arXiv subjects

Gopal Goel

Publications and source records attributed to Gopal Goel.

4 recordsLinked to original sources

arXiv2021

Counting weighted maximal chains in the circular Bruhat order

The totally nonnegative Grassmannian $\mathrm{Gr}(k,n)_{\geq0}$ is the subset of the real Grassmannian $\mathrm{Gr}(k,n)$ consisting of points with all nonnegative Pl\"ucker coordinates. The circular Bruhat order is a poset isomorphic to the face poset of A. Postnikov's (2005) positroid cell decomposition of $\mathrm{Gr}(k,n)_{\geq0}$. We provide a closed formula for the sum of its weighted chains in the spirit of J. Stembridge (2002).

math.CO↗

arXiv2020

A Quantized Analogue of the Markov-Krein Correspondence

We study a family of measures originating from the signatures of the irreducible components of representations of the unitary group, as the size of the group goes to infinity. Given a random signature $\lambda$ of length $N$ with counting measure $\mathbf{m}$, we obtain a random signature $\mu$ of length $N-1$ through projection onto a unitary group of lower dimension. The signature $\mu$ interlaces with the signature $\lambda$, and we record the data of $\mu,\lambda$ in a random rectangular Young diagram $w$. We show that under a certain set of conditions on $\lambda$, both $\mathbf{m}$ and $w$ converge as $N\to\infty$. We provide an explicit moment generating function relationship between the limiting objects. We further show that the moment generating function relationship induces a bijection between bounded measures and certain continual Young diagrams, which can be viewed as a quantized analogue of the Markov-Krein correspondence.

math.PR↗

arXiv2018

Critical groups of iterated cones

Let G be a finite graph, and let G_n be the n-th iterated cone over G. We study the structure of the critical group of G_n arising in divisor and sandpile theory.

math.CO↗

arXiv2018

Discrete Derivative Asymptotics of the $\beta$-Hermite Eigenvalues

We consider the asymptotics of the difference between the empirical measures of the $\beta$-Hermite tridiagonal matrix and its minor. We prove that this difference has a deterministic limit and Gaussian fluctuations. Through a correspondence between measures and continual Young diagrams, this deterministic limit is identified with the Vershik-Kerov-Logan-Shepp curve. Moreover, the Gaussian fluctuations are identified with a sectional derivative of the Gaussian free field.

math.PR↗