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 $(1^n)$?