#### IMSc Webinar

#### Neutron Stars as tools to probe gravitational physics, Saturation of refined Littlewood-Richardson coefficients

#### Manjari Bagchi, Mrigendra Singh Kushwaha

*Neutron stars are extremely dense undead stars. These objects emit electromagnetic waves due to their strong magnetic fileds and fast rotations and sometimes seen as "Pulsars". Due to their extreme density, the gravitational fields around neutron stars are so strong that general relativistic effects become significant and hence pulsars can be used as laboratories to test various theories of gravity. In this talk, I will describe two such uses of pulsars. First, I will mention how a pulsar in a binary system with a black hole can help establish or rule out some of alternative gravity theories. Second, I will describe how a number of pulsars can be used to detect low frequency gravitational waves through "Pulsar Timing Array" experiment. I will also mention our group's contribution in this international experiment, and its future.*

Let $\lambda$, $\mu$ and $

u$ be integer partitions with at most $n$ parts each. The Littlewood-Richardson (LR) coefficient $c_{\lambda,\mu}^{

u}$ is the multiplicity of the irreducible representation $V(

u)$ in the decomposition of the tensor product $V(\lambda)\otimes V(\mu)$ of irreducible polynomial representations of $GL_n$. For each permutation $w$ in $S_n$, the $w$-refined LR coefficient $c_{\lambda,\mu}^{

u}(w)$ is the multiplicity of $V(

u)$ in the decomposition of the so-called Kostant-Kumar submodule $K(\lambda,w,\mu)$ of the tensor product.

The saturation problem asks whether $c_{\lambda,\mu}^{

u}(w) >0$ given that $c_{k\lambda,k\mu}^{k

u}(w) >0$ for some $k \geq 2$. We show that this is true when the permutation $w$ is $312$-avoiding or $231$-avoiding, by adapting the beautiful combinatorial proof of the LR-saturation conjecture due to Knutson and Tao.

This is joint work with K.N. Raghavan and Sankaran Viswanath.

Done