IMSc Webinar

Multiplicity of trivial and sign representations of $S_n$ in hook-shaped representations of $GL_n$.

Sridhar P Narayanan

IMSc

Let $W_\lambda$ be an irreducible representation of $GL_n$ (for
partition $\lambda$ with $\leq n$ parts). Let $V_\mu$ be an irreducible
representation of $S_n$ (for partition $\mu \vdash n$). Then $$W_\lambda=
\sum_{\mu \vdash n} r_{\lambda \mu} V_\mu.$$
The coefficients $r_{\lambda\mu}$ are the restriction coefficients. The
restriction problem is to find combinatorial objects that these coefficients
count. We find such objects when $\lambda$ is the hook shape and $\mu=(n)$
or $\mu= (1^n)$ using the theory of character polynomials and a simple
sign-reversing involution.