#### 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.

Done