#### Alladi Ramakrishnan Hall

#### RSK Correspondence and Representation Theory

#### Arghya Sadhukhan

##### IMSc

*In literature, a model of a representation $\pi$, typically irreducible, of*

a finite group $G$ is an embedding of $\pi$ in a multiplicity free induced

representation, typically induced from an one dimensional representation of

a subgroup of $G$. Around 1990, Klyachko and Inglis et all independently

found an explicit model for complex representations of the symmetric groups

$S_n$. We will see how this discovery follows naturally from our

considerations revolving Schur Weyl duality and a symmetric function

identity, known to even Issai Schur in the 1900. We will start off by

introducing $(GL_m, GL_n)$ duality from the viewpoint of Robinson Schensted

Knuth correspondence, which in turn proves that the $(1^n)$ weight space of

$V_{\lambda}(n)$, the irreducible representations of $GL_n$ associated to

the partition $\lambda \vdash n$ is the Specht module $Sp_{\lambda}$ for

$S_n$, a fact which is in fact equivalent to Schur Weyl duality. Then we

will deduce the model from a known fact about $GL_n$ representations and

eventually see how this leads us to the model as expounded in an article by

Kodiyalam et all. Finally we will sketch some computations, using Schur Weyl

duality and Sage, to the following question: What happens if we pick up in

$V_{\lambda}(n)$ some weight space other than $(1^n)$?

Done