak{l}}_{p}$ otherwise. Let $\mathcal{P}$ denote the category of strict polynomial functors defined over k. We describe a categorical $\mathfrak{g}$-action on $\mathcal{P}$ (in the sense of Chuang and Rouquier) categorifying the Fock space representation of $\mathfrak{g}$.

Jiuzu Hong, Antoine Touzé, Oded Yacobi

Pieri algebras and Hibi algebras in representation theory

Abstract

A class of algebras that unify a variety of calculations in the representation theory of classical groups is discussed. Because of their relation to the classical Pieri Rule, these algebras are called double Pieri algebras. A generalization of the standard monomial theory of Hodge is developed for double Pieri algebras, that uses pairs of semistandard tableaux, rather than a single one. SAGBI theory and toric deformation are key tools. The deformed double Pieri algebras are described using a doubled version of Gelfand–Tsetlin patterns. The approach allows the discussion to avoid dealing with relations between generators.

Roger Howe

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

Abstract

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 energy spectrum of each of these representations is studied and related to plane wave solutions. The steady state solutions whose luminosity (energy) satisfies Planck’s Black Body Radiation Law are described in terms of this analysis. The unitary representations have notable properties. In particular they have positive or negative energy, they are of type $A_{\mathfrak{q}}(\lambda )$ and are quaternionic. Physical implications of the results are explained.

Bertram Kostant, Nolan R. Wallach

Representations with a reduced null cone

Abstract

Let G be a complex reductive group and V a G-module. Let $\pi: V \rightarrow V/\!\!/G$ be the quotient morphism defined by the invariants and set $\mathcal{N}(V ):=\pi ^{-1}(\pi (0))$. We consider the following question. Is the null cone $\mathcal{N}(V )$ reduced, i.e., is the ideal of $\mathcal{N}(V )$ generated by G-invariant polynomials? We have complete results when G is $\mathop{\mathrm{SL}}\nolimits _{2}$, $\mathop{\mathrm{SL}}\nolimits _{3}$ or a simple group of adjoint type, and also when G is semisimple of adjoint type and the G-module V is irreducible.

Hanspeter Kraft, Gerald W. Schwarz

M-series and Kloosterman–Selberg zeta functions for ℝ-rank one groups

Abstract

For an arbitrary Lie group G of real rank one, we give a formula for the Fourier coefficient $D_{\chi ^{{\prime}}}^{\chi }(\xi,\nu )$ of the M-series (a type of Poincaré series) defined in [17], in terms of Kloosterman–Selberg zeta functions $\zeta _{\chi,\chi ^{{\prime}},\xi }(\mu )$. As a consequence we show that the meromorphic continuation of $\zeta _{\chi,\chi ^{{\prime}},\xi }(\nu )$ to $\mathbb{C}$ follows from the meromorphic continuation of the M-series. We also give a description of the pole set in the region $\mathop{Re}\nolimits \nu \geq 0$.

Roberto J. Miatello, Nolan R. Wallach

Ricci flow and manifolds with positive curvature

Abstract

This is an expository article based on the author’s lecture delivered at the conference Lie Theory and Its Applications in March 2011, UCSD. We discuss various notions of positivity and their relations with the study of the Ricci flow, including a proof of the assertion, due to Wolfson and the author, that the Ricci flow preserves the positivity of the complex sectional curvature. We discuss the examples of Wallach of the manifolds with positive pinched sectional curvature and the behavior of Ricci flow on some examples. Finally we discuss the recent joint work with Wilking on the manifolds with pinched flag curvature and some open problems.

Lei Ni

Remainder formula and zeta expression for extremal CFT partition functions

Abstract

We derive a remainder formula for the coefficients of modular invariant partition functions of extremal conformal field theories of central charge c = 24k, where k is a positive integer. The formula encodes, in particular, asymptotics of these coefficients and it provides for additional corrections to Bekenstein–Hawking black hole entropy. We also relate these partition functions to a Patterson–Selberg zeta function. More generally, when c is divisible by 8 we relate this zeta function to vacuum characters of affine E ₈ and $E_{8} \times E_{8}$.

Floyd L. Williams

Principal series representations of infinite-dimensional Lie groups, I: Minimal parabolic subgroups

Abstract

We study the structure of minimal parabolic subgroups of the classical infinite-dimensional real simple Lie groups, corresponding to the classical simple direct limit Lie algebras. This depends on the recently developed structure of parabolic subgroups and subalgebras that are not necessarily direct limits of finite-dimensional parabolics. We then discuss the use of that structure theory for the infinite-dimensional analog of the classical principal series representations. We look at the unitary representation theory of the classical lim-compact groups U(∞), SO(∞) and Sp(∞) in order to construct the inducing representations, and we indicate some of the analytic considerations in the actual construction of the induced representations.

Joseph A. Wolf

Titel: Symmetry: Representation Theory and Its Applications
herausgegeben von: Roger Howe
Markus Hunziker
Jeb F. Willenbring
Verlag: Springer New York
Electronic ISBN: 978-1-4939-1590-3
Print ISBN: 978-1-4939-1589-7
DOI: https://doi.org/10.1007/978-1-4939-1590-3