Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group

We consider the asymptotic behavior as $n\to\infty$ of the spectra of random matrices of the form $$\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}\rho_n ((k,k+1)),$$ where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian and the matrices $\rho_n((k,k+1))$ are obtained by applying an irreducible unitary representation $\rho_n$ of the symmetric group on $\{1,2,...,n\}$ to the transposition $(k,k+1)$ that interchanges $k$ and $k+1$ [thus, $\rho_n((k,k+1))$ is both unitary and self-adjoint, with all eigenvalues either +1 or -1]. Irreducible representations of the symmetric group on $\{1,2,...,n\}$ are indexed by partitions $\lambda_n$ of $n$. A consequence of the results we establish is that if $\lambda_{n,1}\ge\lambda_{n,2}\ge...\ge0$ is the partition of $n$ corresponding to $\rho_n$, $\mu_{n,1}\ge\mu_{n,2}\ge >...\ge0$ is the corresponding conjugate partition of $n$ (i.e., the Young diagram of $\mu_n$ is the transpose of the Young diagram of $\lambda_n$), $\lim_{n\to\infty}\frac{\lambda_{n,i}}{n}=p_i$ for each $i\ge1$, and $\lim_{n\to\infty}\frac{\mu_{n,j}}{n}=q_j$ for each $j\ge1$, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean $\theta Z$ and variance $1-\theta^2$, where $\theta$ is the constant $\sum_ip_i^2-\sum_jq_j^2$ and $Z$ is a standard Gaussian random variable.Comment: Published in at http://dx.doi.org/10.1214/08-AOP418 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sign variation, the Grassmannian, and total positivity

The totally nonnegative Grassmannian is the set of k-dimensional subspaces V of R^n whose nonzero Pluecker coordinates all have the same sign. Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that V is totally nonnegative iff every vector in V, when viewed as a sequence of n numbers and ignoring any zeros, changes sign at most k-1 times. We generalize this result from the totally nonnegative Grassmannian to the entire Grassmannian, showing that if V is generic (i.e. has no zero Pluecker coordinates), then the vectors in V change sign at most m times iff certain sequences of Pluecker coordinates of V change sign at most m-k+1 times. We also give an algorithm which, given a non-generic V whose vectors change sign at most m times, perturbs V into a generic subspace whose vectors also change sign at most m times. We deduce that among all V whose vectors change sign at most m times, the generic subspaces are dense. These results generalize to oriented matroids. As an application of our results, we characterize when a generalized amplituhedron construction, in the sense of Arkani-Hamed and Trnka (2013), is well defined. We also give two ways of obtaining the positroid cell of each V in the totally nonnegative Grassmannian from the sign patterns of vectors in V.Comment: 28 pages. v2: We characterize when a generalized amplituhedron construction is well defined, in new Section 4 (the previous Section 4 is now Section 5); v3: Final version to appear in J. Combin. Theory Ser.

Discussion of "Statistical Inference: The Big Picture" by R. E. Kass

Discussion of "Statistical Inference: The Big Picture" by R. E. Kass [arXiv:1106.2895]Comment: Published in at http://dx.doi.org/10.1214/11-STS337A the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Moment curves and cyclic symmetry for positive Grassmannians

We show that for each k and n, the cyclic shift map on the complex Grassmannian Gr(k,n) has exactly $\binom{n}{k}$ fixed points. There is a unique totally nonnegative fixed point, given by taking n equally spaced points on the trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is even). We introduce a parameter q, and show that the fixed points of a q-deformation of the cyclic shift map are precisely the critical points of the mirror-symmetric superpotential $\mathcal{F}_q$ on Gr(k,n). This follows from results of Rietsch about the quantum cohomology ring of Gr(k,n). We survey many other diverse contexts which feature moment curves and the cyclic shift map.Comment: 18 pages. v2: Minor change

A unified model for the spectrophotometric development of classical and recurrent novae: the role of asphericity of the ejecta

There is increasing evidence that the geometry, and not only the filling factors, of nova ejecta is important in the interpretation of their spectral and photometric developments. Ensembles of spectra and light curves have provided general typographies. This Letter suggests how these can be unified.The observed spread in the maximum magnitude - rate of decline (MMRD) relation is argued to result from the range of opening angles and inclination of the ejecta, and not only to their masses and velocities. The spectroscopic classes can be similarly explained and linked to the behavior of the light curves. The secondary maximum observed in some dust forming novae is a natural consequence of the asphericity. Neither secondary ejections nor winds are needed to explain the phenomenology. The spectrophotometric development of classical novae can be understood within a single phenomenological model with bipolar, although not jet-like, mass ejecta. High resolution spectropolarimetry will be an essential analytical tool.Comment: 4 pages, no figures; accepted for A&A Letters (in press

The Galactic Center Isolated Nonthermal Filaments as Analogs of Cometary Plasma Tails

We propose a model for the origin of the isolated nonthermal filaments observed at the Galactic center based on an analogy to cometary plasma tails. We invoke the interaction between a large scale magnetized galactic wind and embedded molecular clouds. As the advected wind magnetic field encounters a dense molecular cloud, it is impeded and drapes around the cloud, ultimately forming a current sheet in the wake. This draped field is further stretched by the wind flow into a long, thin filament whose aspect ratio is determined by the balance between the dynamical wind and amplified magnetic field pressures. The key feature of this cometary model is that the filaments are dynamic configurations, and not static structures. As such, they are local amplifications of an otherwise weak field and not directly connected to any static global field. The derived field strengths for the wind and wake are consistent with observational estimates. Finally, the observed synchrotron emission is naturally explained by the acceleration of electrons to high energy by plasma and MHD turbulence generated in the cloud wake.Comment: Uses AAS aasms4.sty macros. ApJ (in press, vol. 521, 20 Aug